Electrically tuneable exciton-polaritons through free electron doping in monolayer WS$_2$ microcavities

We demonstrate control over light-matter coupling at room temperature combining a field effect transistor (FET) with a tuneable optical microcavity. Our microcavity FET comprises a monolayer tungsten disulfide WS$_2$ semiconductor which was transferred onto a hexagonal boron nitride flake that acts as a dielectric spacer in the microcavity, and as an electric insulator in the FET. In our tuneable system, strong coupling between excitons in the monolayer WS$_2$ and cavity photons can be tuned by controlling the cavity length, which we achieved with excellent stability, allowing us to choose from the second to the fifth order of the cavity modes. Once we achieve the strong coupling regime, we then modify the oscillator strength of excitons in the semiconductor material by modifying the free electron carrier density in the conduction band of the WS$_2$. This enables strong Coulomb repulsion between free electrons, which reduces the oscillator strength of excitons until the Rabi splitting completely disappears. We controlled the charge carrier density from 0 up to 3.2 $\times$ 10$^{12}$ cm$^{-2}$, and over this range the Rabi splitting varies from a maximum value that depends on the cavity mode chosen, down to zero, so the system spans the strong to weak coupling regimes.


Introduction
Exciton-polaritons are hybrid bosonic quasiparticles that have properties of both of their constituents: material properties of excitons, responsible for strong non-linear interactions; and optical properties of photons, that make them low mass compared with bare excitons. These quasiparticles have enabled the observation of cold atom physics including Bose-Einstein condensation on a chip, 1 superfluidity, 2-4 and topological polaritons. 5 At the same time, owing to their hybrid light-matter nature, exciton-polaritons may provide the strong non-linearities needed to deliver the strongly interacting photons 6 sought by quantum technologies. 7 This scenario could be significantly enriched by the ability of novel quantum materials to support room temperature operation of exciton-polaritons and the addition of unique functionalities, such as on-chip tuneable strength of the light-matter interaction.
Atomically thin semiconductors, part of the family of transition metal dichalcogenides (TMDs), exhibit a range of unique optical properties, which makes them one of the most promising material systems for enabling quantum photonics. 8 Prominent optical transitions in TMDs are widely dominated by excitons with a surprisingly large binding energy 9 making them viable for room temperature quantum electrodynamics. Additionally, due to their large exciton oscillator strength, TMDs support strong light-matter interactions even in their atomically thin form. [10][11][12][13][14][15] However, the tuneability of light-matter interactions in TMDbased devices, which would enable further control over non-linear interactions, is still in its infancy and further development is needed.
To investigate both tuneability and control over the light-matter coupling of excitonpolaritons we designed 16 and fabricated tuneable Fabry-Pérot microcavities with a TMDbased transistor embedded within the microcavity. We chose the TMD material tungsten disulfide, WS 2 , which exhibits the largest oscillator strength, and the smallest damping factor among TMDs of chemical composition MX 2 (M = W, Mo, X = S, Se). 17 These properties make WS 2 the optimum semiconductor TMD for strong coupling measurements of exciton-polaritons at visible frequencies, at room temperature, and in microcavities of moderate quality factor, Q ≈ 100. In our device, we increase free carrier density of a WS 2 monolayer, which produces a strong Coulomb screening of the free electrons that populate the conduction band. 18 This screening reduces the oscillator strength of neutral excitons, which therefore weakens the coupling strength of exciton-polaritons. 19 Electrical control of light-matter coupling has been demonstrated in a monolythic microcavity filled with carbon nanotubes, 20 where near infrared excitons were strongly coupled to a single microcavity mode. By applying a range of gate voltages the system was observed to cross reversibly from the weak to the strong coupling regime. Recently, this phenomenon was observed in the visible range using a monolythic TMD microcavity, 21 showing similar results. In our study we significantly extend the work by applying gate voltages sufficient to reach the saturation regime of the light-matter coupling strength, characterized by the vacuum Rabi splitting. For sufficiently negative values of the gate voltage, the Rabi splitting saturates to its maximum value, whilst for sufficiently positive values of the gate voltage (high electron carrier density), the Rabi splitting was completely eliminated. These observations allow us to explain the functional dependence of the Rabi splitting on the gate voltage, and to probe the nature of the Coulomb screening process.
In this report we studied a WS 2 microcavity in which the TMD excitons are strongly coupled to microcavity modes that can be chosen from the third-order to the fifth-order by controlling the cavity length in a tuneable way. We observed a dependence of the Rabi splitting on the electrically-controlled density of free electrons, a density that ranges from 0 up to 3×10 12 cm −2 . Over this range the Rabi splitting varies from a maximum value (which depends on the order of the chosen cavity mode) down to zero so that the hybrid system spans the strong to weak coupling regimes.

WS 2 -based field effect transistor
We first studied the WS 2 -based field effect transistor without a microcavity. The transistor consisted of a Van der Waals heterostructure composed by WS 2 placed on top of a hexagonal boron nitride (hBN) flake (more details in the experimental section). The WS 2 /hBN heterostructure was placed on a silver (Ag) film, which acts as a microcavity mirror (bottom mirror) as well as the gate contact of the transistor while WS 2 was kept as a ground contact  with the neutral exciton transition of the 1L-WS 2 labelled X o . In this figure we also observe the appearance of a weak transmittance dip at a wavelength of 632 nm, which we associate with a negatively charged exciton, a negative trion, which is labelled as X − . We observe a weakening in the transmission minimum associated with the neutral excitons as we sweep the gate voltage from -5 V to +5 V. We also observed that the feature associated with the negative trions strengthens. To clarify this observation we extract the change in the transmission associated with X o and X − as a function of V G . These data are shown in figure 1(d) where values below one indicate how much the transmission is reduced in strength, whilst values above one indicate how much the transmission increases in strength. We observed no significant electrical current through the hBN barrier in this range of voltages (see supporting information figure S5), which suggests that the free electrons are accumulated on the WS 2 which then leads to a reduction in the excitation rate of neutral excitons due to strong Coulomb repulsion between free carriers filling the conduction band. 23

Tuneable microcavity
A tuneable microcavity was completed by adding a second silver mirror (top mirror) close to the top of the WS 2 transistor, as shown in figure 2(b). For the microcavity we collected transmission spectra as a function of the thickness of the air gap between the hBN spacer and the top silver mirror in a region where there is no WS 2 . These transmission spectra are then compared to the modelled spectra from which we extract the experimental value for the thickness of the air gap (for details on modelling see the supporting information), as shown in the left panels of figure 2(a). Here we identify the second-, third-, and fourth-order cavity modes. We observe excellent stability of our tuneable system, and very good quality of the microcavity modes compared to modelling. We repeated these measurements in a region of the microcavity where the 1L-WS 2 was present, keeping the gate voltage fixed at 0 V. These measurements were compared to the model and are shown in the right panels of 2(a). Here we observe a splitting of the cavity modes at a wavelength of 622 nm, associated with the neutral exciton transition of WS 2 . The mode splitting of the three cavity modes observed is greater than that of both the empty cavity mode and the exciton line widths (36 and 35 meV respectively), demonstrating that the system is in the strong coupling regime. We fit the calculated transmittance of a strongly coupled 1L-WS 2 -microcavity with an air gap of 787 nm, which corresponds to the third-order cavity mode, to the experimental observation as  Experimental and calculated transmittance of a strongly coupled WS 2 microcavity. Strong coupling was achieved for a cavity air gap of 787 ± 3 nm, which corresponds to the third-order cavity mode. significantly larger than values reported for WS 2 on SiO 2 or Al 2 O 3 substrates. 12,21 The origin of this discrepancy could be the difference in the dielectric environment provided by the hBN flake in our devices, and also to the fact that hBN provides an atomically smooth surface (roughness lower than 0.5 nm as measured by atomic force microscopy) which may reduce surface scattering and trapping effects in the WS 2 /hBN interface.

Electrical control of the light-matter coupling
We next discuss the evolution of the cavity mode splitting in the strong coupling regime for different gate voltages ranging from -8 V up to +8 V. To study the exciton and photon nature of the polariton bands we performed a two coupled oscillator model analysis of the experimental data, from which we found the Hopfield coefficients as described in the supporting information. We focused this analysis on the splitting of the third-order cavity mode, and cavity photons can also be observed as a reduction of the Rabi splitting as a function of the gate voltage, as shown in figure 4. In this figure we show the transmittance peak splitting for (a) the third, (b) the fourth, and (c) the fifth-order cavity modes as a function of the gate voltage. These modes correspond to fixed air gaps, being 787 ± 3 nm, 1097 ± 3 nm, and 1408 ± 3 nm respectively. For the three modes we observe a dependence of the Rabi splitting on the gate voltage; the Rabi splitting reaches a saturation value when sufficiently negative voltages are applied to the transistor. In addition the maximum splitting is lower for higher order cavity modes, as expected due to the weaker fields associated with the higher-order modes (see also supporting information figure S6). From these measurements we obtain the Rabi splitting as a function of the gate voltage for the three cavity modes as shown in figure   5(a). In this figure we also indicate the exciton line width, which is equal or larger to any of the three cavity mode widths, indicating that the system is in the strong coupling regime when the Rabi splitting is greater than the exciton line width.

Rabi splitting and density of free electrons
We now focus our attention on obtaining the dependence of the oscillator strength of WS 2 as we change the gate voltage, a dependence that we can relate to an increase of the density of free electrons in the WS 2 . To do this we analyse the splitting of the third-order cavity mode.
We first calculate the transmittance spectrum as shown in figure 2(c) with an oscillator strength of f 0 = 2.6, which results in a Rabi splitting of 60 meV that corresponds to the splitting of the third-order mode for a gate voltage of 0 V. Prior to this measurements, we applied a sufficiently positive gate voltage to remove any excess of free electrons in WS 2 , so   Figure 4: Transmittance spectra of a WS 2 microcavity for different values of the gate voltage, and for a fixed air gap: (a) 787 ± 3 nm, (b) 1097 ± 3 nm, and (c) 1408 ± 3 nm, which correspond to the third, fourth and fifth cavity modes respectively.
where the saturation number N s is defined as the number density of carriers at which the oscillator strength is reduced to a half of the initial value f 0 . The value of N s can be estimated for any 2-dimensional system that allows for the excitation of excitons and free carriers either in the conduction or the valence band. For our analysis we have used N s = 1/8.3πa EB extracted from, 24 where a EB is the effective exciton-Bohr radius of WS 2 , which has been taken from reference 25 to be a value of 2 nm corresponding to the X 0 exciton transition.
These results are shown in figure 5(b), where we also include the saturation function as described in equation (1). The density of free electrons N found in this way agrees well with what is expected in a TMD system that behaves below the level of Mott transition, 18,26 which has been estimated to occur at a density above 10 13 cm −2 . In our system, we found the saturation density to be 9.6×10 11 cm −2 , and in the range of voltages applied, the density of free electrons in the WS 2 ranges from 0 up to 3.2 × 10 12 cm −2 .

Summary and conclusions
In summary, we studied continuous control over the light-matter coupling of exciton-polaritons in a WS 2 microcavity at room temperature. We have achieved this by controlling the charge carrier density in the semiconductor material. Due to the Coulomb interaction the free carriers repel each other and screen the oscillator strength of excitons in WS 2 . A specially designed WS 2 based field effect transistor was built on one of the mirrors of the microcavity, 16 then a second mirror was placed on top of the transistor leaving an air gap which we were able to control with a precision of ±3 nm in our tuneable system. This tuneability allowed us to study the polariton bands as a function of the thickness of the air gap, and we accessed different cavity modes ranging from the second-order to the fifth-order. For each of the cavity modes, at the resonant condition with WS 2 excitons forming exciton-polaritons, we were able to record transmittance spectra over a range of gate voltages, from -8 V up to +8 V, observing a dependence of the Rabi splitting as a function of the gate voltage.
Over this range of voltages we calculated the variation of the oscillator strength in the TMD film, extracted from modelling the transmittance spectra to match the measured values of the Rabi splitting. We attributed the change in the oscillator strength of WS 2 to Coulomb screening by the free electrons populating the conduction band of the TMD, we then were able to extract the density of free carriers in the conduction band, which ranges from 0 cm −2 , at a gate voltage of 0 V, up to 3.2 × 10 12 cm −2 , for a gate voltage of +8 V. Over this range of density of free carriers, the Rabi splitting ranges from a maximum value, which depends on the cavity mode chosen, down to zero, so the system spans the strong to weak coupling regimes. This change in the Rabi splitting could also be extended to p-type semiconductors with a strong excitonic resonance by injecting holes in the valence band of the material.

Tuneable microcavity setup
The tuneable microcavity was completed by a top mirror placed on a piezo-electric stage

Optical and electrical measurements
Transmission measurements where performed in a confocal setup. The light source (Thorlabs R SLS201L/M) was fibre coupled, collimated and focused by x10 objective lenses through the bottom mirror. To collect the transmitted light through the top mirror a x50 objective lens was used (Olympus R SLMPLN50X), which was then focused on a fibre coupled spectrome-ter (OceanOptics R Flame). The gate voltage was applied by a Keithley R 2450 SMU; gate voltage, and tunnelling current were recorded for the different values of the cavity length.
The top mirror was placed on a piezo-stage (Thorlabs R MAX312D, MDT630B) which allows for a control of the cavity length with a precision of ± 3 nm. Transmittance, piezo-stage voltage, and gate voltage measurements were synchronised on LabView R .