Confinement of long-lived interlayer excitons in WS2/WSe2 heterostructures

<jats:title>Abstract</jats:title><jats:p>Interlayer excitons in layered materials constitute a novel platform to study many-body phenomena arising from long-range interactions between quantum particles. Long-lived excitons are required to achieve high particle densities, to mediate thermalisation, and to allow for spatially and temporally correlated phases. Additionally, the ability to confine them in periodic arrays is key to building a solid-state analogue to atoms in optical lattices. Here, we demonstrate interlayer excitons with lifetime approaching 0.2 ms in a layered-material heterostructure made from WS<jats:sub>2</jats:sub> and WSe<jats:sub>2</jats:sub> monolayers. We show that interlayer excitons can be localised in an array using a nano-patterned substrate. These confined excitons exhibit microsecond-lifetime, enhanced emission rate, and optical selection rules inherited from the host material. The combination of a permanent dipole, deterministic spatial confinement and long lifetime places interlayer excitons in a regime that satisfies one of the requirements for simulating quantum Ising models in optically resolvable lattices.</jats:p>

section_content: ince the demonstration of Bose-Einstein condensation 1, 2, ultracold atoms have played a central role in the study and realisation of macroscopic-scale quantum phenomena, including Mott transition 3, superfluidity 4 and many-body localisation 5. Dipolar particles have gained attention as they introduce long-range anisotropic interactions to the exploration of new states of quantum matter [6] [7] [8]. Atomic realisations of dipolar ensembles include Rydberg atoms [9] [10] [11], high magneticmoment atoms [12] [13] [14] [15] [16] and ultracold polar molecules [17] [18] [19]. Excitonpolaritons in semiconductors are an analogous platform to atoms [20] [21] [22] [23] [24] [25] [26]. For semiconductors, long-range interactions can be achieved via spatially indirect excitons, where electrons and holes have finite separation 27, 28 yielding a permanent electric dipole moment [29] [30] [31]. Most progress in realising such states has been in AlGaAs/GaAs double quantum wells 30, [32] [33] [34] [35], while layered materials heterostructures of transition metal dichalcogenide (TMD) monolayers 36 are a promising alternative [37] [38] [39] [40]. TMD monolayers are semiconductors offering optical access to orbital, spin and valley degrees of freedom 41. The difference in band energies of two TMD monolayers can be exploited to create type-II interlayer excitons 42, 43 with static electric dipole. Moreover, their optical transition strength, energy, and selection rules can be engineered by controlling the layer separation 44 and their relative stacking angle [45] [46] [47] [48] [49] [50]. 

The potential of TMD heterostructure devices to explore many-body effects is evidenced by reports of excitonic condensation 51 and Mott-Hubbard physics with charge carriers in moiré superlattices [52] [53] [54]. The periodicity of these superlattices, typically up to a ~10-nm characteristic length scale, is determined by the interlayer lattice constant mismatch and stacking angle mismatch. Confining interlayer excitons in an arbitrary potential energy landscape independent of the TMD lattice opens up the prospect of exploiting the long-range nature of their dipole-dipole interactions, as well as their optical accessibility to explore many-body physics, such as quantum spin Ising models. Such a system could be viewed as a solid-state analogue to quantum simulators using Rydberg atoms 55 and molecules 56 in optical lattices or tweezers arrays. At the cost of reduced coherence time and site-to-site homogeneity in a solid-state device, they offer larger arrays and shorter refresh times. 

For long-range interactions to emerge, excitons do not have to be indistinguishable, but their decay rate must be lower than the interaction rate between neighbouring lattice sites 56. In an array with single-site optical addressability, the confining traps must be spaced by an optically resolvable distance ~0. 5 μm. At this spacing, interlayer excitons, with an electric dipole of ~0. 6 nm ⋅ q e (~29 Debye) 40, have a dipole-dipole coupling rate of 1 MHz. This places a lifetime requirement of at least 1 μs on excitons. Here, we show the deterministic confinement of interlayer excitons with lifetimes exceeding 1 μs. This opens up the prospect of implementing confined interlayer excitons in quantum simulation applications.
section_name: S

section_content: Optical characterisation of the device. Figure 1a is an illustration of our devices, fabricated by exfoliating WSe 2 and WS 2. With the aim of creating long-lived excitons, we limit non-radiative decay 57, which often dominates exciton loss 51, 58, by working with TMDs that have low point-defect density of 10 9 -10 10 cm -2 as estimated from X-Ray Diffraction (XRD) and Scanning Transmission Electron Microscopy (STEM) measurements (see Supplementary Note S1 and Supplementary Figs. S1 and S2 ). We transfer WSe 2 and WS 2 monolayers onto a SiO 2 /Si substrate patterned with an array of nanopillars 220-250 nm tall and 4 μm apart (see Methods and Supplementary Note S2). For the first device (device A), we choose an interlayer stacking angle of ~50°, away from high-symmetry angles of 0°or 60°, in order to introduce an interlayer momentum mismatch that reduces the direct optical exciton recombination rate 59. Five distinct locations are labelled in Fig. 1a (L1-L5 ). These are marked by circles in the optical image of device A shown in Fig. 1b, where the WSe 2 and WS 2 monolayers are outlined in red and green, respectively, yielding a large region (>160 μm 2 ) of WS 2 /WSe 2 heterostructure. 

Figure 1c shows a photoluminescence (PL) map taken with continuous-wave (CW) 2. 33-eV (532-nm) laser excitation at a temperature T = 4 K. This energy is greater than both WSe 2 and WS 2 monolayer optical bandgaps of 1. 73 60 and 2. 08 eV 61, respectively. Consequently, PL emission is observed from both WSe 2 and WS 2 monolayers and WS 2 /WSe 2 heterostructure regions. The spectral emission from the flat WSe 2 and WS 2 monolayer regions (L1 and L2 in Fig. 1b ) is shown in Figs. 1d (i ) and (ii), respectively. The PL emission between 1. 65 and 2. 0 eV in panel (i) (filled in green) is the intralayer exciton recombination in monolayer WS 2 61, 62, and that between 1. 6 and 1. 75 eV in panel (ii) (filled in red) is the intralayer exciton recombination in monolayer WSe 2 60, 63. Figure 1d panel (iii) shows the spectrum from the flat part of the heterostructure, L3 in Fig. 1b. A lowerenergy feature at 1. 4 eV (filled in yellow) is found only in the heterostructure region and is consistent with interlayer exciton emission from stacked heterostructures reported previously 50, 64. Figure 1d (iv) is the PL spectrum from a monolayer WSe 2 on a nanopillar, displaying a ~50-fold brighter emission with respect to flat monolayer WSe 2 and a sub-meV full-width at halfmaximum emission peak. Given that monolayer WSe 2 on such nanopillars results in the strong quantum confinement of intralayer excitons 62, 65, we look for similar signatures in the interlayer emission spectra. Figure 1d (v) is the PL spectrum of the heterostructure on a nanopillar. We observe a 20-fold brighter interlayer emission in the low optical excitation regime (<1 μW), and sharp spectral features (~2 meV or less full-width at halfmaximum) between 1. 35 and 1. 50 eV on eight nanopillar locations, in contrast to the weak and spectrally broad emission from the flat heterostructure. These remain after a full thermal cycle with only minor spectral variations (see Supplementary Note S10 and Supplementary Figs. S17-S19 ). 

To isolate interlayer excitations from any effects coming from the monolayers, such as free charge carriers 66 or intralayer excitons 67, we use an optical excitation energy of 1. 50 eV for all remaining measurements. This energy is at the onset of the interlayer exciton PL spectrum, Fig. 1d (iii), and below the optical bandgaps of WSe 2 and WS 2 monolayers. Figure 1e displays the PL intensity map under 1. 50 eV excitation. The emission is only observed from the heterostructure region with both monolayer regions remaining dark, demonstrating that only interlayer excitons are generated. 

Confined and delocalised interlayer excitons. Figure 2a shows the interlayer exciton PL spectra from the flat WS 2 /WSe 2 heterostructure for 0. 05-μW (black curve), 1-μW (red curve) and 100-μW (blue curve) laser excitation power, P, at 4 K. The spectrally integrated PL intensity of the interlayer exciton with respect to P is shown in Fig. 2b. While the interlayer exciton emission starts with a linear dependence on P, it converges to a P 0. 3 scaling for P > 0. 3 μW. Sublinear behaviour was also previously reported for MoSe 2 /WSe 2 heterostructures 36, 68. This can be caused by density-dependent mechanisms, such as exciton-exciton annihilation, dipolar repulsion and phase-space filling. The interlayer exciton energy also undergoes a blue-shift with increasing P (see Supplementary Fig. S5 for device A and Supplementary Fig. S15 for device B), which is typically attributed to dipolar interactions 36, 68, 69. 

Figure 2c presents PL spectra from the heterostructure on a nanopillar for P = 0. 05 μW (black curve), 1 μW (red curve) and 100 μW (blue curve). For the three emission peaks highlighted in green, grey and blue we plot in Fig. 2d the integrated PL intensity as a function of P. For the analysis of emission peaks at other nanopillar locations in devices A and B, see Supplementary Figs. S7 and S16, respectively. Each emission peak shows a linear Pdependence, followed by saturation, consistent with previous reports of interlayer exciton localisation 68, 70. This behaviour is characteristic of a quantum-confined system 71, 72, and contrasts that of delocalised interlayer excitons in the flat heterostructure (Fig. 2a, b ). The fact that these emission peaks saturate at different powers (see Supplementary Note S4) indicates multiple confining potentials at the nanopillar, possibly because the heterostructure does not conform smoothly to the nanopillar (see Supplementary Fig. S4a ) 62. The emission peaks are spectrally stable within 1 meV over a minute timescale (see Supplementary Note S5) and unchanged after repeated excitation power-dependent measurements (see Supplementary Fig. S9 ). In our time series, some emission peaks show intensity anticorrelation (see Supplementary Fig. S10 ), often referred to as spectral jumps, which is another signature of quantum-confined systems. 

Optical selection rules of confined interlayer excitons. Figure 3a and b presents confined interlayer exciton PL spectra above a nanopillar (L5 of Fig. 1b ) at 4 K, for out-of-plane magnetic field B from 0 to 9 T. The excitation is linearly polarised and the collection is right-hand σ + (left-hand σ -) circularly polarised in panel a (panel b). The red-shifting Zeeman state from each b Spectrally integrated intensity of interlayer exciton PL as a function of excitation power, P. The dashed grey curve follows a linear dependence in P, while the solid grey curve follows P 0. 3. Sub-linear PL intensity is also seen in device B, see Supplementary Fig. S15. The error bars are smaller than the data points. c PL spectra from the WS 2 /WSe 2 heterostructure on a nanopillar at 0. 05 μW (black curve), 1 μW (red curve) and 100 μW (blue curve). d Spectrally integrated PL intensity of the three emission peaks shaded with green, grey and blue bands in c as a function of P. Data (filled squares) are colour-coded to the spectral bands of c. Solid curves are fits to data using the saturation function A × P n /(P sat + P n ), from where we determine n = 0. 94 ± 0. 12, 1. 06 ± 0. 08, 1. 04 ± 0. 06 and P sat = 0. 1, 0. 6, 0. 9 μW for the green, grey and blue data, respectively. The error bars are smaller than the data points. For the saturation curves of sharp emission peaks on device B see Supplementary Fig. S16. emission peak is only seen for σ + detection, while the blueshifting Zeeman state is only seen for σ -detection. This demonstrates that the two magnetic configurations of confined interlayer excitons sustain optical selection rules with an in-plane transition dipole moment 73. 

The distribution of Zeeman splitting with respect to B across different nanopillars is given in Fig. 3c as a blue-shaded region, where the data corresponding to the smallest and largest measured g-factor (11. 9 and 15. 4) are plotted in blue and red circles, respectively. The similarity in g-factors, mean value 13. 2 and standard deviation 1. 1, across the nanopillar confining traps and for different emission peaks (see Supplementary Fig. S8 ) suggests that the confined excitons have the same microscopic origin. The variation in g-factors, similar to that observed in confined intralayer excitons in WSe 2 monolayers 72, [74] [75] [76], is one order of magnitude larger than observed for moiré-confined interlayer excitons 49. Our average g-factor agrees with the expected g-factor of 14 [77] [78] [79] for electrons and holes residing in opposite valleys, corresponding to the K 0 À K transition. This is consistent with our heterostructure alignment close to 60°. 

Figure 3d presents the PL spectra under σ + excitation at B = 0 T for σ + (black) and σ -(red) circularly polarised collection. The overlap of the two spectra demonstrates that excitation polarisation is not maintained for confined interlayer excitons, and the same is observed in delocalised excitons (see Supplementary Note S3 and Supplementary Fig. S6a ). One dominant mechanism for the loss of exciton valley polarisation is the electron-hole exchange interaction 80. The polarisation retention diminishes when the exchange interaction rate γ EI exceeds the exciton decay rate γ d. Relative to intralayer excitons, the increased electron-hole separation of interlayer excitons reduces γ EI and γ d by the same factor and, consequently, angle-aligned heterostructures exhibit valley polarisation 49, 64 similar to monolayers 63, typically >10%. In our device, the momentum mismatch introduced by our large stacking angle further suppresses γ d, while leaving γ EI unaffected, resulting in a suppression of valley polarisation. 

We also confirm that the g-factor of the delocalised exciton has the same sign as the confined excitons. To do this, we monitor the circular polarisation of the delocalised interlayer excitons under magnetic field (see Supplementary Note S3 and Supplementary Fig. S6b ). In a positive (negative) magnetic field the σ + (σ -) emission becomes dominant irrespective of excitation polarisation, owing to exciton thermalisation to the lower-energy Zeeman state 63, 81. 

Lifetimes of confined and delocalised interlayer excitons. Figure 4 presents lifetime measurements on interlayer excitons under 1. 5-eV laser excitation with ~3 ps pulses at 4 K (see Methods). Figure 4a shows an example emission intensity histogram of delocalised interlayer excitons (grey bars) of device A as a function of time after excitation, and an exponential fit (solid red curve) reveals a decay time of 180. 6 ± 3. 5 μs. Across the flat heterostructure region, the average lifetime is 175 ± 5 μs. The spatial and momentum separation of electrons and holes of interlayer excitons, and the use of low defect density WSe 2 and WS 2 monolayers, are likely responsible for these values. In device B, which has a smaller twist angle (<7°), the lifetime drops to 10 ± 2 μs for delocalised excitons, highlighting the role of momentum mismatch in exciton recombination (see Supplementary Fig. S15d ). The lifetime of delocalised excitons shows a strong excitation dependence: increasing the excitation power reduces the exciton decay time (see Supplementary Note S6 and Supplementary Fig. S11 ), as expected from density-dependent interactions and loss channels, even in the regime where PL intensity depends linearly on the excitation power (P < 0. 3 μW in Fig. 2b ). 

Figure 4b is a lifetime measurement of the confined interlayer exciton emission peak at 1. 39 eV from Fig. 3d (L5 in Fig. 1b ). We fit a biexponential function of the form A s e Àt=τ s + A l e Àt=τ l, where τ l (τ s ) is the long (short) lifetime and A l (A s ) is the amplitude of the exponential with long (short) lifetime. This yields τ s = 59. 4 ± 0. 9 ns and τ l = 389. 1 ± 1. 2 ns. This biexponential behaviour, albeit 3d ). The solid red curve is a biexponential fit with two lifetimes τ s = 59. 4 ± 0. 9 ns and τ l = 389. 1 ± 1. 2 ns. The inset presents a three-level model with coupling between a bright state b j i and a shelving state s j i, as discussed in Supplementary Note S8. c τ s (dark grey bars) and τ l (light grey bars) values extracted from biexponential fits to the lifetime measurements at the eight nanopillar locations in device A (indexed a-h). Error bars are fitting errors. The average value of τ s (τ l ) across all nanopillars is 80 ns (2 μs). The inset maps the physical location of the nanopillars on the WS 2 /WSe 2 heterostructure region. The reduction of lifetime for confined interlayer excitons as compared to delocalised is also seen in device B, see Supplementary Figs. S15d and S16d. 

with varying lifetimes, is observed throughout the entire spectral range for confined interlayer excitons (see Supplementary Note S7 and Supplementary Fig. S12 ). Figure 4c is a summary of the spectrally integrated lifetimes from the eight nanopillar confining traps. The extracted τ s and τ l are shown in Fig. 4c in black and grey bars, and lie in the ~10-175 ns and ~0. 4-4 μs ranges, respectively. The reduction in interlayer exciton lifetime after confinement accompanies an enhancement in PL brightness (also for device B, see Supplementary Note S9), suggesting a modified oscillator strength under localisation. This likely arises from the relaxation of stacking-angle-induced momentum mismatch of the exciton transition under confinement that otherwise inhibits the recombination of delocalised interlayer excitons 82, 83. 

One source of the observed biexponential decay is the presence of two excited states as shown in the energy level diagram of Fig. 4b : an optically active state, b j i, and an energetically similar shelving state that is dark, s j i. Lifetime measurements from 4 to 70 K (see Supplementary Fig. S13a-c ) fitted with a three-level model, and diminishing PL intensity with increasing temperature (see Supplementary Fig. S13d ), indicate that s j i is higher in energy than b j i (see Supplementary Note S8). We assume the radiative rate for the bright state, Γ r, to be temperature independent and the coupling rates between bright and shelving states, Γ bs (bright to shelving), Γ sb (shelving to bright), and the non-radiative decay rate Γ nr to be phonon mediated, hence temperature dependent. At 4 K we find that 1/τ s ≈ Γ r and 1/τ l ≈ Γ nr + Γ bs, with average values across device A of 13 MHz and 500 kHz, respectively. The non-radiative loss rate is lower than all other rates for the confined interlayer excitons up to 70 K (see Supplementary Note S8). The rate Γ bs increases with temperature and matches Γ sb at 40 K, indicating that s j i is ~5 meV higher in energy than b j i, comparable to the calculated ~10-29 meV conduction band splitting at the K-point of WS 2 84 monolayer. A higher shelving state energy is also supported by a diminishing PL intensity with increasing temperature (see Supplementary Fig. S13d ). These measurements indicate that the shelved state is the spin-forbidden K 0 À K transition.
section_name: Results and discussion

section_content: In summary, we demonstrated deterministic confinement of long-lived interlayer excitons across multiple sites in devices with different stacking angles. The confined interlayer exciton lifetime up to ~4 μs is not limited by non-radiative decay and is sufficient to observe dipole-mediated interactions in an optically resolvable lattice with a spacing of ~0. 5 μm. The next step is employing resonant excitation toward transform-limited linewidths [85] [86] [87] [88] and coherent optical control of excitons, leading ultimately to the detection of site-to-site interaction-induced spectral shifts. Further work will also focus on developing a full account of the photophysics of these confined interlayer excitons, including the influence of stacking angle, interlayer spacing, and electrostatic gating of the devices. Other confining geometries, such as ridges and rings, may also offer the opportunity to probe many-body phenomena in one dimension.
section_name: Conclusions

section_content: Fabrication. TMD bulk crystals were synthesised through flux zone growth 89. Precursor powders were sourced from Alfa Aesar. Additional electrolytic purification is implemented to achieve purity ≥99. 9999%. After purification, the powders were analysed using secondary ion mass spectroscopy to confirm the absence of metal impurities. Compared to crystals grown by chemical vapour transport, flux growth produces samples with much lower point and topological defects. 

Flakes were then prepared by micromechanical cleavage 90 on Nitto Denko tape, then exfoliated again for transfer to a polydimethylsiloxane (PDMS) stamp placed on a transparent glass slide, allowing bidirectional inspection of the flakes under an optical microscope. Optical contrast was utilised to identify monolayers prior to transfer 91. We obtained monolayer flakes with lateral dimensions ~50 μm. 

Substrates with arrays of silica nanopillars, 100 nm in diameter and 220-250 nm high, were prepared by direct-write lithography 62. The substrates underwent wet cleaning (1 min ultrasonication in acetone and isopropanol) and were subsequently exposed to an oxygen-assisted plasma at 10 W for 60 s to remove impurities and contaminants from the surface. The WSe 2 monolayer flakes were then stamped on the nanopillars with a micro-manipulator 62, 92. A WS 2 monolayer was then deposited on the WSe 2 monolayer following the same stamping procedure. In both steps the PDMS stamp was removed after depositing the monolayer. Raman spectroscopy (see Supplementary Fig. S3a ), second-harmonic generation (see Supplementary Fig. S3b ) and atomic force microscopy (see Supplementary Fig. S4 ) confirm the monolayer nature of the constituents, the twist angle and the interlayer spacing, respectively. 

Photoluminescence measurements. All PL measurements were taken at 4 K in a 9 T closed-cycle cryostat (Attodry 1000 from Attocube Systems AG). Excitation and collection pass through a confocal setup with the device in reflection geometry. We used CW illumination from either a 2. 33 eV laser (Ventus 532 from Laser Quantum Ltd. ) or a Ti:Sapphire laser (Mira 900 from Coherent Inc. ) at 1. 58 or 1. 50 eV. All spectra were corrected for the spectral transfer function of the setup, measured with a broadband incandescent source. The PL signal was sent to a 150line grating spectrometer (Princeton Instruments Inc. ) or an avalanche photodiode for PL maps (PerkinElmer Inc. ). 

Lifetime measurements. To perform lifetime measurements, we excite the samples every ~10 μs for the confined interlayer excitons, and up to ~1 ms for interlayer excitons, with ~3 ps pulses from a Ti:Sapphire laser (Mira 900 from Coherent Inc. ) tuned to 785 nm (1. 58 eV) or 825 nm (1. 50 eV). An acousto-optic modulator (AOM) down-samples the 76 MHz laser repetition rate to the kHz-MHz range required to measure up to hundreds μs lifetimes. A time-to-digital converter (quTAU from qutools GmbH) with 81 ps timing resolution collects start-stop histograms with 'start' triggered by the AOM pulse-picking and 'stop' triggered by an avalanche photodiode (APD) output stemming from the single-photon detection of interlayer exciton PL. The converter remains idle until a subsequent 'start' signal arrives. To ensure there is no susceptibility to spurious artifacts from startstop measurements, we determine the lifetime using photon time-tagging, and calculate the lifetimes in post-processing for a subset of locations. Both methods resulted in consistent lifetimes for confined interlayer excitons. Time-tagging is required for the measurement of delocalised interlayer exciton lifetimes, due to the count rate being comparable to, or even lower than, the APD dark counts, as well as being at least one order of magnitude smaller than the rate of pulsed excitations.
section_name: Methods

section_name: Acknowledgements

section_content: We acknowledge funding from the EU Quantum and Graphene Flagships ; ERC grants PEGASOS, Hetero2D and GSYNCOR; EPSRC Grants EP/K01711X/1, EP/K017144/1, EP/N010345/1 and EP/L016057/1 ; the Faraday Institution FIRG001 ; and DOE, NSF DMR 1552220, DMR 1904716, and NSF CMMI 1933214. D. M. K. acknowledges support of a Royal Society University Research Fellowship URF\R1\180593.
section_name: Acknowledgements

section_name: Data availability

section_content: The datasets generated and analysed during the current study are available from the corresponding authors upon reasonable request.
section_name: Data availability

section_content: The computer codes used for data analysis during the current study are available from the corresponding authors upon reasonable request.
section_name: Code availability

section_name: Author contributions

section_content: A. R. -P. M., D. M. K., A. C. F. and M. A. conceived and managed the project; Y. Q., M. B. and S. T. provided the bulk WS 2 and WSe 2 crystals and performed the XRD and STEM measurements; P. L. and M. L. provided the nanopillar substrates; I. P., A. R. C. and G. W. fabricated the devices; A. R. -P. M., D. M. K., I. P., C. P., M. F., E. M. A., L. S. and M. A. performed the optical measurements and analysed the results. All authors participated in the discussion of the results and the writing of the manuscript.
section_name: Author contributions

section_content: The authors declare no competing interests.
section_name: Competing interests

section_name: Additional information

section_content: The online version contains supplementary material available at https://doi. org/10. 1038/s42005-021-00625-0. 

Correspondence and requests for materials should be addressed to A. C. F. or M. A. 

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section_name: Supplementary information