Two-dimensional metal-chalcogenide films in tunable optical microcavities

Quasi-two-dimensional (2D) films of layered metal-chalcogenides have attractive optoelectronic properties. However, photonic applications of thin films may be limited owing to weak light absorption and surface effects leading to reduced quantum yield. Integration of 2D films in optical microcavities will permit these limitations to be overcome owing to modified light coupling with the films. Here we present tunable microcavities with embedded monolayer MoS2 or few monolayer GaSe films. We observe significant modification of spectral and temporal properties of photoluminescence (PL): PL is emitted in spectrally narrow and wavelength-tunable cavity modes with quality factors up to 7400; PL life-time shortening by a factor of 10 is achieved, a consequence of Purcell enhancement of the spontaneous emission rate. This work has potential to pave the way to microcavity-enhanced light-emitting devices based on layered 2D materials and their heterostructures, and also opens possibilities for cavity QED in a new material system of van der Waals crystals.


Microcavity fabrication and optical properties Thin film fabrication
Monolayer MoS 2 and thin sheets of GaSe have been obtained by mechanical cleavage of bulk crystals. GaSe films were deposited straight from the wafer dicing tape on the flat DBR substrate, whereas the MoS 2 films were first deposited on a polymer layer and then transferred onto the flat DBR using standard transfer techniques. 1 Fig.1(a) shows a microscope image of a thin film of MoS 2 . Areas with a single-and multiple-monolayer thicknesses can be distinguished on the graph from different shades of green color. The thickness of the films was further verified using atomic force microscopy. As seen on the graph the lateral dimensions of the single-monolayer part of the film exceed 50 μm by 50 μm. For GaSe films, flakes with sizes varying from 10 μm to 50 μm could be achieved. Our study is focused on relatively thick GaSe films with thicknesses ranging from 30 nm to 100 nm. We find that the PL intensity in GaSe films increases dramatically with the increase of the film thickness.

Mirror design and fabrication
The open-access microcavity fabrication consists of a two-step process 2,3 . First, the templates of concave mirrors are fabricated using a focused ion beam (FIB) machine (FIB200 from FEI). In this process, gallium ions are fired onto a precisely selected position of a silica substrate for a certain period of time referred to as the 'dwell time'. By adapting the dwell time as a function of the position of the ion beam, we create concave templates with various radii of curvature ranging from 1.7 μm to 25 μm onto a single chip. The template dimensions are measured by AFM. The smallest optically active cavities obtained so far had a radius of curvature of 5.6 μm. The efficiency of the FIB approach relies on the smallest achievable ion beam diameter, which in our case was down to 5 nm. The rms roughness of the template surface was found to be below 1 nm. During the second step, both the substrate with the concave mirrors and another flat silica substrate are coated with dielectric distributed Bragg Concave features with different radii of curvature are milled into a quartz substrate using focused ion beam, and are then covered with ten SiO 2 /TiO 2 quarterwavelength layer pairs, forming a distributed Bragg reflector. (c) PL spectrum measured in a microcavity containing a thin GaSe film (43 nm). A high Q-factor of around 7400 is observed for a longitudinal mode. The inset shows the same PL spectrum in a wide range of wavelength, where other modes with non-zero transverse mode numbers (m, n =0) are observed.
reflectors (DBRs) comprising ten layers of SiO 2 /TiO 2 (with refractive indexes 1.4 and 2.1, respectively) with layer thicknesses tuned to achieve maximum reflectivity at 650 nm. A microscope image of a typical substrate with concave mirrors is shown in Fig.1(b). This mode has a wavelength λ q,m,n = 603.7nm. Here q=15 and m, n=0 (see Eq.1 in the main text for the dependence of the resonant wavelength on q, m and n). The inset also shows the modes with q=14 and m, n =0 observed in a broader spectral range. The measurements are carried out at a temperature of 4.2K using laser excitation at 532 nm in a cavity with L cav =3.25μm and the radius of curvature of the top mirror R c =25μm.

FDTD calculations of the cavity modes Collection optics considerations
The ratio of observed intensities for the structure with and without the top mirror depends to some extent on the collection optics since the angular distribution of emission for a given wavelength of light is different for the two cases. The objective lens which collects the light is 7.5mm above the sample. This is sufficiently far from the sample compared to the wavelength of light and to the experimentally measured spot sizes that it can be considered to be in the far field. The collection efficiency is then entirely determined by the lens numerical aperture.
The clear aperture of the objective is 4.5mm so only light emitted within ± 16.7 degrees is collected. To calculate the fraction of light collected it is necessary to know the power radiated per unit solid angle as a function of observation angle in the far field, the so-called radiation patterns. 4 Then the power emitted within the collection range of the lens may be compared with the total emitted power. Finally, comparison of the fraction of collected power may be made between the cases with and without top mirror to obtain the collection enhancement due to directionality of the cavity modes.  radiating in free space may be expanded in a basis of plane-waves. In the presence of the mirror the total upwards-radiating field (towards the collection lens) is the coherent sum of the radiation field emitted upwards by the dipole and the reflection from the mirror of the field emitted downwards by the dipole. The free-space fields are separated into components with TE and TM polarisation with respect to the planar multilayer and the amplitude reflection coefficients are calculated using a transfer matrix technique. Figure 2 shows the sum of radiation patterns due to x and y polarised dipoles for the case of dipoles in free space and directly on top of the DBR. For zero angle the DBR reflective phase is close to zero so the reflection reinforces the upward propagating wave and the upwards emission is enhanced with respect to free space. At larger angles the reflective phase increases so that the reflection begins to interfere destructively with the directly radiated field. For angles greater than 90 degrees the radiation is into the substrate.

Radiation pattern for electric dipoles on a flat DBR
Very little power is radiated into modes close to 180 degrees because the DBR reflects them.
For angles greater than 20 degrees from the negative z-axis, however, the DBR becomes ineffective and light is lost into the substrate. To obtain total power radiated in the range of polar angles 0 ≤ θ ≤ θ max the radiation patterns must be integrated with azimuthal and polar collection angles according to P is the differential element of solid angle. Overall, 49% of the total radiation is emitted in the upwards direction and 9% is emitted within ±16.7 degrees. This compares to only 3% emitted within ±16.7 degrees for dipoles in free space.

Modes in a cavity with a concave DBR
When the top mirror is present a cavity is formed which confines electromagnetic modes at certain frequencies in all three dimensions. At these resonant frequencies the fields and hence the angular spread of the emission are determined by the cavity geometry. Simulations of the electromagnetic fields associated with the resonant cavity modes were performed with the finite-difference time-domain (FDTD) method, using a freely available software package. 6 The fields were first determined in the near-field zone close to the cavity. Simulations were performed in a cylindrical geometry on a two-dimensional grid of radial r and axial position z. This allowed much faster simulation times than a full three-dimensional calculation, which was necessary due to the rather small 10 nm grid resolution used to accurately represent the cavity layers and top mirror curvature.
The separation between the upper and lower mirrors was first set to the experimentally estimated value and then refined in the following way. At the start of the simulation a broad frequency spectrum of electromagnetic radiation was excited by an electric dipole current source with short Gaussian temporal profile positioned on top of the lower DBR at r = 400nm. The electromagnetic energy flux passing through a box surrounding the structure (denoted by dotted lines in Fig. 3(a)) was collected for a sufficiently long time to allow all the energy to leave the simulation region. The flux was Fourier transformed to obtain a spectrum of the radiation emitted by the structure. Such a spectrum is shown in Figure 3(b). The sharp peaks identify resonant cavity modes and correspond to modes with a range of longitudinal (z-direction) and transverse (radial) quantization numbers. To find the fundamental (zero transverse quantization number) mode the simulation was repeated a number of times using a narrow-band excitation centered on each cavity mode in the spectrum. The cavity modes have a high quality factor Q and so decay much more slowly than other transient fields caused by the excitation. After several decay-times of the chosen cavity mode the remaining electric and magnetic fields may be considered to have an approximately single-frequency harmonic time dependence and to represent the spatial dependence of the chosen cavity mode. Simulations were run for between one and three decay times τ = Q/ω in order to reach this condition. After this the electric and magnetic fields at all points in the simulation volume were output. Figures 3(c-d) show the spatial profiles of the time-averaged electric intensity E · E * for two of the modes in the spectrum corresponding to the longitudinal mode (c) and a mode with non-zero radial quantization number (d). The mirror separation was then adjusted slightly to bring the fundamental mode close to the experimental wavelength before the fundamental mode field profile was recalculated.

Effect of the concave DBR on the radiation pattern
Angular profiles for radiation from several cavities from Table 1 Table 1 structure C5. It has been shown experimentally that the quality-factor in hemispherical cavities increases with increasing mirror separation up to a critical value where it begins to decrease. 3 This behaviour was attributed to a loss of mode stability which, in a purely geometrical picture, is where some rays at higher angles become able to exit the resonator on each round trip. Geometrical arguments predict that this occurs when the mirror separation is greater than the radius of curvature. 3 However, these arguments ignore the finite diameter of the curved section of the mirror. In the real system there is a sharply discontinuous interface between curved and planar regions which may cause scattering with a strength which depends on the local field amplitude. Qualitatively, the angular spread of the mode is dictated by the radius of curvature. The spatial extent of the beam at the top mirror will be proportional to this angular spread and the mirror separation. When the separation is large the spatial beam size will overlap the discontinuous region leading to scattering. It is likely that this scattering is the cause of the observed energy loss from the confined mode into sideways propagating modes and also of the complicated angular emission profile above the structure. The two effects together tend to reduce the fraction of power collected by the objective lens.
Finally, the fractions of total emitted power collected by the objective may be compared between the cavities and the case where there is no mirror (see main text and also Methods).
For the cavities close to the experimental parameters C1 and C3 we expect to collect 5.3 and 2.2 times more of the total emission than in the case with no top mirror. This agrees with the qualitative notion that the larger cavity should give a more directional beam and so more of the emission should be collected.

Calculation of the Purcell enhancement
Further to the angular distribution, the FDTD method employed above can be used to calculate the effective mode volumes and Q-factors for the optical modes in the microcavity.
We now discuss how these may be used to make an estimate of the Purcell factor. The enhancement of the spontaneous emission rate of an emitter at position r 0 due to a cavity may be calculated using the standard formula 7 Here λ is the vacuum wavelength, r 0 is the position of the field maximum in the cavity and n is refractive index. The effective mode volume is given by Here r = (r, φ, z) is position in space and (r) |E(r)| 2 = (r) (E(r) · E * (r)) is the electric energy density. Since the full electromagnetic fields of the cavity modes are calculated as a function of position by the FDTD simulations the effective mode volumes are obtained simply by numerically performing the integral in equation 2. We obtain volumes of 6.46, 3.10 and 1.61 µm 3 for the cavities C1, C2 and C3 respectively.
Our FDTD calculations also give central wavelengths and Q-factors for the modes. These are obtained by examining the field in the cavity using a harmonic inversion technique. 6 For the cavities C1, C2 and C3 we obtain Q 1 =11000, Q 2 =7700 and Q 3 =4700 respectively which lead to Purcell factors of F P 1 =40.7, F P 2 =59.4 and F P 3 =69.8.
These Purcell factors provide an upper limit to the enhancement which may be achieved in a cavity. The actual ratio of spontaneous emission rates observed in an experiment will depend on the spectral and spatial overlap of the emitter with the optical mode which effect the local density of states and magnitude of the vacuum field respectively. 8 Accurate modeling of the experimentally observed cw PL enhancement and lifetime shortening in time-resolved measurements also requires the knowledge of the non-radiative decay rates in the 2D film.