Engineering radiative coupling of excitons in 2D semiconductors: supplementary material

The resonance energy and the transition rate of atoms, molecules, and solids were understood as their intrinsic properties in classical electromagnetism. It was later realized that these quantities are linked to the radiative coupling between the transition dipole and photon modes. Such effects can be greatly amplified in macroscopic many-body systems from virtual photon exchange between dipoles, but are often masked by inhomogeneity and pure dephasing, especially in solids. Here, we observe in both absorption and emission spectroscopy the renormalization of the exciton resonance and the radiative decay rate in transition metal dichalcogenides monolayers due to their radiative interactions. Tuning the photon mode density near the monolayer, we demonstrate control of cooperative Lamb shift, radiative decay, and valley polarization of the excitons as well as control of the charged exciton emission. Our work establishes a technologically accessible and robust experimental system for engineering cooperative matter–light interactions.


Sample preparation
Typical dry transfer method based on polymer stamping is applied to fabricate the encapsulated MoSe2 and WSe2 monolayer encapsulated with hexagonal boron nitrides(hBN) flakes. MoSe2, WSe2 and hBN were exfoliated from the highquality crystals on 285nm SiO2/silicon substrates. A polyethylene terephthalate stamp is used to pick up the layers in sequence at 50⁰C under microscope operation. The entire stamp along with the encapsulated TMDC is transferred onto a double-side polished sapphire wafer by melting the PET at 110⁰C. The PET stamp is later dissolved in a methylene dichloride solution.

Reflectance and photoluminescence spectroscopy
The encapsulated MoSe2 or WSe2 on a sapphire substrate is clamped on a copper plate with a through hole. A 2-by-2 millimeter distributed-bragg-reflector(DBR) with stopband from 640 to 800nm is mounted on a piezo-electric stage and positioned inside the through hole. Before the optical measurements, the DBR is pushed against the sapphire substrate leaving a small distance around 2-3 microns between the encapsulated TMDC and the mirror. During measurements, we step the piezo-electric stage to record the mirror-distance dependence and measure photoluminescence and reflectance spectra at the same sample position. All optical measurements are done in a cryostat with sample temperature at 50 Kelvin.
Differential reflectance and photoluminescence of the encapsulate TMD are measured. A femtosecond Ti:sapphire laser around 750nm was focused onto the sample via a 50× long working distance(0.42 numerical aperture) and the reflected light is selected by an aperture equivalent to ~2μm spot size in diameter. The light was collected into a grating spectrometer with a spectral resolution of 0.2 nm and recorded by a charge-coupled device. The reflectance contrast spectrum was determined by Rs/Rm(ω) , where Rs(ω) and Rm(ω) are the reflected intensity from the mirror with and without the encapsulated monolayer. All reflection measurements are done with laser intensity of 50nW/μm2 to prevent light-induced nonlinearity. This laser power corresponds ~8×10 9 excitons/cm 2 when the exciton absorption is the strongest(anti-node). For photoluminescence, either 532nm or 633nm excitation laser is used to excite the sample. The emission is collected by the same optical path with a color filter to extinct the excitation laser.

3.Mirror distance calibration
To calibrate the monolayer-to-mirror distance for each piezo step, we utilize the interference pattern between the DBR mirror and the sapphire substrate when they are sufficiently far apart. FigureS1A shows the reflection spectra of the sapphire-DBR system normalized to the spectrum measured with sapphire only. The fringes observed within the DBR stopband(640 to 800nm) comes from the interference pattern between the sapphire reflection and the DBR reflection. The wavelengths of the neighboring reflection minima are related to the mirror distanced by the following equation: This allows us to estimate dat different piezo positionsfordsufficiently large to allow at least two fringes in the stopband. The calibration results are plotted in FigureS1B, showing a linear relation between d and the piezo step. This allows us to extrapolate the relation to smaller d for measurements shown in the main text.

Renormalization of exciton resonance energy and linewidth
The general Hamiltonian describing the collective excitations of Wannier excitons in quantum wells is considered in [1,2]. In this reference, the renormalized resonance energy and linewidth due to the coupling between free-space photons and the collective excitations are shown as Ω and γ in the equation (27) in [1]. Here, we provide an effective Hamiltonian including an excitonic transition and light-matter interactions when the thin film is placed in front of a perfect mirror: where b , b are creation and annihilation operators for an exciton, ℏω the excitonic resonance energy, d the transition dipole for the exciton, E(r) and B(r) the electric and magnetic field operator at position r, and L the location of the exciton dipole moment. The electric field in front of a mirror forms a standing wave pattern, which is also imposed on the vacuum fluctuations. Therefore the electric field operator E(z) = i dkα sin(kz) a , k > 0. Note that we assume α, the normalization constant, is independent of k since only the mode around exciton resonance is considered. Inserting the electric field operator into the Hamiltonian, we obtain equation (1) with g= α|d|/h. To solve the Hamiltonian, we make the ansatz of wavefunction: |ψ(t)〉 = X(t)|exc, 0〉 + dkP (t) |g, k〉 where |exc〉, |g〉 denote the one exciton and ground state of the monolayer,|0〉,|k〉 denote the vacuum state and one photon in mode k state of the radiation field ,and X(t), P (t) are the corresponding coefficients. The time-dependent Schrodinger equation, i | ( )〉 = H|ψ(t)〉, gives the dynamics of X(t), P (t): where X (t), P (t) are the excitonic and photonic wavefunction in the rotating frame. Solving the differential equation: = − γ 2 X (t) + γ 2 e X (t − τ)Θ(t − τ) whereτ = 2L/c, γ = ℏ and Θ(t) is the heavenside step function. For steady state, we can ignore Θ(t) and focus on the asymptotic solution for t → ∞. The differential equation becomes a simple exponential decay X (t) = − (1 − e )X (t) in this limit, leading to a solution for excitonic wavefucntion: with , defined in equation3(a) and (b).