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Abstract
Strong light–matter interactions based on two-dimensional excitons formed in quantum materials such as monolayer transition-metal dichalcogenides have become a major subject of research in recent years. Particularly attractive is the extraordinarily large oscillator strength as well as binding energy of the excitonic quasiparticles in these atomically-thin crystal lattices. Numerous theoretical studies and experiments have been devoted to the exploration of the excitonic systems that could be exploited in future nano-scaled optoelectronic devices. To obtain unique insight into the exciton’s characteristics in an archetype monolayer quantum material, we directly measure the quasiparticle energy–momentum dispersion for the first time optically. Our results for h-BN encapsulated single-layer WSe2 clearly indicate an emission regime with a dispersion in the meV range in within the light cone at cryogenic temperatures. The amount of dispersion agrees well with calculations for an exciton–polariton based on the material’s monolayer exciton, or energetic modifications caused by exciton exchange interactions predicted for this material family. The measurable dispersion slightly weakens for elevated excitation densities, whereas at elevated temperatures, it even becomes immeasurable. The obtained reduction in dispersion is attributed to an enhanced role of uncorrelated charge carriers as well as the formation of phonon sidebands above 100 K.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Two-dimensional excitons formed in quantum materials such as monolayer transition-metal dichalcogenides and their strong light–matter interaction have attracted unrivalled attention by the research community due to their extraordinarily large oscillator strength as well as binding energy, and the inherent spin–valley locking [1,2]. Semiconducting few-layer and monolayer materials with their sharp optical resonances such as WSe2 have been extensively studied and envisioned for applications in the weak [3–5] as well as strong light–matter coupling [6–9] regimes, for effective nano-laser operation with various different structures [10–14], and particularly for valleytronic nanophotonics motivated by the circular dichroism [15,16]. Many of these applications, which may benefit heavily from the two-dimensional electronic quasiparticle’s properties in such films, require controlling, manipulating and first of all understanding the nature of the optical resonances that are attributed to exciton modes. While theory [17–19] and previous experiments [20–23] have provided unique methods to the characterization and classification efforts regarding the band structure and optical modes in 2D materials, the quasiparticles’ energy–momentum dispersion is still under exploration.
In solids, fundamental excitation of matter is characterized by the formation of hydrogen-like quasiparticles [24] consisting of an excited electron Coulomb-bound to a defect electron—a “hole” in the valence band. When coupling to a photon mode is impossible—either due to the absence of light states at high phase-space vectors or for dipole-forbidden transitions—bare excitons are found in the lattice of the host medium, also known as dark excitons. In contrast, the optical resonance of this quasiparticle state, which is a consequence of exciton–polariton formation, has its peculiar energy (for simplicity it is often just referred to as the exciton peak): it is on the one hand less energetic than the direct band-gap transition, due to the binding energy of the electronically-neutral bound system, which can be described in the two-particle picture with a free-particle dispersion and excitonic effective mass. On the other hand, the optical transitions occurring only at very low momenta within the range of light states have a shifted energy with respect to the bare exciton resonance, since the bright (i.e. optically visible) exciton is usually discussed in the picture of polaritons (see [25–29] and references therein). The shift of the longitudinal branch compared to the commonly discussed transversal exciton strongly depends on the oscillator strength. It can range from 0.08 meV for GaAs [30] over 60 meV or even 1 eV for transitions in molecular crystals [31]. The polaritons represent the quantization of the polarization—i.e. they are the eigenstates (basis vectors) of the light–matter Hamiltonian [32]— and are mixed states of photon modes and the relevant material resonance—note, here the coupling with the ordinary light cone (photon states) in the material is discussed in contrast to cavity–polaritons that are the product of coupling with a cavity mode. In a typical photoluminescence scenario, the excitation takes place in the upper polariton branch (UPB) in within the light cone [26,33,34]. From there, the excitation scatters/relaxes to lower lying states in the upper or lower polariton branch (LPB). Caused by momentum conservation of the parallel part with respect to the matter–air interface [9,35,36], only polaritons in within the light cone can be emitted to free space if phonon-assisted emission is neglected [34].
The resulting “optical band gap” is very characteristic for absorption and emission properties of semiconductor quantum structures [37,38], particularly in the case of two-dimensional (2D) transition-metal dichalcogenides (TMDCs) with extraordinary binding energies on the order of 0.5 eV in the monolayer (ML) regime [19,20,39,40] owing to the strong quantum confinement and optical band gaps in the visible to near-infrared spectral region (see [41], Mak et al. and references therein). Monolayer TMDCs such as WSe2, furthermore, feature a direct band gap configuration at two equivalent time-symmetry-inverted momentum-space valleys, K and K’, and a strong oscillator strength of electron–hole pairs (about two orders bigger than for conventional quantum wells, e.g. GaAs [42,43]). On top, spin–valley locking gives rise to circular dichroism and helicity-dependent light–matter interaction with the respective valleys. Comprehensibly, owing to the fact that room temperature resonances resemble excitonic behaviour with considerable binding energies, these materials have become very attractive for various applications ranging from photonics [41] to valleytronics [15,16,44].
Owing to the fact that emission out of the exciton line does not necessarily correspond to the decay of the expected Coulomb-bound particles [32], care needs to be taken when interpreting signatures that can be attributed to exciton modes. It is understood that the presence of excitonic resonances lets the electron–hole plasma excited in semiconductors radiate out of the exciton mode as well, even when there is no exciton population [32,45–47]. This can be understood in a sense that the excess energy compared to the exciton ground state is given to the remaining plasma [48,49], thus becoming only meaningful if an ensemble is considered. The build-up of an (incoherent) 2D-exciton population on ps-to-ns timescales can be probed effectively using ultrafast optical-pump–THz/IR-probe schemes, which can cover the intra-excitonic transitions (see [22,50]), as a means of verification of formed excitons (composite quasiparticles) and timescale investigations [22]. Complementary density-dependent probing of the absorption edge is used for the investigation of a transition from a Mott-insulator state to an electron–hole-plasma with eyes towards population inversion and band-gap renormalization [21]. However, hydrogen-like transitions and density-dependent binding-energy reduction or bleaching in the material are not unambiguous signatures for the presence of free excitons in monolayers, but clear features of excitonic populations. Therefore, another method is needed to directly evidence their presence, and in particular their polaritonic nature within the light cone.
An investigation of the linearity of charge-carrier-density-dependent photoluminescence intensity can give a hint of qualitatively different dynamics and exciton species. For example, a linear regime indicates free excitons [51–53]—in contrast to a sublinear behaviour for defect or bound states [51]. However, this type of experiment is not unambiguous, as for example biexcitons and plasma share the same predicted linearity factor [54,55]. Previously, temperature-dependent studies on monolayer WSe2 involving also hexagonal-boron-nitride (h-BN) encapsulated monolayers indicated such transition from a linear to a sublinear regime by a temperature-dependent decrease of the linearity factor above 100 K [56]. A similar experiment showed a change in the linewidth behaviour as a function of the temperature with a change from a linear to exponential increase above 100 K [57,58] that has been attributed to the formation of phonon sidebands [57,59,60]—or in the language of quasiparticles the occurrence of polaronic effects. Furthermore, a kink in the input-output linearity factor at higher excitation densities near to the Mott transition is found for TMDCs caused by a different band renormalization at K and Σ point [61,62].
Here, we demonstrate the first-of-its-kind direct optical measurement of the dispersion of the exciton resonance—with nearly parabolic centre-of-mass-momentum dependence—in a 2D semiconductor. Thereby, at cryogenic temperatures and low excitation densities, a dispersion in the range of 1–2 meV within the light cone and an effective exciton mass m* in the range of 10−3 to ${10^{ - 4}}{m_e}$ (electron mass) can be extracted for encapsulated WSe2. While this value disagrees with commonly stated values of 0.29 ${m_e}$ for excitons [18,63–65], Qiu et al. in 2015 [66] and other groups [67,68,69] predicted a mixing of the two degenerate excitons, formed by carriers at the K and K’ point, due to exchange interaction into one parabolic and one linear branch at centre-of-mass momentum $Q \approx 0$. The predicted dispersion of the upper branch for TMDCs in within the light cone is close to our observed values for WSe2.
However, here another source that could contribute to the dispersion is additionally taken into consideration for the explanation of observed features owing to the rather parabolic dispersion profiles. As we will discuss in the following, the strong oscillator strength of the monolayer exciton, together with the small dephasing rate at cryogenic temperature and low excitation densities, could give rise to polaritonic effects in the light-cone region. It is worth noting that such dispersion measurements have not even been performed explicitly for ordinary planar semiconductor quantum structures owing to the expectation that curvatures in the case of typical effective quantum-well-exciton masses ${\approx} > $ 0.5 me would be too shallow for any observations, not considering polariton formation. Furthermore, we investigate and discuss the behaviour of a 2D-confined freely-propagating Coulomb-bound (Wannier-like [46]) two-particle system with in-plane momentum dispersion as a function of temperature, excitation densities, and in relation to quasi-resonant and off-resonant excitation.
In our cryogenic optical spectroscopy measurements, a temperature-dependent transition from a free-exciton to a regime dominated by polaronic effects attributed to the previously reported phonon sidebands above 100 K is indicated. We also see a small excitation-density-dependent increase of the effective mass at elevated densities. This could be explained by the additional presence of electron–hole plasma or a sufficiently high amount of pump-induced phonons in the steady state. Ultimately, choosing an off-resonant above-(electronic)-band-gap injection of hot carriers as excitation scheme, the measured dispersion becomes flat and represents a plasma-dominated emission at the exciton resonance. These changes in the energy–momentum dispersion are in line with expectations for the three regimes and highlight the potential of our experiment, which set stringent requirements to the optical setup and the sample in order to be successful.
2. Theoretical considerations
For a free 2D-exciton with in-plane centre-of-mass motion, its energy can be described by the relation:
Owing to this dispersion relation and energy as well as momentum conservation, emitted photons resulting from the radiative collapse of free excitons of certain mass and momentum will carry information about the particle via its own momentum and energy (schematically shown in Fig. 1(b)). Decomposing the wave vector k of the photon into an out-of-plane k⊥ and in-plane component k||, the angle of emission in vacuum ${\theta _0}$ with respect to the normal is directly linked to the transverse momentum of the photon k|| via the following equation:
Fig. 1. Micrograph of the sample and schematics of the experiment: (a) Microscopy image of the h-BN-WSe2-h-BN stack on an SiO2/Si substrate. The sandwiched monolayer region is indicated by an arrow. The dotted line indicates WSe2, the dashed line the top h-BN. Inset: Schematic 2D heterostructure. (b) Schematic exciton propagation in a 2D lattice with in-plane momentum q|| and emission of photons with wave vector k, with the corresponding projection of detected angles in the Fourier space and corresponding in-plane dispersion relation.
To achieve the desired signatures experimentally, care has to be taken with the choice of the measurement parameters. For a maximized optical collection of emitted angles, which correspond to the in-plane momentum by Eq. (4), a numerical aperture NA = 0.6 is chosen for the employed microscope objective, while the working distance must enable sample studies behind a cryostat window. By Fourier-space projection, the imaging optics of the setup project angles up to ${\theta _o} ={\pm} 37^{\circ }$ onto the imaging monochromator’s entrance-slit plane. In contrast to angle-resolved photo-electron spectroscopy (ARPES), which resolves mainly the electronic band structure, but can provide indirect evidence of the exciton dispersion [73], which is not easily understood, angle-resolved photoluminescence spectroscopy provides an effective access to the optical dispersion, i.e. the polaritonic exciton’s dispersion. While electron energy loss spectroscopy can indeed also measure exciton dispersions using a fundamentally different approach, for example in bulk TMDCs [65,74] and hBN [75], up to our knowledge no measurement on a monolayer has been obtained, yet. However, only when the spectral lines are very narrow (i.e. small dephasing rate) and the optical resolution in E and k is sufficiently high, an exciton–polariton branch or the predicted big interaction-mediated exciton splitting at finite k, respectively the coexisting dispersions of two exciton branches, can be detected in Fourier-space-resolved spectra on the order of (sub-)meV angle-dependent energy changes. Such a (linear) exchange-energy-affected dispersion feature of TMDC excitons with meV changes could also facilitate observation of the polariton-branch curvature.
3. Experiment
3.1 Sample preparation
Tungsten diselenide (WSe2) bulk single crystals were grown in an excess selenium flux (defect density: 5×1010/cm2) (see [76]). For encapsulated samples, monolayer WSe2 and h-BN were first exfoliated from bulk single crystals onto SiO2. For WSe2, the SiO2 substrate was first exposed to an O2 plasma step before exfoliation. Monolayers and thin h-BN were both identified by optical contrast using a microscope. Afterwards, a dry stacking technique using polypropylene carbonate (PPC) on PDMS was used to pick up and stack h-BN/WSe2 layers. First a top layer of h-BN is picked up at 48 degrees C, then WSe2, and finally the bottom layer of h-BN. After each h-BN pickup step the PPC is briefly heated to 90 degrees C to re-smooth the PPC and ensure a clean wave front. For transferring the stack onto a clean substrate (∼290-nm SiO2 on Si), the substrate is first heated to 75 degrees C, the stack is then put into contact, and gradually heated to 120 degrees C. Afterwards, the PPC/PDMS is lifted and the substrate is immersed in chloroform and rinsed with IPA to remove polymer residue. Atomic-force microscopy confirmed a total stack thickness of ∼40 nm (∼10 nm + ∼30 nm for the encapsulating top and bottom h-BN, respectively). The resulting h-BN-encapsulated monolayer system is shown in Fig. 1(a).
3.2 Optical measurements
Angle-resolved micro-photoluminescence (µ-PL) measurements were performed using a self-built confocal optical microscope with Fourier-space imaging capabilities (Fig. 1(b), similar to [35,36]). The sample was mounted in an evacuated (∼10−7 mbar) helium-flow cryostat, which was placed under the 40x microscope (NA 0.6) objective of the µ-PL setup. All data shown are time-integrated spectra. For quasi-resonant excitation, a continuous-wave-operated titanium-sapphire laser (SpectraPhysics Tsunami) was used emitting light at 693 nm. A high aspect-ratio long-pass filter with specified edge at 700 nm (Thorlabs FELH 700) was used to block the laser light. Off-resonant pumping was achieved with a 445-nm continuous-wave diode laser. For detection, a nitrogen-cooled charge-coupled device (CCD) behind an imaging monochromator (Princeton Instruments Acton SP2300) was used, using two-dimensional chip read-out. Dispersion curves that are intensity (counts per integration time) over wavelength (long axis) over momentum (short axis) were recorded via the exposed CCD area.
For our investigations, an h-BN sandwiched high-quality WSe2 monolayer on an SiO2/Si substrate (see Fig. 1(a)) was mounted in an optical micro-cryostat for spectroscopy at temperatures down to 10 K. From such high-quality sandwich stack, a very narrow linewidth of the excitonic photoluminescence is obtained at cryogenic temperatures which is essential for the linewidth and detectability of the energy-momentum dispersion in practical measurements.
For k-space-resolved spectroscopy, the Fourier-space image of the sample’s emission is projected onto the monochromator entrance slit via a set of lenses. The light cone in the medium (vacuum) amounts to |k|||∼ 34 (8.7) µm−1, the maximum detectable angle of ± 37° corresponds to |k|||∼5.2 µm−1 in WSe2. The investigated excitation densities were set by a neutral density filter wheel (discrete steps). Detection of the µ-PL signal from the sample took place behind a spatially-filtering aperture in the real-space projection plane of the confocal microscope, which selects a spot of ∼1 µm diameter. The laser-spot diameter amounts to approximately 2 µm.
Each line of the acquired 2D-image has been fitted individually by Gaussian fit curves. Even though Lorentzian fitting is possible at low temperatures and quasi-resonant condition, a Gaussian profile is used as a substitute to a purely Lorentzian profile since small inhomogeneous broadening is still present especially at higher temperatures and for the sake of direct comparison between off-resonantly and quasi-resonantly excited PL data.
An excitation-density-dependent study clearly shows a curved dispersion of the WSe2 exciton mode which is slightly reduced in curvature at elevated densities. Figures 2(a) and 2(b) depict dispersion curves measured at 15 K for the continuous-wave (cw) excitation densities 0.1 and 3.6 kW/cm2, respectively. Here, even at the highest (cw safe-to-pump) power density, a parabolic behaviour can be deduced using the simple free-particle in-plane dispersion relation with effective masses (Eq. (1)). Dashed lines represent fitted parabolic dispersion curves corresponding to the extracted effective masses. As neither pump-dependent shifts nor biexciton peaks are present over the probed one and a half orders of magnitude densities, the carrier densities can be estimated to be in the linear regime (cf. Fig. 3(d)). If one further assumes a constant absorption of 1 percent [18] at the quasi-resonant excitation energy and an exciton decay time of 10 ps [56] one can estimate the steady-state carrier density to range from 3×108 to 1.1×1010 cm−2. Figure 2(c) shows an overview on the density-dependent changes obtained in this density region.
Fig. 2. Density-dependent dispersion changes of the WSe2 exciton mode: (a) and (b) depict energy–momentum-dispersion curves measured at 15 K in false-colour intensity representation in arbitrary units (linear colour scale from white to dark blue corresponding to minimum to maximum signal), for the excitation densities 0.1 and 3.6 kW/cm2, respectively. (c) An overview on the density-dependent changes is given for the measurement range, showing a reduction of the curvature that corresponds to higher effective masses. (d) shows a corresponding linewidth diagram with increased width towards higher momenta. Dashed lines represent fitted parabolic dispersion curves corresponding to the extracted effective masses labelling the data. Dispersion and linewidth data are offset for clarity vertically by 1.0 meV and 0.5 meV, respectively. Error bars represent fit errors of Gaussian fits to the individual line spectra.
Fig. 3. Pump-scheme-dependent Fourier-space spectra of the WSe2 exciton mode: Photoluminescence dispersion curves measured at very low temperatures and for similar excitation densities are compared for quasi-resonant and off-resonant excitation ((a), 15 K with 3.6 kW/cm2 and, (b), 10 K with 2 kW/cm2, respectively). The dashed line in (a) represents the parabolic dispersion of a free particle following the peak positions extracted for each line-spectrum individually by a Gauss fit, whereas in (b) a flat dispersion is indicated by a dotted line. Insets represent corresponding integrated line spectra without angle-resolution. (c) shows the extracted Gaussian-fit linewidth of both measurements in a comparison as a function of the in-plane momentum. (d) From the input–output curve of the ground-state (k = 0) photoluminescence under quasi-resonant excitation, one can extract a linearity factor of 1, indicating the presence of free excitons over the investigated density range.
In photoluminescence, a slight reduction of the curvature is obtained for increased densities that corresponds to an increase of the effective mass m* = meff me from 6.6×10−4 to 7.2×10−4 me as labelled in the plot. Additionally, reflection-contrast spectra present dispersion measurements recorded in a white-light-reflection configuration, resembling a zero-density-regime probe of the dispersion relation (see Fig. 5), which feature similar effective masses. Relative effective-mass fit certainty is about 5–10%. The density-dependent PL trend can be expected for a transition from a regime with (negligibly) low fraction of electron–hole plasma to one with an increased fraction, leading to slightly higher optically-retrieved effective masses at higher pump-densities under quasi-resonant excitation conditions. Simultaneously, no change in the emission-linewidth can be recognized within this density range, as data displayed in a waterfall diagram with constant offset of 0.5 meV in Fig. 2(d) show.
Here, the extracted Gaussian linewidth as a function of the momentum is displayed. One can clearly see a smaller linewidth at the centre of the light cone of about 1.7 meV (full-width-at-half-maximum corresponds to 4.0 meV) compared to those closer to its boundaries (about 2.1 meV, FWHW of 4.9 meV). This could be as the states with vanishing centre-of-mass momentum, i.e. resting excitons, are less influenced by inhomogeneous broadening due to spatial fluctuations. In the following, the measured dispersion picture needs to be interpreted in the exciton–polariton picture to explain such light effective masses and correspondingly strong dispersion curvatures. Alternatively, a strong increase of the exchange interaction similar to the predication for MoS2 could also lead to such masses.
Remarkably, but not surprisingly, the excitation conditions strongly influence the emission regime obtained for the 2D system. While near-resonant optical excitation with the titanium-sapphire laser in continuous-wave mode enabled us to detect dispersion characteristics for free excitons (exciton–polaritons) in our 2D-material system, high-energy off-resonant excitation with a blue continuous-wave laser diode led to a completely lost (or irresolvable) angle-dependence of the emission out of the excitonic resonance.
The choice of these two excitation modes had to serve experimental necessities. On the one hand, strong suppression of the laser light in optical spectra at close-to-resonant excitation conditions below the electronic band gap were needed which was achieved only for narrow-line continuous-wave operation together with a high-performance long-pass filter. On the other hand, blue light was needed for strictly off-resonant laser wavelengths beyond the tunability range of the titanium-sapphire laser. A comparison of the Fourier-space spectra at about 15 K for both excitation schemes is shown in Fig. 3.
For quasi-resonant and off-resonant pumping, the results are very distinct, as Fig. 3 demonstrates. In the left plot, it can be seen that excitation with the titanium-sapphire laser at 693 nm (1.79 eV) results in a measurable dispersion at 15 K with angle-dependent Gaussian-fit linewidths (σ) between 1.7 and 2.1 meV, i.e. FWHM ranging from 4.0 to 4.9 meV (full-width-at-half-maximum corresponds to $2\sqrt {2ln(2)} \sigma $). It is worth noting that the theoretical homogeneous linewidth for similar TMDCs was predicted to be a few meV (Selig et al. 2016 [58], Dey et al. 2016 [77]), whereas the experimentally-determined extrapolated homogeneous linewidth value for WSe2 at 0 K has been reported to be 1.6 meV (Moody et al. 2015 [78]). Thus, the low-temperature FWHM in this work as low as <4.0 meV around k = 0 is very close to the homogeneous-linewidth limit. In contrast, off-resonant excitation with a 445-nm diode laser at similar pump densities leads to a straight line in the Fourier-space-resolved emission spectrum at the same temperature level (10 K), with angle-independent linewidths of about 3 meV (FWHM ∼ 7 meV). The densities for both excitation energies amount to approx. 1010 cm−2. With the same integration times and exposure conditions used, signal levels in Figs. 3(a) and 3(b) are correlated. If one assumes 10 times more absorption for the off-resonant excitation scheme, the density in Fig. 3(b) would be a factor of 5.6 larger than in Fig. 3(a) which is not indicated by the count rates. However, linewidth extraction (Fig. 3(c)) indicates clear differences in the emission regime due to a dominating role of uncorrelated charge-carriers in the case of off-resonant excitation. Regarding density-dependent investigations, we assume from the linearity factor of 1 in the input-output characteristics of the excitonic resonance (Fig. 3(d)) that a free-excitonic regime is present for the whole pump-density range corresponding to the deduced excitation densities between 108 cm−2 for the lowest power used and 1010 cm−2 for the highest power used here. We neglect renormalization of bands owing to the small carrier concentrations obtained by our cw pumping and the lack of density-dependent energy and linewidth changes (cf. Fig. 2).
It is important to note that continuous-wave pumping provides steady-state photoluminescence from the investigated excitonic mode at equilibrium conditions of the electronic and lattice system. Thereby, any time-resolved density-dependent effects, which could arise from population build-up, relaxation and cooling processes and temporal heat dissipation into the lattice, are ruled out. Moreover, intensity levels for time-integrated acquisition are good enough under continuous-wave pumping for imaging spectroscopy probing the Fourier space. In contrast, time-integrated spectroscopy with pulsed irradiation would lead to a smearing out of the exciton dispersion due to the dynamics and varying particle densities throughout the emission pulse, as every single time window in the ps scale would constitute a different excitation regime.
A clear dispersion is also obtained in temperature-dependent optical measurements for temperatures up to 100 K in photoluminescence and in reflection contrast. Figure 4(a) shows an energy–momentum dispersion measured for monolayer WSe2 PL at 15 K for quasi-resonant pumping at 3.6 kW/cm2. Similar data for 100 and 150 K are shown in Figs. 4(b) and 4(c), respectively. An emission spectrum measured at 100 K with off-resonant pumping at 2 kW/cm2 is shown in Fig. 4(d), which features a flat dispersion in contrast to the quasi-resonant excitation scenario in Fig. 4(b). Figure 4(e) summarizes the temperature series for quasi-resonant excitation. The error bars of the dispersion data points (also in Fig. 2(c)) correspond to the Gauss-fit uncertainty of the wavelength-resolved line fits. Dashed lines represent fitted parabolic dispersion curves corresponding to the extracted effective masses labelling the data. Similar results are obtained in reflection contrast measurements. Figure 5 shows a corresponding temperature series, for which a flattening of the curvature with effective masses increasing from 3×10−4 to 7×10−4 me can be observed. These values represent a zero-density excitation regime and are not affected by density-related effects on the curvature as discussed below. However, a non-zero curvature remains visible at higher temperatures, whereas in contrast, PL dispersion becomes unmeasurable due to the role of phonon sidebands above 100 K.
Fig. 4. Free-exciton emission characteristics in temperature-dependent optical dispersion measurements: (a) Energy–momentum dispersion (false-colour intensity representation, similar to Fig. 2) for monolayer WSe2 obtained at 15 K for quasi-resonant pumping. Similar data for 100 and 150 K is shown in (b) and (c), respectively. For comparison to the corresponding quasi-resonant case in (b), an emission spectrum measured at 100 K with off-resonant pumping is displayed in (d). (e) summarizes the dispersion temperature series for quasi-resonant excitation at 15 K, 77 K, 100 K and 150 K. Error bars represent fit errors of Gaussian fits to the individual line spectra. Dashed lines represent fitted parabolic dispersion curves corresponding to the extracted effective masses labelling the data. Linewidth data shown in semi-logarithmic scale. No offsets applied.
Fig. 5. Fourier-space resolved white-light reflection-contrast measurements: (a) and (b) show the first derivative of measured reflection-contrast spectra for the above discussed encapsulated WSe2 at 10 K and 150 K, respectively. For clarity, a parabolic fit to the centre frequency is displayed into the contour charts. Dots indicate the extracted resonance positions along the measured dispersion from line fits (1st derivative of a Lorentzian profile). (c) Temperature-dependent energy–momentum dispersion extracted from measured spectra with fitted parabolic dispersion. The average effective mass is 3×10−4 ${m_e}$.
4. Discussion
In the following, a discussion regarding the attribution of the dispersion behaviour to underlying effects is provided: Therefore, we need to look into (a) the loss of dispersion due to plasma and polaron effects, (b) exchange-interaction effects and (c) polaritonic effects.
While non-resonantly excited emission exhibits no curvature at all, the unambiguous occurrence of a curved dispersion up to about 100 K for quasi-resonant excitation can be either attributed to the presence of polaritons based on free excitons in the TMDC below 100 K, a strong exchange interaction or a combined effect. Its light character gives the exciton–polariton a strongly reduced effective mass (average effective mass is 3×10−4 me) in comparison to bare bound electron-hole pairs (0.29 me) [18]. At the same time a lifting of degeneracy of the exciton branches could also lead to smaller effective masses similar to the striking features of the optical resonance at cryogenic temperatures and low densities observed here.
However, above 100 K, the recorded photoluminescence signal cannot be attributed to purely radiative recombination of a free excitonic feature. Indeed, when considering phonon sidebands, the disappearance can be understood—as will be discussed later on—thereby shedding light on the nature of those species that are observed at elevated temperatures. This finding improves our general understanding to that extent that most of the excitonic population at room temperature in monolayer WSe2 (and likely all other 2D semiconductors of the same family) cannot be thought of as free excitons, but have to be considered with their phonon sidebands in the picture of polarons [57,79].
The loss of curvature for angle-resolved data for off-resonant pumping, however, can be explained by the electron–hole-plasma dominance in the monolayer’s emission. In contrast to the free exciton with centre-of-mass propagation in the plane of 2D materials, which are supposed to exhibit a measurable dispersion as polaritons owing to their large oscillator strength, electron-hole plasma emission out of the exciton resonance cannot exhibit any dispersion according to the theoretical considerations below. Optical recombination for unbound electron-hole pairs takes statistically place for all possible angles at the detected energies in consideration of the equations in the polaron picture below after Feldtmann et al. [79] (or similarly described in the exciton picture [59,79,80]). Equations (6)–(8) are a steady-state solution to the semiconductor luminescence equations including the phonon sidebands (SLE) [57,59,79], a commonly used many-body approach for modelling emission of semiconductors that is valid for a low excitation density. It describes the photoluminescence intensity at energy $E = \hbar \omega $ and momentum q:
Equation (7) describes the intensity for a certain phonon sideband. While below 100K the occupation numbers of the optical phonons are nearly zero, making the sideband negligible for the bright excitons (still noticeable for the dark states [59]). However, for temperatures above 100 K, a clear sign of phonon sidebands formation for bright states can be seen in temperature-depended studies [57,58]. From 10 K to 100 K, mainly contributions by acoustic phonons are of relevance. It is clear from the last part of Eq. (7) that the formation of phonon sidebands leads to the activation of excitonic/polaritonic states outside the light cone [81]. As can be seen from the denominator, the formation of phonon sidebands lifts the locking between q and Q, that can be assumed at low temperatures. Therefore, the dispersion cannot be easily accessed by means of angle-resolved photoluminescence spectroscopy.
The nominator in the last part of Eq. (7) together with Eq. (8) shows that the emission of plasma at the excitonic resonance or a phonon sideband with momentum q is not linked to the momentum of the carriers k, but to the difference of momentum of the hole and electron. As the sum goes over all possible combinations of uncorrelated holes and electrons, no dispersion (i.e. flat profile in measurements) can be obtained statistically for a pure electron–hole plasma. In the case of a mixed regime of excitons and uncorrelated charge carriers, an effectively smaller signature of dispersion can be expected.
To take the possibility of an observation of a split excitonic dispersion according to the aforementioned model represented by Eqs. (2) and (3) into consideration, experimental data from Fig. 4(a) is displayed with the results of the dispersion model of Qiu et al. [66] (dashed lines) in Fig. 6(a), using their model values for MoS2. The predicted exchange-interaction-caused dispersion modification—an effect of the lifted degeneracy in the form of a splitting at finite k—on the order of 1 meV within the setup’s detectable light-cone is already close to our observed dispersion for WSe2. Although calculations were reported by Qiu et al. [66] for suspended MoS2, the result is qualitatively most likely and well believably transferable to WSe2 subject of this work (quantitatively a similar magnitude can be expected), as indicated by the calculation of Deilmann et al. [69]. The obtained dispersion-related energy shifting in the experimental data is slightly bigger than that predicted for the upper exciton branch in MoS2 [66,67]. This difference can originate from a higher value of the exchange interaction (A) for WSe2 (see [69]). Deilmann et al. have calculated the dispersion for all four common TMDCs for monolayer as well as for bilayer [69]. From their plot, an increase of roughly a factor of two can be estimated for the Tungsten family compared to their Molybdenum counterparts. However, while the recorded momentum-dependent photoluminescence could be a consequence of such effect, in the following, another source of dispersion is taken into account for the explanation of observed features owing to the rather parabolic dispersion profiles.
Fig. 6. Dispersion scenarios: (a) Experimental data from Fig. 4(a) with the model of Qiu et al.66 (dashed lines) using their model values for MoS2. (b) Numerically-obtained solution of the upper exciton–polariton branch according to Eq. (5) with an underlying parabolic exciton dispersion (m* = 1.4 me from Ref. [66]). (c) Solution similar to (b) for an underlying exciton having one parabolic and one linear branch [66]. In both cases, damping was neglected. The dotted curve in the inset of the simulation plots is a parabolic fit with an effective mass of 2.0×10−3 me respectively 1.2×10−3 me.
In the next step, the possibility and magnitude of the influence of polaritonic effects will be discussed. Therefore, Eq. (5) was solved for two scenarios: (i) a double-degenerate exciton resonance (i.e. parabolic dispersion related to exciton mass) neglecting the exchange interaction and—for the sake of comparison—(ii) a combined situation described by Eqs. (2) and (3) (i.e. branches non-degenerate, featuring one linear and one parabolic branch); the values for the parameters are taken from Qiu et al. [66]). The dephasing rate was neglected, its oscillator strength was taken as $f = 0.64e{V^2}$. This resembles almost half of the value (1.9 eV2) determined in Li et al. [42]. This is necessary because of the use of two branches instead of one (distribution of the oscillator strength onto both branches), to effectively reach the same amount of absorbance. Furthermore, a small decline of the oscillator strength (about 20%) upon encapsulation is predicted [18]. In fact, for a larger value of the oscillator strength, the exciton–polariton curvature would be even increased due to a stronger normal-mode splitting. The background dielectric was also estimated from Li et al. [42] as ${{\epsilon }_b} = 15$. The polariton results in case (i) and (ii) can be seen in Figs. 6(b) and 6(c), respectively. The horizontal axis represents Re{k}, while the energy–momentum contour plot’s profile with linear false-colour scaling represents Im{k}. These calculations have to be understood as exemplary polariton behaviour based on estimations for the parameters of a monolayer-TMDC exciton. Here, the upper polariton branch (relative to the photoluminescence energy at zero momentum) clearly approaches the light cone in WSe2 (black line), while the vacuum light cone (blue line at lower momentum) is also displayed. The setup detection limit is indicated as dashed line. The calculated solutions were fitted with a parabolic dispersion like the experimental data. From the curvature around the centre of the light cone, a mass of about 10−3${m_e}$ and lighter than that could be expected.
In summary, these theoretical considerations explain the relatively strong curvature of the excitonic resonance in the range of 15 to 100 K when considering the modes as exciton–polaritons or assuming a strong underlying exchange-interaction-related exciton mode splitting. If the combination of both is considered, even stronger k-dependent energy shifts can be expected. Also, the lack of dispersion in angle-resolved measurements for off-resonant pumping schemes is supported by the SLE by the role of uncorrelated charge carriers, whereas above 100 K the polaron picture comes into play. Continuous-wave quasi-resonant pumping, however, directly leads to the matter excitation corresponding to exciton (exciton–polariton) formation and is evidenced by the optically measured dispersions in time-integrated spectra representing the system in its steady-state configuration. Although reports in the literature have discussed light–exciton interactions for in-plane guided modes in bulk TMDCs [82], our ML results are directly obtained within the light cone for both PL and reflectance contrast measurements.
5. Conclusion
The presented work has focused on energy–momentum dispersion studies for monolayer WSe2 using our direct optical measurements of Fourier-space-resolved spectra, which confirm the presence of free excitons with distinct dispersion features. The low-temperature measurable dispersion on the meV scale renders the excitonic feature a uniquely light-weight quasiparticle with effective mass on the order of 10−3 to 10−4 electron masses, which would make the system highly attractive for Bose-Einstein-condensation studies at cryogenic temperatures. This surprisingly strong dispersion could arise in two scenarios. On the one hand, for 2D excitons with high oscillator strength, polaritonic effects that can drastically reduce the effective mass as supported by our calculations cannot be neglected, provided that the samples are nearly only homogenously broadened. On the other hand, the role of electronic exchange interactions is considered to be another possible source of strong dispersion. However, here an almost twice as strong exchange interaction would be necessary compared to the prediction for MoS2, as indicated for WSe2 in the literature. In addition, a combined case could be possible. Further experiments and calculation are needed to clarify the exact origin.
For the characterized optical resonance, we also show a density- and temperature-dependent transition from a free-quasi-particle-dominated regime to an intermediate phase with plasma and phonon contributions, respectively, for an archetype 2D material. Changes of the energy–momentum dispersions’ curvatures were unambiguously and astonishingly measured via angle-resolved photoluminescence for high-quality WSe2 encapsulated between thin h-BN providing excellent excitonic linewidth conditions for such an unrivalled study, stimulating further experiments and theoretical considerations.
Appendix A: The polariton picture and solution for polaritons with dephasing
From the link between the real and imaginary parts of the complex frequency-dependent refractive index $n(\omega ) + i\kappa $ and the real and imaginary parts of the momentum ${\mathop{\rm Re}\nolimits} \{ k\} = n(\omega ){\omega _0}/c$ and ${\mathop{\rm Im}\nolimits} \{ k\} = \kappa (\omega ){\omega _0}/c$, respectively (after Eq. (5.8) in Klingshirn [26]), the dispersion relation of exciton–polaritons can be readily obtained, whereas the upper branch starts at the energy of the longitudinal exciton ${\omega _l}$ and the light branch asymptotically approaches the Lorentz oscillator frequency ${\omega _0}$. The observation of polariton dispersion can be regarded as a clear indicator of a close-to homogeneous-linewidth regime, where the harmonic-oscillator description for the emitter holds true and its coupling to the field is strongest.
The exemplary result of Eq. (5) for the case without damping has been performed for a ML TMDC exciton with typical literature parameters (see Fig. 4), once using two doubly-degenerate parabolic exciton branches Fig. 4(b) and once using the model with exchange-interaction effects by Qiu et al. [66] Fig. 4(c). For chosen parameters, the splitting ${\omega _L} - {\omega _0}$ amounts to 31 meV, which could be even higher for free-standing monolayers with optimal parameters given the naturally large oscillator strength of 2D excitons. A similar result of Eq. (5) with else equal parameters for the case of small damping is shown in Fig. 7. Due to a dephasing rate ${\gamma _x} = 0.5meV$, the dispersion is slightly increased compared to the case of no damping; here, the splitting ${\omega _L} - {\omega _0}$ amounts to 25 meV. If the solution is fitted in a similar fashion compared to the solution without damping $({\gamma _x} = 0)$, effective masses as low as 1×10−3 me can be obtained for the considered cases of Fig. 4. If the kink visible in Fig. 7 in the zero-momentum region (zoomed in inset) due to dephasing is considered, even smaller effective masses can be deduced which match those in the experiment. With and without damping, the exchange-interaction model clearly enhances the slope of the polariton dispersion around the centre of the light cone.
Fig. 7. Polariton solution with dephasing: (a) Simulation using two doubly-degenerate parabolic exciton branches and (b) using the model with exchange-interaction effects by Qiu et al. [66]. In both cases, a dephasing value of 0.5 meV was used. The results in (a) clearly match the descriptions and dispersions in Klingshirn [26], whereas in (b) a stronger curvature results from the underlying linear upper exciton branch. Due to the dephasing, the overall dispersion is increased compared to Fig. 6. The dotted line in the inset represents a parabolic fit with 17×10−4 (a) and 10×10−4 me(b). Light cones of the setup, vacuum and material are indicated by dashed, solid blue and solid black lines, respectively.
Appendix B: Temperature-dependent photoluminescence measurements
Additional measurements of the same kind were performed months later with smaller temperature steps in quasi-resonantly excited photoluminescence (cf. Fig. 8). Unfortunately, an unnoticed slight tilt in the mount-angle of the monochromator’s CCD camera has caused a minor tilt in these luminescence dispersion data (c.f. flat lines at 130 and 150 K). As can be seen from the photoluminescence measurement, the observed dispersion clearly flattens above 100 K. Similar to previous measurements, the linewidth broadens significantly. Here, the initial low-temperature linewidth is already slightly above 3 meV (FWHM ∼ 7 meV) due to aging effects for the 2D stack.
Fig. 8. Repeated Fourier-space resolved photoluminescence temperature series: The diagram shows the extracted dispersion (left) similar to Fig. 4 together with the extracted linewidth (right). No offsets are applied to the colour-coded linearly scaled data. The evolution of the extracted effective mass is indicated by labels to the measured dispersions and their parabolic fit curves.
To demonstrate further repeatability, another freshly prepared sample has been measured in our setup. For the quasi-resonant measurement, again a curvature of a comparable magnitude is found. Similar to previous measurements, the clear dispersion vanished above 100 K (cf. Fig. 9).
Funding
Deutsche Forschungsgemeinschaft (RA2841/5-1, SFB1083); Philipps-Universität Marburg; Deutscher Akademischer Austauschdienst; National Science Foundation (DMR-1420634).
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