Greatly enhanced light emission of MoS2 using photonic crystal heterojunction

We present theoretical study on developing a one-dimensional (1D) photonic crystal heterojunction (h-PhC) that consists of a monolayer molybdenum disulfide (MoS2) structure. By employing the transfer matrix method, we obtained the analytical solution of the light absorption and emission of two-dimensional materials in 1D h-PhC. Simultaneously enhancing the light absorption and emission of the medium in multiple frequency ranges is easy as h-PhC has more modes of photon localization than the common photonic crystal. Our numerical results demonstrate that the proposed 1D h-PhC can simultaneously enhance the light absorption and emission of MoS2 and enhance the photoluminescence spectrum of MoS2 by 2–3 orders of magnitude.

Two-dimensional (2D) transition metal dichalcogenides (TMDCs), such as MoS 2 and WSe 2 , are direct-gap semiconductor 2D materials with excellent optical properties and are thus considered one of the best materials for future optoelectronic devices [1][2][3][4][5][6][7] . The light absorption and emission of 2D TMDCs per unit mass are much higher than that of traditional semiconductor materials. 2D TMDCs typically have a thickness of less than 1 nm, and their light absorption and emission are weak, thus limiting their applications in optoelectronic devices. However, due to the benefits of the ultra-thin channel, 2D TMDCs can be combined with optical microstructures, such as photonic crystals, microcavities, and surface plasmas, which effectively enhance the light absorption  and emission 8,9,[32][33][34][35][36][37][38][39][40][41] due to the optical localization in these structures. Lien et al. 8 and Serkan et al. 9 used surface plasmas or optical multilayers to enhance the light absorption and emission of MoS 2 or WSe 2 , thus enhanced the photoluminescence (PL) of MoS 2 or WSe 2 by 10-30 times.
To further enhance the light emission and absorption of 2D TMDCs, we investigated the effect of photonic crystal heterojunction (h-PhC) on the light absorption and emission of MoS 2 . Similar to the semiconductor heterojunction, h-PhC comprises photonic crystals (PhC) with different lattice constants or shapes 42 . Earlier studies have found that h-PhC that comprise different PhCs can obtain strong light localization in several frequency ranges 42,43 . On the basis of these findings, one can propose a h-PhC that is formed by different PhCs at intervals to form a multimode high-speed optical waveguide.
Thus, if 2D TMDCs are combined with h-PhC, the strong light localization of h-PhC in multiple frequency ranges can simultaneously enhance the light emission and absorption of 2D TMDCs. We therefore conducted a detailed study on a h-PhC which consists of 1D PhCs with two kinds of crystal lattices that form an h-PhC microcavity structure. To thoroughly understand the light absorption and emission in h-PhC, we first identified the analytical solution of the light absorption and emission of MoS 2 in h-PhC. The findings indicate that h-PhC can enhance the light absorption and emission of MoS 2 and enhance the PL spectrum of MoS 2 by 2-3 orders of magnitude, which has a promising prospect and important application value in fluorescent probe, 2D LED, etc. The analytical solution can be used not only for the light absorption and emission in h-PhC but also for the calculation of other 1D PhC-2D materials composite structures.

Model and Theory
The structure of h-PhC is shown in Fig. 1 44 . λ λ ′ = × 10 6 , λ is wavelength of the input light beams, and the thicknesses of the A 1 and A 2 layers are λ 10 / (4 × 1.53) and λ 20 /(4 × 1.53), respectively. λ 10  To model the absorption of MoS 2 in this structure, the transfer matrix method is used first 46,22 . In the l-th layer, the electric field of the TE mode light beam with incident angle θ t is given by where k l = k lr + ik li is the wave vector of the incident light, e z is the unit vectors in the z direction, and x i is the position of the l-th layer in the x direction. And the magnetic field of the TM mode in the l-th layer is given by The electric (magnetic) fields of TE (TM) mode in the (l + 1)-th and l-th layer are related by the matrix utilizing the boundary condition is the complex dielectric permittivity, and d 1 is the thickness of the l-th layer. Thus, the fields in the (l + 1)-th layer are related to the incident fields by the transfer matrix To thoroughly describe the light absorption and emission of MoS 2 in h-PhC, improve the computational speed to optimize the structure, and help scholars who are not familiar with the transfer matrix method for computing, we obtained the analytical solutions of the light absorption and emission of MoS 2 in h-PhC using the transfer matrix method. Since the transfer matrix of the electric fields of TE mode and the transfer matrix of magnetic fields of TM mode have the same form, we only show the analytical solution of the TE mode. First, for a N-period PhC in air, the transfer matrix can be written as 47 .
is the propagation angle in the C 1 and C 2 layer. Similar, we can get the transfer matrix of the lower part PhC, The transfer matrix of the C 1 layer is 11 .
is the microcavity length, k cx is the wave vector of the light in the C 1 or C 2 layer. Taking the approximation that e ik d 1  . Thus, the total transfer matrix of the C 1 , C 2 , and MoS 2 layer is The total transfer matrix of the h-PhC is We can get the matrix element where r t and t t are the reflection amplitude and transmission amplitude of the exit DBR mirror, respectively. r b is the reflection amplitude of the back DBR mirror, z ol is the distance between the monolayer MoS 2 and back DBR mirror. When the pumping and outgoing lights are on the same side of the h-PhC, r r , is the optical length of microcavity, n c is the permittivity of the C 1 and C 2 layer. By using the Fourier transform, the emitted radiation from the top DBR mirror in the frequency domain can be written as

DBRt t t b ol t b t o c t b b t ol oc
Neglecting the changes of the spontaneous time, integral of Eq. (16), the emission intensity can be calculated by 48,49 .

Results
We first calculated the absorption and the relative radiation intensity of MoS 2 when the pumping and outgoing lights are on the same side of the h-PhC. The optimized parameters are as follows: λ 10 = 730 nm, λ 20 , N 1 = 6, and N 2 = 7. The incident angle is θ i = 48°. The pumping light is in TE mode. The outgoing light is vertically emitted. The calculation results are shown in Fig. 2. Two strong absorption peaks . L c = n c d c is the microcavity optical path, m 0 is a positive integer, and θ θ ′ = arcsin n / i c is the propagation angle of light in the defective layer. Thus, when the incident angle increases, the resonance wavelength moves in the short-wave direction, the reflectivities of the PhCs on both sides increases, the travel path of light in MoS 2 increases, and the maximum absorption can reach 0.8 or more. PL is enhanced by approximately 3 orders of magnitude.
We also calculated the positive incidence of the pumping light and the absorption and relative radiation intensity of MoS 2 when the pumping and outgoing lights are on opposite sides of the h-PhC. According to the experiment 9 , in our calculation, the wavelength of the pumping light is 488 nm, and the wavelength of the outgoing light approaches 660 nm. We calculated the corresponding parameters by optimizing the h-PhC structure under different pumping light incidences as follows: when the pumping light is normally incident and the pumping and outgoing lights are on the same side of the h-PhC, λ 10 = 580 nm, λ 20 = 610 nm, , N 1 = 7, and N 2 = 7. When the pumping light is normally incident and the pumping and outgoing lights are on opposite sides of the h-PhC, λ 10 = 630 nm, λ 20 570 nm, d 11 nm , N 1 = 7, and N 2 = 7. When the pumping light is obliquely incident and the pumping and outgoing lights are on opposite sides of the h-PhC, λ 10 = 660 nm, λ 20 = 580 nm, , N 1 = 7, N 2 = 7, and θ i = 30°. The detailed calculation results are shown in Fig. 4. Regardless of whether the pumping and outgoing lights are on the same or different sides of the h-PhC, the absorption of MoS 2 is higher when the pumping light is obliquely incident. A strong local touch in the vicinity of two wavelengths can be easily obtained as change in incidence angle can adjust the resonant wavelength. When the pumping light is normally incident and the pumping and outgoing lights are on opposite sides of the h-PhC, the absorption of MoS 2 is high because if MoS 2 in the microcavity structure obtains strong absorption and emission, the reflectivity of the rear reflector should be higher but the reflectivity of the front reflector should not be excessively high 19 . The bandgap width of the PhC is not big enough due to the large difference between the wavelengths of the pumping and outgoing lights. The pumping and outgoing lights on different sides realize this goal easily.
For comparison, we calculated the light absorption and emission of MoS 2 in a homojunction. The optimized structural parameters are as follows: when the pumping light is obliquely incident and the pumping and outgoing lights are on the same side, λ λ = =660 nm 10 20 , , N 1 = 5, N 2 = 7, and θ i = 42°. When the pumping light is normally incident and the pumping and outgoing lights are on the same side, λ λ = =680 nm 10 20 , d 4 nm , N 1 = 5, and N 2 = 7. When the pumping light is obliquely incident and the pumping and outgoing lights are on opposite sides, λ λ = =670 nm 10 20 , d 0 nm , N 1 = 6, N 2 = 5, and θ i = 54°. When the pumping light is normally incident and the pumping and outgoing lights are on opposite sides: λ λ = =650 nm 10 20 ,  is obliquely incident and the pumping and outgoing lights are on opposite sides, the absorption is the largest (approximately 0.33) but the outgoing light enhancement is low. If light emission is enhanced using longer PhC cycles than those used in the current study, the light absorption of MoS 2 will decrease. However, this case does not happen in h-PhC.
Finally, we discuss the effect of light localization and the feasibility of the experiment. The effect of light localization: We used the Q value to judge the strength of light localization. The larger the Q value, the higher the light localization and light absorption and emission intensity of MoS 2 . However, the higher the Q value, the smaller the full width at half maximum of the spectral line and the narrower the absorption and PL spectra. Excessively narrow absorption and emission spectra are not conducive to practical application. Moreover, when the Q value is high, the microcavity affects the transition time of the exciton. Thus, in Optimizing the parameters, we choose N 1 ≤ 7 and N 2 ≤ 7.  The feasibility of the experiment: PhC and 2D materials composite structures (particularly 2D materials-PhC microcavity) were created [14][15][16] . Compared with the existing structure, this structure only changes the lattice constant of the upper and lower reflectors of the PhC microcavity. Therefore, the experiment is completely achievable.

Conclusion
We studied the effect of 1D h-PhC on the light absorption and emission of monolayer MoS 2 and obtained the solutions of both light absorption and emission in 1D PhC-2D materials composite structures. h-PhC has more models of photon localization than common PhC, which enhances the light emission and absorption of MoS 2 simultaneously, and increases the PL spectrum of MoS 2 by 2-3 orders of magnitude. When the pumping light is obliquely incident and the pumping and outgoing lights are on opposite sides of the h-PhC, it is easier to enhance the light absorption and emission of MoS 2 at the same time. The analytical solution can be used not only for light absorption and emission in h-PhC but also applicable for other 1D PhC-2D materials composite structures. This study has a promising prospect and important application value in 2D material-based optoelectronic devices.

Methods
The modified transfer matrix method is used to model the absorption of monolayer MoS 2 in the photonic crystal micro-cavity.