Ten-valley excitonic complexes in charge-tunable monolayer WSe$_2$

Abstract

Excitons dominate the optical response of two-dimensional (2D) semiconductors. Strong interactions produce peculiar excitonic complexes, which provide a testing ground for exciton and quantum many-body theories. Here, we report a hitherto unobserved many-body exciton that emerges upon filling both the K and Q valleys of WSe$_2$. We optically probe the exciton landscape using charge-tunable devices with unusually thin dielectrics that facilitate doping up to several $10^{13}$ cm$^{-2}$. We observe the emergence of the thermodynamically stable complex when 10 valleys are electrostatically filled. We gain insight into its physics using magneto-optical measurements. Our results are well-described by a model where the number of distinguishable Fermi seas interacting with the photoexcited electron-hole pair defines the complex's behavior. In addition to expanding the repertoire of excitons in 2D semiconductors, this complex could probe the limit of exciton models and answer open questions about screened Coulomb interactions in 2D semiconductors.

Figures (9)

• Figure 1: Device structure and gate-dependent optical response of WSe2. a, Gate-dependent reflection contrast spectra of a WSe2 monolayer taken at 4K showing neutral (around $V\mathrm{_{G}} \sim \qty{0}{\volt}$), negatively charged (positive $V\mathrm{_{G}}$), and positively charged excitons (negative $V\mathrm{_{G}}$). b, Close-up view on the negatively charged regime from (a) revealing a variety of excitonic complexes: the exchange-split singlet and triplet trions ($X^{-}_{\mathrm{S,T}}$), the hexciton ($H$), the oxciton ($O$), and another many-body exciton ($M$). Arrows mark the onset of the filling of the lower conduction band (CB) valleys at K/K' (green), the upper CB valleys at K/K' (orange), and the lower CB valleys at Q/Q' (purple). Inset top right: reflection contrast spectra from the same dataset over the transition from $O$ to $M$. Inset bottom left: the Brillouin zone of the 2D WSe2 crystal with labels to show the $\Gamma$ point, the K/K' points and the Q/Q' points right in between the latter two. c, Band diagram schematics of the neutral exciton and negatively charged excitons for increasing electron doping. Initially, electrons start filling the lower spin-orbit split K/K' valleys as the density increases, sequentially promoting the formation of singlet and triplet trion/tetron-, hexciton-, and oxciton complexes. Eventually, the Fermi level reaches the Q/Q' valley band edge, giving rise to an even larger $M$ exciton involving charges from both the K/K’ and Q/Q’ valleys. The complexes are depicted as the binding between the photoexcited e-h pair and Fermi particle-hole excitations of the distinguishable Fermi seas, wherein the CB holes of the Fermi seas move together and are correlated with the complex.
• Figure 1: Gate-dependent optical response of WSe2 control sample. a, Gate-dependent reflection contrast spectra of our control WSe2 device recorded at 4K, the inset shows a schematic of the dual-gated WSe2 device with labelled hBN thicknesses. Due to the asymmetry in the hBN spacer thicknesses the bottom gate can be biased more than the top gate. Therefore the voltage on the y-axis is the voltage applied to the top gate, where the voltage at the bottom gate is $V_{\mathrm{b}}=1.4\cdot V_\mathrm{t}$. b, Close-up view of the negatively charged regime from (a) revealing the same excitonic complexes as observed in Fig. \ref{['fig:1device']}: the exchange-split trions ($X^{-}_{\mathrm{S,T}}$), the hexciton ($H$), the oxciton ($O$), and the many-body complex ($M$). The start of the filling of the lower K/K' valleys, the upper K/K' valleys, and the Q/Q' valleys are marked with a green, orange, and purple dashed line, respectively. Notably, the energy at which the Q/Q' valleys reside ($\Delta_{\mathrm{KQ}}$) in this control sample, is different from the value of the main sample shown in Fig. \ref{['fig:2qvalley']}a. This is evidenced by the different ratios of the electron densities at which $O$ appears (filling of the upper K/K' valleys) and the density at which $M$ appears (filling of the Q/Q' valleys). This ratio is given by $(V_{\mathrm{Q}}-V_{0})/(V_{\mathrm{uK}}-V_{0})$, in which $V_{0}$, $V_{\mathrm{uK}}$ and $V_{\mathrm{Q}}$, are the voltages at which the filling starts of the lower valleys at K/K', the upper valleys at K/K' and the valleys at Q/Q' respectively, which yields 1.81 and 1.96 for the main sample shown in Fig. \ref{['fig:2qvalley']}a and the control sample shown here respectively. Filling in these ratios in Extended Data Fig. \ref{['fig:S4Charge_dens_calc']}c an ∼ 8 increase in $\Delta_{\mathrm{KQ}}$ is found for the control sample with respect to the main sample. We attribute this difference to the different dielectric environments due to different hBN thicknesses, which influences the Q/Q' valleys stronger than the K/K' valleys.
• Figure 2: Filling of the Q valley and comparison with WS2. a, The electron-doped side of the 4K gate-dependent reflection contrast measurement of WSe2 is shown on the left side in which horizontal dashed lines mark the onset of the filling of the lower K/K' valleys (green), upper K/K' valleys (orange) and lower Q/Q' valleys (purple). Abutting this data is a calculation of the distribution of carriers in the different CB valleys as a function of the overall carrier density by minimizing the total energy of the electron gas (kinetic and exchange). The charge density scales are matched by calibrating the reflection contrast data using magneto-optic experiments (see Methods). The onsets of the resonances $O$ and $M$ are perfectly reproduced by using $\Delta_\mathrm{c} = \qty{12}{\milli\electronvolt}$ for the spin-orbit splitting in the K/K' valleys ren_measurement_2023kapuscinski_rydberg_2021, and $\Delta_{\mathrm{KQ}} = \qty{30}{\milli\electronvolt}$ for the energy difference between the lower CB valleys of K and Q. b, The same as in (a) but for a WS2 sample with comparable hBN dielectric thicknesses to those in the main WSe2 sample (5.3 and 3 for the top and bottom dielectric respectively). To reach the maximum density while avoiding breakdown, we apply $V\mathrm{_{G}}$ to the top gate and $0.8 V\mathrm{_{G}}$ to the bottom gate. Horizontal dashed lines mark the onset of the filling of the lower and upper K/K' valleys in green and orange, respectively. The right-hand side shows a calculation of the distribution of charges in the different CB valleys in the WS2 monolayer. The gate-voltage dependent charge density of the reflection contrast data was determined based on the calibration of the WSe2 sample in (a), and adapted using a simple capacitor model (see Methods for details). The filling of the K/K' valleys shows great similarity with WSe2, but contrastingly the filling of the Q/Q' valleys would only happen at the experimentally inaccessible electron density of ∼ 4e13□cm because $\Delta_{\mathrm{KQ}} = \qty{81}{\milli\electronvolt}$ in a WS2 monolayerkormanyos_k_2015.
• Figure 2: Fitting of the gate-dependent reflection contrast of the main WSe2 device. Low-temperature reflection contrast data, presented in Fig. \ref{['fig:1device']}, are fitted using a dispersive Lorentzian function (see Methods) for the voltage range of 2V to 6V, making fitting with a single dispersive Lorentzian feasible. In this range, the hexciton ($H$), the oxciton ($O$), and the many-body complex ($M$) are observed without spectral overlap of other resonances. a, Example reflection contrast spectra with the corresponding fits overlayed as dashed red lines. b, A heatmap of the recorded reflection contrast spectra with the center energies of the dispersive Lorentzian fits plotted in dark yellow on top. The fits accurately follow the recorded resonances and confirm the red-shift, blue-shift, red-shift character of the $H$, $O$, and $M$ excitons, respectively. The orange (purple) horizontal dashed line marks the transition from $H$ to $O$ ($O$ to $M$) exciton. c, The amplitude of the resonance resulting from the fits. The shaded region represents the fitting error as given by the least-squares method. Importantly, we observe no significant change in the amplitude for any of the three resonances. d, The full width at half maximum of the resonances resulting from the fits. We observe an almost constant width for the $H$ exciton, which agrees with the composite excitonic states model, because it is an optimal complex with a distinct photoexcited electron-hole pair. Starting from the transition from $H$ to $O$, we observe a continuous broadening that also extends to the $M$ exciton. In addition there are steps in the width of the resonance when transitioning from $H$ to $O$ and from $O$ to $M$. These observations are expected for $O$ and $M$, as they are complexes with an indistinct photoexcited electron-hole pair.
• Figure 3: Magneto-optics of WSe2. a,b, Gate-dependent reflection contrast measurements performed at 8T magnetic field and a mixing chamber temperature of ∼ 20mK, resolved for right-handed circularly polarized light $\sigma^{+}$ (a) and left-handed circularly polarized light $\sigma^{-}$ (b). Four gating regimes are marked with colored brackets and are labelled I-IV, which sequentially show $H$, $O$, $O$ and $M$ excitons for the $\sigma^{+}$ measurement and $H$, $H$, $O$ and $M$ excitons for the $\sigma^{-}$ measurement. c, A plot of the four energetically lowest, resolvable, local minima, per spectrum, extracted from (a) and (b) using a peak-detection method described in the Methods. Left (right) ticks in panels (a), (b), and (c) correspond to the voltage (density) scale on the far left (right) of the figure. d, Bandstructure models explaining the observed resonances in panels (a) and (b) for the different gating regimes I-IV. In I, the lower K- and K' valleys are filled and hexcitons form for both $\sigma^{+}$ and $\sigma^{-}$. The additionally observed resonances in (a) and (b) are due to Landau levels in the upper K- and K' valleys. In II, the upper K' valley is partially filled and hosts a Fermi sea. Therefore $O$ is observed for $\sigma^{+}$ yet $H$ is still observed for $\sigma^{-}$, now with a different slope than in I, indicated with a black arrow in (b). As the Landau levels in the upper K' valley fill, resonances disappear for $\sigma^{-}$ due to Pauli blocking. In III, all CB valleys at K and K' are populated, both $\sigma^{+}$ and $\sigma^{-}$ show $O$ excitons, and the systematic disappearance of resonances due to the filling of Landau levels. In IV, in addition to the K/K' valleys, the Q/Q' valleys are populated, allowing for the formation of $M$. No Landau level reminiscent oscillations are observed for $M$. We note that Landau quantization also occurs in the VB and the lower CB valleys at K and K' liu_landau-quantized_2020li_many-body_2022wang_observation_2020li_phonon-exciton_2020. Regardless, they do not affect the spectroscopic features we study in this work at a magnetic field of 8; therefore, we have chosen not to include them in our bandstructure drawings shown in (c) to improve clarity.