Interlayer excitons in a bulk van der Waals semiconductor

Quadrupolar and Dipolar Excitons in Bilayer 2H-MoSe_{2}

We report the experimental observation of quadrupolar exciton states in the reflectance contrast spectrum of 2H-stacked bilayer MoSe_{2}. The application of a vertical electric field results in a quadratic energy redshift of these quadrupolar excitons, in contrast to the linear energy splitting observed in the coexisting dipolar excitons within the bilayer MoSe_{2}. We perform helicity-resolved reflectance contrast measurements to investigate the spin and valley configurations of the quadrupolar exciton states as a function of applied vertical electric and magnetic fields. Comparing our results with a phenomenological coupled-oscillator model indicates that the electric- and magnetic-field dependence of the quadrupolar exciton states can be attributed to the intra- and intervalley hybridization of spin-triplet interlayer excitons with opposite permanent dipole moments, mediated by interlayer hole tunneling. These results position naturally stacked MoSe_{2} bilayers as a promising platform to explore electric-field-tunable many-body exciton phenomena.

Infrared Magnetopolaritons in MoTe_{2} Monolayers and Bilayers

MoTe_{2} monolayers and bilayers are unique within the family of van der Waals materials since they pave the way toward atomically thin infrared light-matter quantum interfaces, potentially reaching the important telecommunication windows. Here, we report emergent exciton polaritons based on MoTe_{2} monolayers and bilayers in a low-temperature open microcavity in a joint experiment-theory study. Our experiments clearly evidence both the enhanced oscillator strength and enhanced luminescence of MoTe_{2} bilayers, signified by a 38% increase of the Rabi splitting and a strongly enhanced relaxation of polaritons to low-energy states. The latter is distinct from polaritons in MoTe_{2} monolayers, which feature a bottlenecklike relaxation inhibition. Both the polaritonic spin valley locking in monolayers and the spin-layer locking in bilayers are revealed via the Zeeman effect, which we map and control via the light-matter composition of our polaritonic resonances.

Quadrupolar excitons in MoSe2 bilayers

The quest for platforms to generate and control exotic excitonic states has greatly benefited from the advent of transition metal dichalcogenide (TMD) monolayers and their heterostructures. Among the unconventional excitonic states, quadrupolar excitons-a superposition of two dipolar excitons with anti-aligned dipole moments-are of great interest for applications in quantum simulations and for the investigation of many-body physics. Here, we unambiguously demonstrate the emergence of quadrupolar excitons in natural MoSe2 homobilayers, whose energy shifts quadratically in electric field. In contrast to trilayer systems, MoSe2 homobilayers have many advantages, which include a larger coupling between dipolar excitons. Our experimental observations are complemented by many-particle theory calculations offering microscopic insights in the formation of quadrupolar excitons. Our results suggest TMD homobilayers as ideal platform for the engineering of excitonic states and their interaction with light and thus candidate for carrying out on-chip quantum simulations.

Photoinduced Long-Distance Charge Transfer in Silicanes: The Stacking Matters

The generation of interlayer charge transfer excitons upon photoexcitation is strongly desirable for two-dimensional (2D) materials stacked through van der Waals interactions. In this work, we investigate photoinduced charge transfer in silicanes (SiH) with three typical stackings. A concept of the regional natural hole orbital and its conjugated particle orbital is developed to characterize excited states in solids. This method delivers bonding information about excited states and explains the formation of certain types of states in nanomaterials. Utilizing this tool, we demonstrate that SiH in the 1H and 6R stackings exhibits an interlayer charge transfer distance that reaches ∼10 Å under violet and near-ultraviolet radiation. The charge transfer is attributed to the interlayer overlap between orbitals at the conduction band minimum, which is disfavored by the 3R stacking. Our findings suggest a new and feasible approach for tuning the optoelectronic properties of Group 14 2D materials by altering their stackings.

Interlayer and Moiré excitons in atomically thin double layers: From individual quantum emitters to degenerate ensembles

Abstract

Interlayer excitons (IXs), composed of electron and hole states localized in different layers, excel in bilayers composed of atomically thin van der Waals materials such as semiconducting transition-metal dichalcogenides (TMDs) due to drastically enlarged exciton binding energies, exciting spin-valley properties, elongated lifetimes, and large permanent dipoles. The latter allows modification by electric fields and the study of thermalized bosonic quasiparticles, from the single particle level to interacting degenerate dense ensembles. Additionally, the freedom to combine bilayers of different van der Waals materials without lattice or relative twist-angle constraints leads to layer-hybridized and Moiré excitons, which can be widely engineered. This article covers fundamental aspects of IXs, including correlation phenomena as well as the consequence of Moiré superlattices with a strong focus on TMD homo- and heterobilayers.

Graphical abstract

Revealing the Optical Transition Properties of Interlayer Excitons in Defective WS2/WSe2 Heterobilayers

Manipulation of physical properties in multidimensional tunable moiré superlattice systems is a key focus in nanophotonics, especially for interlayer excitons (IXs) in two-dimensional materials. However, the impact of defects on IXs remains unclear. Here, we thoroughly study the optical properties of WS2/WSe2 heterobilayers with varying defect densities. Low-temperature photoluminescence (PL) characterizations reveal that the low-energy IXs are more susceptible to defects compared to the high-energy IXs. The low-energy IXs also show much faster PL quenching rate with temperature, faster peak width broadening rate with laser power, shorter lifetime, and lower circular polarization compared to the low-energy IXs in the region with fewer defects. These effects are attributed to the combined effects of increased electron scattering, exciton-phonon interactions, and nonradiative channels introduced by the defects. Our findings aid in optimizing moiré superlattice structures.

Long-Lived Interlayer Excitons in WS2/Ruddlesden-Popper Perovskite van der Waals Heterostructures

Two-dimensional (2D) transition metal dichalcogenides (TMDs) and perovskites hold substantial promise for various optoelectronic applications such as light emission, photodetection, and energy harvesting. However, each of these materials possesses certain limitations that can be overcome by synergistically combining them to form heterostructures, thereby unveiling intriguing optical properties. In this study, we present an uncomplicated technique for crafting a van der Waals (vdW) heterojunction comprising monolayer WS2 and a Ruddlesden-Popper (RP) perovskite, namely (TEA)2PbI4. By utilizing ultrafast transient absorption (TA) spectroscopy, we explored the charge carrier dynamics within the WS2/(TEA)2PbI4 heterostructure. Our findings uncover a type-II band alignment in the heterostructure, facilitating rapid (within 260 fs) hole transfer from WS2 to the perovskite and leading to the formation of interlayer excitons (IXs) with a much longer lifetime (728 ps). This strategic approach has the potential to contribute to the development of hybrid systems aimed at achieving high-performance optoelectronic devices.

Highly tunable ground and excited state excitonic dipoles in multilayer 2H-MoSe2

The fundamental properties of an exciton are determined by the spin, valley, energy, and spatial wavefunctions of the Coulomb-bound electron and hole. In van der Waals materials, these attributes can be widely engineered through layer stacking configuration to create highly tunable interlayer excitons with static out-of-plane electric dipoles, at the expense of the strength of the oscillating in-plane dipole responsible for light-matter coupling. Here we show that interlayer excitons in bi- and tri-layer 2H-MoSe2 crystals exhibit electric-field-driven coupling with the ground (1s) and excited states (2s) of the intralayer A excitons. We demonstrate that the hybrid states of these distinct exciton species provide strong oscillator strength, large permanent dipoles (up to 0.73 ± 0.01 enm), high energy tunability (up to ~200 meV), and full control of the spin and valley characteristics such that the exciton g-factor can be manipulated over a large range (from -4 to +14). Further, we observe the bi- and tri-layer excited state (2s) interlayer excitons and their coupling with the intralayer excitons states (1s and 2s). Our results, in good agreement with a coupled oscillator model with spin (layer)-selectivity and beyond standard density functional theory calculations, promote multilayer 2H-MoSe2 as a highly tunable platform to explore exciton-exciton interactions with strong light-matter interactions.

Giant Faraday rotation in atomically thin semiconductors

Faraday rotation is a fundamental effect in the magneto-optical response of solids, liquids and gases. Materials with a large Verdet constant find applications in optical modulators, sensors and non-reciprocal devices, such as optical isolators. Here, we demonstrate that the plane of polarization of light exhibits a giant Faraday rotation of several degrees around the A exciton transition in hBN-encapsulated monolayers of WSe2 and MoSe2 under moderate magnetic fields. This results in the highest known Verdet constant of -1.9 × 107 deg T-1 cm-1 for any material in the visible regime. Additionally, interlayer excitons in hBN-encapsulated bilayer MoS2 exhibit a large Verdet constant (VIL ≈ +2 × 105 deg T-1 cm-2) of opposite sign compared to A excitons in monolayers. The giant Faraday rotation is due to the giant oscillator strength and high g-factor of the excitons in atomically thin semiconducting transition metal dichalcogenides. We deduce the complete in-plane complex dielectric tensor of hBN-encapsulated WSe2 and MoSe2 monolayers, which is vital for the prediction of Kerr, Faraday and magneto-circular dichroism spectra of 2D heterostructures. Our results pose a crucial advance in the potential usage of two-dimensional materials in ultrathin optical polarization devices.

Monolayer-like Exciton Recombination Dynamics of Multilayer MoSe2 Observed by Pump-Probe Microscopy

Transition metal dichalcogenides (TMDCs) have garnered considerable interest over the past decade as a class of semiconducting layered materials. Most studies on the carrier dynamics in these materials have focused on the monolayer due to its direct bandgap, strong photoluminescence, and strongly bound excitons. However, a comparative understanding of the carrier dynamics in multilayer (e.g., >10 layers) flakes is still absent. Recent computational studies have suggested that excitons in bulk TMDCs are confined to individual layers, leading to room-temperature stable exciton populations. Using this new context, we explore the carrier dynamics in MoSe2 flakes that are between ∼16 and ∼125 layers thick. We assign the kinetics to exciton-exciton annihilation (EEA) and Shockley-Read-Hall recombination of free carriers. Interestingly, the average observed EEA rate constant (0.003 cm2/s) is nearly independent of flake thickness and 2 orders of magnitude smaller than that of an unencapsulated monolayer (0.33 cm2/s) but very similar to values observed in encapsulated monolayers. Thus, we posit that strong intralayer interactions minimize the effect of layer thickness on recombination dynamics, causing the multilayer to behave like the monolayer and exhibit an apparent EEA rate intrinsic to MoSe2.

Tunable Interlayer Delocalization of Excitons in Layered Organic-Inorganic Halide Perovskites

Layered organic-inorganic halide perovskites exhibit remarkable structural and chemical diversity and hold great promise for optoelectronic devices. In these materials, excitons are thought to be strongly confined within the inorganic metal halide layers with interlayer coupling generally suppressed by the organic cations. Here, we present an in-depth study of the energy and spatial distribution of the lowest-energy excitons in layered organic-inorganic halide perovskites from first-principles many-body perturbation theory, within the GW approximation and the Bethe-Salpeter equation. We find that the quasiparticle band structures, linear absorption spectra, and exciton binding energies depend strongly on the distance and the alignment of adjacent metal halide perovskite layers. Furthermore, we show that exciton delocalization can be modulated by tuning the interlayer distance and alignment, both parameters determined by the chemical composition and size of the organic cations. Our calculations establish the general intuition needed to engineer excitonic properties in novel halide perovskite nanostructures.

Pressure Dependence of Intra- and Interlayer Excitons in 2H-MoS2 Bilayers

The optical and electronic properties of multilayer transition metal dichalcogenides differ significantly from their monolayer counterparts due to interlayer interactions. The separation of individual layers can be tuned in a controlled way by applying pressure. Here, we use a diamond anvil cell to compress bilayers of 2H-MoS2 in the gigapascal range. By measuring optical transmission spectra, we find that increasing pressure leads to a decrease in the energy splitting between the A and the interlayer exciton. Comparing our experimental findings with ab initio calculations, we conclude that the observed changes are not due to the commonly assumed hydrostatic compression. This effect is attributed to the MoS2 bilayer adhering to the diamond, which reduces the in-plane compression. Moreover, we demonstrate that the distinct real-space distributions and resulting contributions from the valence band account for the different pressure dependencies of the inter- and intralayer excitons in compressed MoS2 bilayers.

Stress-Tuned Optical Transitions in Layered 1T-MX2 (M=Hf, Zr, Sn; X=S, Se) Crystals

Optical measurements under externally applied stresses allow us to study the materials' electronic structure by comparing the pressure evolution of optical peaks obtained from experiments and theoretical calculations. We examine the stress-induced changes in electronic structure for the thermodynamically stable 1T polytype of selected MX2 compounds (M=Hf, Zr, Sn; X=S, Se), using the density functional theory. We demonstrate that considered 1T-MX2 materials are semiconducting with indirect character of the band gap, irrespective to the employed pressure as predicted using modified Becke-Johnson potential. We determine energies of direct interband transitions between bands extrema and in band-nesting regions close to Fermi level. Generally, the studied transitions are optically active, exhibiting in-plane polarization of light. Finally, we quantify their energy trends under external hydrostatic, uniaxial, and biaxial stresses by determining the linear pressure coefficients. Generally, negative pressure coefficients are obtained implying the narrowing of the band gap. The semiconducting-to-metal transition are predicted under hydrostatic pressure. We discuss these trends in terms of orbital composition of involved electronic bands. In addition, we demonstrate that the measured pressure coefficients of HfS2 and HfSe2 absorption edges are in perfect agreement with our predictions. Comprehensive and easy-to-interpret tables containing the optical features are provided to form the basis for assignation of optical peaks in future measurements.

Ultrafast pseudospin quantum beats in multilayer WSe2 and MoSe2

Layered van-der-Waals materials with hexagonal symmetry offer an extra degree of freedom to their electrons, the so-called valley index or valley pseudospin, which behaves conceptually like the electron spin. Here, we present investigations of excitonic transitions in mono- and multilayer WSe₂ and MoSe₂ materials by time-resolved Faraday ellipticity (TRFE) with in-plane magnetic fields, B_∥, of up to 9 T. In monolayer samples, the measured TRFE time traces are almost independent of B_∥, which confirms a close to zero in-plane exciton g factor g_∥, consistent with first-principles calculations. In contrast, we observe pronounced temporal oscillations in multilayer samples for B_∥ > 0. Our first-principles calculations confirm the presence of a non-zero g_∥ for the multilayer samples. We propose that the oscillatory TRFE signal in the multilayer samples is caused by pseudospin quantum beats of excitons, which is a manifestation of spin- and pseudospin layer locking in the multilayer samples.

Screening of Excitons by Organic Cations in Quasi-Two-Dimensional Organic-Inorganic Lead-Halide Perovskites

Interlayer organic cations in quasi-two-dimensional halide perovskites are a versatile tuning vehicle for the optoelectronic properties of these complex systems, but chemical intuition for this design route is yet to be established. Here, we use density functional theory, the GW approximation, and the Bethe-Salpeter equation approach to understand the contribution of the organic cation to the quasiparticle band gap and exciton binding energy of layered perovskites. We show that organic cations in quasi-two-dimensional perovskites contribute significantly to the dielectric screening in these systems, countering quantum confinement effects on the quasiparticle band gap and the exciton binding energy. Using a simple electrostatics model inspired by parallel-plate capacitors, we decouple the organic cation and inorganic layer contributions to the effective dielectric constants and show that dielectric properties of layered perovskites are broadly tunable via the interlayer cation, providing a direct means of tuning photophysical properties for a variety of applications.

Strong Substrate Strain Effects in Multilayered WS2 Revealed by High-Pressure Optical Measurements

The optical properties of two-dimensional materials can be effectively tuned by strain induced from a deformable substrate. In the present work we combine first-principles calculations based on density functional theory and the effective Bethe-Salpeter equation with high-pressure optical measurements to thoroughly describe the effect of strain and dielectric environment onto the electronic band structure and optical properties of a few-layered transition-metal dichalcogenide. Our results show that WS2 remains fully adhered to the substrate at least up to a -0.6% in-plane compressive strain for a wide range of substrate materials. We provide a useful model to describe effect of strain on the optical gap energy. The corresponding experimentally determined out-of-plane and in-plane stress gauge factors for WS2 monolayers are -8 and 24 meV/GPa, respectively. The exceptionally large in-plane gauge factor confirms transition metal dichalcogenides as very promising candidates for flexible functionalities. Finally, we discuss the pressure evolution of an optical transition closely lying to the A exciton for bulk WS2 as well as the direct-to-indirect transition of the monolayer upon compression.

Temperature Dependence of the Indirect Gap and the Direct Optical Transitions at the High-Symmetry Point of the Brillouin Zone and Band Nesting in MoS2, MoSe2, MoTe2, WS2, and WSe2 Crystals

Following the rise of interest in the properties of transition metal dichalcogenides, many experimental techniques were employed to research them. However, the temperature dependencies of optical transitions, especially those related to band nesting, were not analyzed in detail for many of them. Here, we present successful studies utilizing the photoreflectance method, which, due to its derivative and absorption-like character, allows investigating direct optical transitions at the high-symmetry point of the Brillouin zone and band nesting. By studying the mentioned optical transitions with temperature from 20 to 300 K, we tracked changes in the electronic band structure for the common transition metal dichalcogenides (TMDs), namely, MoS₂, MoSe₂, MoTe₂, WS₂, and WSe₂. Moreover, transmission and photoacoustic spectroscopies were also employed to investigate the indirect gap in these crystals. For all observed optical transitions assigned to specific k-points of the Brillouin zone, their temperature dependencies were analyzed using the Varshni relation and Bose-Einstein expression. It was shown that the temperature energy shift for the transition associated with band nesting is smaller when compared with the one at high-symmetry point, revealing reduced average electron-phonon interaction strength.

Unusually large exciton binding energy in multilayered 2H-MoTe2

Although large exciton binding energies of typically 0.6–1.0 eV are observed for monolayer transition metal dichalcogenides (TMDs) owing to strong Coulomb interaction, multilayered TMDs yield relatively low exciton binding energies owing to increased dielectric screening. Recently, the ideal carrier-multiplication threshold energy of twice the bandgap has been realized in multilayered semiconducting 2H-MoTe₂ with a conversion efficiency of 99%, which suggests strong Coulomb interaction. However, the origin of strong Coulomb interaction in multilayered 2H-MoTe₂, including the exciton binding energy, has not been elucidated to date. In this study, unusually large exciton binding energy is observed through optical spectroscopy conducted on CVD-grown 2H-MoTe₂. To extract exciton binding energy, the optical conductivity is fitted using the Lorentz model to describe the exciton peaks and the Tauc–Lorentz model to describe the indirect and direct bandgaps. The exciton binding energy of 4 nm thick multilayered 2H-MoTe₂ is approximately 300 meV, which is unusually large by one order of magnitude when compared with other multilayered TMD semiconductors such as 2H-MoS₂ or 2H-MoSe₂. This finding is interpreted in terms of small exciton radius based on the 2D Rydberg model. The exciton radius of multilayered 2H-MoTe₂ resembles that of monolayer 2H-MoTe₂, whereas those of multilayered 2H-MoS₂ and 2H-MoSe₂ are large when compared with monolayer 2H-MoS₂ and 2H-MoSe₂. From the large exciton binding energy in multilayered 2H-MoTe₂, it is expected to realize the future applications such as room-temperature and high-temperature polariton lasing.

Organic-2D Material Heterostructures: A Promising Platform for Exciton Condensation and Multiplication

We predict that high temperature Bose-Einstein condensation of charge transfer excitons can be achieved in organic-two-dimensional (2D) material heterostructures, at ∼50-100 K. Unlike 2D bilayers that can be angle-misaligned, organic-2D systems generally have momentum-direct low-energy excitons, thus favoring condensation. Our predictions are obtained for ZnPc-MoS₂ using state-of-the-art first-principles calculations with the GW-BSE approach. The exciton energies we predict are in excellent agreement with recent experiments. The lowest energy charge transfer excitons in ZnPc-MoS₂ are strongly bound with a spatial extent of ∼1-2 nm and long lifetimes (τ₀ ∼ 1 ns), making them ideal for exciton condensation. We also predict the emergence of inter-ZnPc excitons that are stabilized by the interaction of the molecules with the 2D substrate. This novel way of stabilizing intermolecular excitons by indirect substrate mediation suggests design strategies for singlet fission and exciton multiplication, which are important to overcome the Shockley-Queisser efficiency limit in solar cells.

Computational methods for 2D materials modelling

Materials with thickness ranging from a few nanometers to a single atomic layer present unprecedented opportunities to investigate new phases of matter constrained to the two-dimensional plane. Particle-particle Coulomb interaction is dramatically affected and shaped by the dimensionality reduction, driving well-established solid state theoretical approaches to their limit of applicability. Methodological developments in theoretical modelling and computational algorithms, in close interaction with experiments, led to the discovery of the extraordinary properties of two-dimensional materials, such as high carrier mobility, Dirac cone dispersion and bright exciton luminescence, and inspired new device design paradigms. This review aims to describe the computational techniques used to simulate and predict the optical, electronic and mechanical properties of two-dimensional materials, and to interpret experimental observations. In particular, we discuss in detail the particular challenges arising in the simulation of two-dimensional constrained fermions and quasiparticles, and we offer our perspective on the future directions in this field.

The optical response of artificially twisted MoS[Formula: see text] bilayers

Two-dimensional layered materials offer the possibility to create artificial vertically stacked structures possessing an additional degree of freedom-the interlayer twist. We present a comprehensive optical study of artificially stacked bilayers (BLs) MoS[Formula: see text] encapsulated in hexagonal BN with interlayer twist angle ranging from 0[Formula: see text] to 60[Formula: see text] using Raman scattering and photoluminescence spectroscopies. It is found that the strength of the interlayer coupling in the studied BLs can be estimated using the energy dependence of indirect emission versus the A[Formula: see text]-E[Formula: see text] energy separation. Due to the hybridization of electronic states in the valence band, the emission line related to the interlayer exciton is apparent in both the natural (2H) and artificial (62[Formula: see text]) MoS[Formula: see text] BLs, while it is absent in the structures with other twist angles. The interlayer coupling energy is estimated to be of about 50 meV. The effect of temperature on energies and intensities of the direct and indirect emission lines in MoS[Formula: see text] BLs is also quantified.

Experimental and Theoretical Studies of the Electronic Band Structure of Bulk and Atomically Thin Mo1-x W x Se2 Alloys

We present studies focused on the evolution of the electronic band structure of the Mo1-x W x Se2 alloy with the tungsten content, which was conducted by combining experimental and theoretical methods. Employed spectroscopic techniques, namely, photoreflectance, photoacoustic spectroscopy, and photoluminescence, allowed observing indirect and direct transitions at high and beyond high-symmetry points of the Brillouin zone (BZ). Two excitons (A and B) associated with the K point of the BZ were observed together with other optical transitions (C and D) related to band nesting. Moreover, we have also identified the indirect transition for the studied crystals. Obtained energies for all transitions were tracked with a tungsten content and compared with results of calculations performed within density functional theory. Furthermore, based on the mentioned comparison, optical transitions were assigned to specific regions of the BZ. Finally, we have obtained bowing parameters for experimentally observed features, for, i.e., thin-film samples: b(A) = 0.13 ± 0.03 eV, b(B) = 0.14 ± 0.03 eV, b(C) = 0.044 ± 0.008 eV, and b(D) = 0.010 ± 0.003 eV.

Electrical tuning of optically active interlayer excitons in bilayer MoS2

Interlayer (IL) excitons, comprising electrons and holes residing in different layers of van der Waals bonded two-dimensional semiconductors, have opened new opportunities for room-temperature excitonic devices. So far, two-dimensional IL excitons have been realized in heterobilayers with type-II band alignment. However, the small oscillator strength of the resulting IL excitons and difficulties with producing heterostructures with definite crystal orientation over large areas have challenged the practical applicability of this design. Here, following the theoretical prediction and recent experimental confirmation of the existence of IL excitons in bilayer MoS₂, we demonstrate the electrical control of such excitons up to room temperature. We find that the IL excitonic states preserve their large oscillator strength as their energies are manipulated by the electric field. We attribute this effect to the mixing of the pure IL excitons with intralayer excitons localized in a single layer. By applying an electric field perpendicular to the bilayer MoS₂ crystal plane, excitons with IL character split into two peaks with an X-shaped field dependence as a clear fingerprint of the shift of the monolayer bands with respect to each other. Finally, we demonstrate the full control of the energies of IL excitons distributed homogeneously over a large area of our device.

Identification of twist-angle-dependent excitons in WS2/WSe2 heterobilayers

Stacking atomically thin films enables artificial construction of van der Waals heterostructures with exotic functionalities such as superconductivity, the quantum Hall effect, and engineered light-matter interactions. In particular, heterobilayers composed of monolayer transition metal dichalcogenides have attracted significant interest due to their controllable interlayer coupling and trapped valley excitons in moiré superlattices. However, the identification of twist-angle-modulated optical transitions in heterobilayers is sometimes controversial since both momentum-direct (K-K) and -indirect excitons reside on the low energy side of the bright exciton in the monolayer constituents. Here, we attribute the optical transition at approximately 1.35 eV in the WS2/WSe2 heterobilayer to an indirect Γ-K transition based on a systematic analysis and comparison of experimental PL spectra with theoretical calculations. The exciton wavefunction obtained by the state-of-the-art GW-Bethe-Salpeter equation (GW-BSE) approach indicates that both the electron and hole of the exciton are contributed by the WS2 layer. Polarization-resolved k-space imaging further confirms that the transition dipole moment of this optical transition is dominantly in-plane and is independent of the twist angle. The calculated absorption spectrum predicts that the usually called interlayer exciton peak coming from the K-K transition is located at 1.06 eV, but with a much weaker amplitude. Our work provides new insights into understanding the steady-state and dynamic properties of twist-angle-dependent excitons in van der Waals heterostructures.

Valley-Dependent Interlayer Excitons in Magnetic WSe2/CrI3

Heterostructures of two-dimensional transition-metal dichalcogenides and ferromagnetic substrates are important candidates for the development of viable new spin- or valleytronic devices. For the prototypical bilayer of WSe₂ on top of a ferromagnetic layer of CrI₃, we find substantially different coupling of both WSe₂ K-valleys to the sublayer. Besides an energy splitting of a few meV, the corresponding excitons have significantly different interlayer character with charge transfer allowed at the K̅^- point but forbidden at K̅⁺. The different exciton wave functions result in a distinctly different response to magnetic fields with g factors of about -4.4 and -4.0, respectively. By means of ab initio GW/Bethe-Salpeter equation calculations, these findings establish g factors as tool for investigating the exciton character and shedding light on the detailed quantum-mechanical interplay of magnetic and optical properties which are essential for the targeted development of optoelectronic devices.

Proximity control of interlayer exciton-phonon hybridization in van der Waals heterostructures

Van der Waals stacking has provided unprecedented flexibility in shaping many-body interactions by controlling electronic quantum confinement and orbital overlap. Theory has predicted that also electron-phonon coupling critically influences the quantum ground state of low-dimensional systems. Here we introduce proximity-controlled strong-coupling between Coulomb correlations and lattice dynamics in neighbouring van der Waals materials, creating new electrically neutral hybrid eigenmodes. Specifically, we explore how the internal orbital 1s-2p transition of Coulomb-bound electron-hole pairs in monolayer tungsten diselenide resonantly hybridizes with lattice vibrations of a polar capping layer of gypsum, giving rise to exciton-phonon mixed eigenmodes, called excitonic Lyman polarons. Tuning orbital exciton resonances across the vibrational resonances, we observe distinct anticrossing and polarons with adjustable exciton and phonon compositions. Such proximity-induced hybridization can be further controlled by quantum designing the spatial wavefunction overlap of excitons and phonons, providing a promising new strategy to engineer novel ground states of two-dimensional systems.

Here, the authors demonstrate proximity-controlled strong-coupling between Coulomb correlations and lattice dynamics in neighbouring van der Waals materials (WSe₂ and a gypsum layer), creating electrically neutral hybrid exciton-phonon eigenmodes called excitonic Lyman polarons.

Quantitative Nanoinfrared Spectroscopy of Anisotropic van der Waals Materials

Anisotropic dielectric tensors of uniaxial van der Waals (vdW) materials are difficult to investigate at infrared frequencies. The small dimensions of high-quality exfoliated crystals prevent the use of diffraction-limited spectroscopies. Near-field microscopes coupled to broadband lasers can function as Fourier transform infrared spectrometers with nanometric spatial resolution (nano-FTIR). Although dielectric functions of isotropic materials can be readily extracted from nano-FTIR spectra, the in- and out-of-plane permittivities of anisotropic vdW crystals cannot be easily distinguished. For thin vdW crystals residing on a substrate, nano-FTIR spectroscopy probes a combination of sample and substrate responses. We exploit the information in the screening of substrate resonances by vdW crystals to demonstrate that both the in- and out-of-plane dielectric permittivities are identifiable for realistic spectra. This novel method for the quantitative nanoresolved characterization of optical anisotropy was used to determine the dielectric tensor of a bulk 2H-WSe₂ microcrystal in the mid-infrared.

Giant Stark splitting of an exciton in bilayer MoS2

Transition metal dichalcogenides (TMDs) constitute a versatile platform for atomically thin optoelectronics devices and spin-valley memory applications. In monolayer TMDs the optical absorption is strong, but the transition energy cannot be tuned as the neutral exciton has essentially no out-of-plane static electric dipole^1,2. In contrast, interlayer exciton transitions in heterobilayers are widely tunable in applied electric fields, but their coupling to light is substantially reduced. In this work, we show tuning over 120 meV of interlayer excitons with a high oscillator strength in bilayer MoS₂ due to the quantum-confined Stark effect³. We optically probed the interaction between intra- and interlayer excitons as they were energetically tuned into resonance. Interlayer excitons interact strongly with intralayer B excitons, as demonstrated by a clear avoided crossing, whereas the interaction with intralayer A excitons is substantially weaker. Our observations are supported by density functional theory (DFT) calculations, which include excitonic effects. In MoS₂ trilayers, our experiments uncovered two types of interlayer excitons with and without in-built electric dipoles. Highly tunable excitonic transitions with large in-built dipoles and oscillator strengths will result in strong exciton-exciton interactions and therefore hold great promise for non-linear optics with polaritons.

Interlayer Bound Wannier Excitons in Germanium Sulfide

We report a cathodoluminescence (CL) study of layered germanium sulfide (GeS) where we observe a sharp emission peak from flakes covered with a thin hexagonal boron nitride film. GeS is a material that has recently attracted considerable interest due to its emission in the visible region and its strong anisotropy. The measured CL peak is at ~1.69 eV for samples ranging in thickness from 97 nm to 45 nm, where quantum-confinement effects can be excluded. By performing ab initio ground- and excited-state simulations for the bulk compound, we show that the measured optical peak can be unambiguously explained by radiative recombination of the first free bright bound exciton, which is due to a mixing of direct transitions near the Γ-point of the Brillouin Zone and it is associated to a very large optical anisotropy. The analysis of the corresponding excitonic wave function shows a Wannier-Mott interlayer character, being spread not only in-plane but also out-of-plane.

Ab Initio Studies of Exciton g Factors: Monolayer Transition Metal Dichalcogenides in Magnetic Fields

The effect of a magnetic field on the optical absorption in semiconductors has been measured experimentally and modeled theoretically for various systems in previous decades. We present a new first-principles approach to systematically determine the response of excitons to magnetic fields, i.e., exciton g factors. By utilizing the GW-Bethe-Salpeter equation methodology we show that g factors extracted from the Zeeman shift of electronic bands are strongly renormalized by many-body effects which we trace back to the extent of the excitons in reciprocal space. We apply our approach to monolayers of transition metal dichalcogenides (MoS_{2}, MoSe_{2}, MoTe_{2}, WS_{2}, and WSe_{2}) with strongly bound excitons for which g factors are weakened by about 30%.

Light-matter interaction in van der Waals hetero-structures

Even if individual two-dimensional materials own various interesting and unexpected properties, the stacking of such layers leads to van der Waals solids which unite the characteristics of two dimensions with novel features originating from the interlayer interactions. In this topical review, we cover fabrication and characterization of van der Waals hetero-structures with a focus on hetero-bilayers made of monolayers of semiconducting transition metal dichalcogenides. Experimental and theoretical techniques to investigate those hetero-bilayers are introduced. Most recent findings focusing on different transition metal dichalcogenides hetero-structures are presented and possible optical transitions between different valleys, appearance of moiré patterns and signatures of moiré excitons are discussed. The fascinating and fast growing research on van der Waals hetero-bilayers provide promising insights required for their application as emerging quantum-nano materials.

Controlling interlayer excitons in MoS2 layers grown by chemical vapor deposition

Combining MoS2 monolayers to form multilayers allows to access new functionalities. Deterministic assembly of large area van der Waals structures requires concrete indicators of successful interlayer coupling in bilayers grown by chemical vapor deposition. In this work, we examine the correlation between the stacking order and the interlayer coupling of valence states in both as-grown MoS2 homobilayer samples and in artificially stacked bilayers from monolayers, all grown by chemical vapor deposition. We show that hole delocalization over the bilayer is only allowed in 2H stacking and results in strong interlayer exciton absorption and also in a larger A-B exciton separation as compared to 3R bilayers. Comparing 2H and 3R reflectivity spectra allows to extract an interlayer coupling energy of about t⊥ = 49 meV. Beyond DFT calculations including excitonic effects confirm signatures of efficient interlayer coupling for 2H stacking in agreement with our experiments.

Directional Light Emission from Layered Metal Halide Perovskite Crystals

Metal halide perovskites are being increasingly explored for use in light-emitting diodes (LEDs), with achievements in efficiency and brightness charted across the spectrum. One path to further boosting the fraction of useful photons generated by injected electrical charges will be to tailor the emission patterns of devices. Here we investigate directional emission from layered metal halide perovskites. We quantify the proportion of in-plane versus out-of-plane transition dipole components for a suite of layered perovskites. We find that certain perovskite single crystals have highly anisotropic emissions and up to 90% of their transition dipole in-plane. For thin films, emission anisotropy increases as the nominal layer thickness decreases and is generally greater with butylammonium cations than with phenethylammonium cations. Numerical simulations reveal that anisotropic emission from layered perovskites in thin-film LEDs may lead to external quantum efficiencies of 45%, an absolute gain of 13% over equivalent films with isotropic emitters.

The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy

The GW approximation in electronic structure theory has become a widespread tool for predicting electronic excitations in chemical compounds and materials. In the realm of theoretical spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The number of GW calculations in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GW calculations for the first time. We also give an introduction to the many-body theory behind GW, a review of modern applications like molecules and surfaces, and a perspective on methods which go beyond conventional GW calculations. This review addresses chemists, physicists and material scientists with an interest in theoretical spectroscopy. It is intended for newcomers to GW calculations but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques.

Ultrafast transition between exciton phases in van der Waals heterostructures

Heterostructures of atomically thin van der Waals bonded monolayers have opened a unique platform to engineer Coulomb correlations, shaping excitonic^1-3, Mott insulating⁴ or superconducting phases^5,6. In transition metal dichalcogenide heterostructures⁷, electrons and holes residing in different monolayers can bind into spatially indirect excitons^1,3,8-11 with a strong potential for optoelectronics^11,12, valleytronics^1,3,13, Bose condensation¹⁴, superfluidity^14,15 and moiré-induced nanodot lattices¹⁶. Yet these ideas require a microscopic understanding of the formation, dissociation and thermalization dynamics of correlations including ultrafast phase transitions. Here we introduce a direct ultrafast access to Coulomb correlations between monolayers, where phase-locked mid-infrared pulses allow us to measure the binding energy of interlayer excitons in WSe₂/WS₂ hetero-bilayers by revealing a novel 1s-2p resonance, explained by a fully quantum mechanical model. Furthermore, we trace, with subcycle time resolution, the transformation of an exciton gas photogenerated in the WSe₂ layer directly into interlayer excitons. Depending on the stacking angle, intra- and interlayer species coexist on picosecond scales and the 1s-2p resonance becomes renormalized. Our work provides a direct measurement of the binding energy of interlayer excitons and opens the possibility to trace and control correlations in novel artificial materials.

Anisotropic ultraviolet-plasmon dispersion in black phosphorus

By means of momentum-resolved electron energy loss spectroscopy (EELS) coupled with scanning transmission electron microscopy, we have studied the dispersion relation of interband plasmonic modes in the ultraviolet in black phosphorus. We find that the dispersion of the interband plasmons is anisotropic. Experimental results are reproduced by density functional theory, by taking into account both the anisotropy of the single-particle response function, arising from the anisotropic band structure, and the damping. Moreover, our theoretical model also indicates the presence of low-energy excitations in the near-infrared that are selectively active in the armchair direction, whose existence has been experimentally validated by high-resolution EELS (HREELS) in reflection mode.

The quantum-confined Stark effect in layered hybrid perovskites mediated by orientational polarizability of confined dipoles

The quantum-confined Stark effect (QCSE) is an established optical modulation mechanism, yet top-performing modulators harnessing it rely on costly fabrication processes. Here, we present large modulation amplitudes for solution-processed layered hybrid perovskites and a modulation mechanism related to the orientational polarizability of dipolar cations confined within these self-assembled quantum wells. We report an anomalous (blue-shifting) QCSE for layers that contain methylammonium cations, in contrast with cesium-containing layers that show normal (red-shifting) behavior. We attribute the blue-shifts to an extraordinary diminution in the exciton binding energy that arises from an augmented separation of the electron and hole wavefunctions caused by the orientational response of the dipolar cations. The absorption coefficient changes, realized by either the red- or blue-shifts, are the strongest among solution-processed materials at room temperature and are comparable to those exhibited in the highest-performing epitaxial compound semiconductor heterostructures.

Valley-contrasting optics of interlayer excitons in Mo- and W-based bulk transition metal dichalcogenides

Recently, spatially indirect ("interlayer") excitons have been discovered in bulk 2H-MoTe2. They are theoretically predicted to exist in other Mo-based transition metal dichalcogenides (TMDCs) and are expected to be present in W-based TMDCs as well. We investigate interlayer excitons (XIL) in bulk 2H-MoSe2 and 2H-WSe2 using valley-resolved magneto-reflectance spectroscopy under high magnetic fields of up to 29 T combined with ab initio GW-BSE calculations. In the experiments, we observe interlayer excitons in MoSe2, while their signature is surprisingly absent in WSe2. In the calculations, we find that interlayer excitons exist in both Mo- and W-based TMDCs. However, their energetic positions and their oscillator strengths are remarkably different. In Mo-based compounds, the interlayer exciton resonance XIL is clearly separated from the intralayer exciton X1sA and has a high amplitude. In contrast, in W-based compounds, XIL is close in energy to the intralayer A exciton X1sA and possesses a small oscillator strength, which explains its absence in the experimental data of WSe2. Our combined experimental and theoretical observations demonstrate that interlayer excitons can gain substantial oscillator strength by mixing with intralayer states and hence pave the way for exploring interlayer exciton physics in Mo-based bulk transition metal dichalcogenides.

Interlayer Excitons with Large Optical Amplitudes in Layered van der Waals Materials

Vertically stacked two-dimensional materials form an ideal platform for controlling and exploiting light-matter interactions at the nanoscale. As a unique feature, these materials host electronic excitations of both intra- and interlayer type with distinctly different properties. In this Letter, using first-principles many-body calculations, we provide a detailed picture of the most prominent excitons in bilayer MoS₂, a prototypical van der Waals material. By applying an electric field perpendicular to the bilayer, we explore the evolution of the excitonic states as the band alignment is varied from perfect line-up to staggered (Type II) alignment. For moderate field strengths, the lowest exciton has intralayer character and is almost independent of the electric field. However, we find higher lying excitons that have interlayer character. They can be described as linear combinations of the intralayer B exciton and optically dark charge transfer excitons, and interestingly, these mixed interlayer excitons have strong optical amplitude and can be easily tuned by the electric field. The first-principles results can be accurately reproduced by a simple excitonic model Hamiltonian that can be straightforwardly generalized to more complex van der Waals materials.

Interlayer Trions in the MoS2/WS2 van der Waals Heterostructure

Electronic excitations in van der Waals heterostructures can have interlayer or intralayer character depending on the spatial localization of the involved charges (electrons and holes). In the case of neutral electron-hole pairs (excitons), both types of excitations have been explored theoretically and experimentally. In contrast, studies of charged trions have so far been limited to the intralayer type. Here we investigate the complete set of interlayer excitations in a MoS₂/WS₂ heterostructure using a novel ab initio method, which allows for a consistent treatment of both excitons and trions at the same theoretical footing. Our calculations predict the existence of bound interlayer trions below the neutral interlayer excitons. We obtain binding energies of 18/28 meV for the positive/negative interlayer trions with both electrons/holes located on the same layer. In contrast, a negligible binding energy is found for trions which have the two equally charged particles on different layers. Our results advance the understanding of electronic excitations in doped van der Waals heterostructures and their effect on the optical properties.