Control of the Exciton Radiative Lifetime in van der Waals Heterostructures

TL;DR

This paper demonstrates that the exciton radiative lifetime in van der Waals heterostructures can be precisely controlled by adjusting the hBN encapsulation layer thickness, leveraging the Purcell effect to tune emission times over an order of magnitude.

Contribution

It introduces a method to control exciton radiative rates in 2D heterostructures through simple layer thickness modifications, supported by experimental and theoretical analysis.

Findings

Radiative lifetime can be tuned by hBN thickness.

Spontaneous emission time varies up to 10 ps.

Enhanced or inhibited emission observed depending on configuration.

Abstract

Optical properties of atomically thin transition metal dichalcogenides are controlled by robust excitons characterized by a very large oscillator strength. Encapsulation of monolayers such as MoSe$_2$ in hexagonal boron nitride (hBN) yields narrow optical transitions approaching the homogenous exciton linewidth. We demonstrate that the exciton radiative rate in these van der Waals heterostructures can be tailored by a simple change of the hBN encapsulation layer thickness as a consequence of the Purcell effect. The time-resolved photoluminescence measurements together with cw reflectivity and photoluminescence experiments show that the neutral exciton spontaneous emission time can be tuned by one order of magnitude depending on the thickness of the surrounding hBN layers. The inhibition of the radiative recombination can yield spontaneous emission time up to $10$~ps. These results are in very good agreement with the calculated recombination rate in the weak exciton-photon coupling regime. The analysis shows that we are also able to observe a sizeable enhancement of the exciton radiative decay rate. Understanding the role of these electrodynamical effects allow us to elucidate the complex dynamics of relaxation and recombination for both neutral and charged excitons.

Equations

$$\frac{d{n}_{hot}}{dt}=-\frac{n_{hot}}{{\tau }_{relax}},$$

$$\frac{d{n}_X}{dt}=-\frac{n_X}{{\tau }_{rad}}+\frac{n_{hot}}{{\tau }_{relax}},$$

$$I(t)\propto \frac{n_{hot}^{(0)}}{{\tau }_{relax}-{\tau }_{rad}}\left[\exp \left(-\frac{t}{{\tau }_{relax}}\right)-\exp \left(-\frac{t}{{\tau }_{rad}}\right)\right].$$

$$\frac{d{n}_{hot}}{dt}=-\frac{n_{hot}}{{\tau }_{relax}}-\frac{n_{hot}}{{\tau }_{nr}},$$

$$\frac{d{n}_X}{dt}=-\frac{n_X}{{\tau }_{rad}}+\frac{n_{hot}}{{\tau }_{relax}},$$

$${\tau }_{nr}\gg {\tau }_{rad}.$$

$$\frac{1}{{\tau }_{decay}}=\frac{1}{{\tau }_{relax}}+\frac{1}{{\tau }_{nr}}\ll \frac{1}{{\tau }_{rad}}$$

$$\frac{DR}{R}=\frac{R_{ML}-R}{R},$$

$$\begin{gathered} {\hat{T}}_{tot}={\hat{T}}_{\mathrm{Si}{\mathrm{O}}_2\to \mathrm{Si}}{\hat{T}}_{\mathrm{Si}{\mathrm{O}}_2}{\hat{T}}_{\mathrm{hBN}\to \mathrm{Si}{\mathrm{O}}_2}{\hat{T}}_{\mathrm{hBN}}^{\prime } \\ {\hat{T}}_{\mathrm{MoS}{\mathrm{e}}_2}{\hat{T}}_{\mathrm{hBN}}{\hat{T}}_{air\to \mathrm{hBN}}. \end{gathered}$$

$${\hat{T}}_{\mathrm{MoS}{\mathrm{e}}_2}=\frac{1}{t}\left(\begin{array}{cc}{t}^2-r^2& r\\ -r& 1\end{array}\right),$$

$$r=\frac{\mathrm{i}{\Gamma }_0^{\mathrm{hBN}}}{{\omega }_0-\omega -\mathrm{i}({\Gamma }_0^{\mathrm{hBN}}+\Gamma )},t=1+r.$$

$${\hat{T}}_{tot}\left(\begin{array}{c}1\\ r_{tot}(\omega )\end{array}\right)=\left(\begin{array}{c}{t}_{tot}(\omega )\\ 0\end{array}\right).$$

$$\mathcal{A}(\omega )=1-|r_{tot}(\omega ){|}^2-|t_{tot}(\omega ){|}^2/n_{\mathrm{Si}},$$

$$r_{tot}(\omega )=\frac{Z_r}{{\omega }_0^{\mathrm{eff}}-\omega -\mathrm{i}({\Gamma }_0^{\mathrm{eff}}+\Gamma )}+\dots ,$$

$$t_{tot}(\omega )=\frac{Z_t}{{\omega }_0^{\mathrm{eff}}-\omega -\mathrm{i}({\Gamma }_0^{\mathrm{eff}}+\Gamma )}+\dots ,$$

$$\mathcal{A}(\omega )\approx \frac{{\mathcal{A}}_0}{({\omega }_0^{\mathrm{eff}}-\omega {)}^2+({\Gamma }_0^{\mathrm{eff}}+\Gamma {)}^2}.$$

$$\frac{1}{8\pi }\int d\mathbf{r}\{|E_0{|}^2+|B_0{|}^2\}=\frac{\mathcal{V}|E_0{|}^2}{4\pi }=\frac{\hslash \omega }{2},$$

$$|E_0|=|B_0|=\sqrt{\frac{2\pi \hslash \omega }{\mathcal{V}}},$$

$$|A_0|=\sqrt{\frac{2\pi \hslash c^2}{\omega \mathcal{V}}}.$$

$$\hat{V}=-\frac{1}{c}\int d\mathbf{r}\mathbf{j}\mathbf{A},$$

$$⟨\mathrm{exc}|\hat{V}|0⟩=-\sqrt{\mathcal{S}}\frac{e{p}_{cv}}{m_{0}c}{A}_0\phi (0){\delta }_{{\mathbf{q}}_{\parallel },0},$$

$$\frac{1}{{\tau }_0}=2{\Gamma }_0^{\mathrm{vac}}=\frac{2\pi }{\hslash }\sum _{\mathbf{q}}|⟨\mathrm{exc}|\hat{V}|0⟩{|}^2\delta (\hslash cq-\hslash {\omega }_0).$$

$$\sum _{\mathbf{q}}\dots {\delta }_{{\mathbf{q}}_{\parallel },0}=\frac{\mathcal{V}}{\mathcal{S}}\int \frac{d{q}_z}{2\pi }\dots .$$

$${\Gamma }_0^{\mathrm{vac}}=\frac{2\pi q_0}{\hslash }{\left(\frac{e{p}_{cv}\phi (0)}{m_0{\omega }_0}\right)}^2.$$

$$\frac{{\varkappa }_b}{4\pi }\int d\mathbf{r}|E_0{|}^2=\frac{\hslash \omega }{2},|E_0|=\sqrt{\frac{2\pi \hslash \omega }{{\varkappa }_b\mathcal{V}}},|A_0|=\sqrt{\frac{2\pi \hslash c^2}{{\varkappa }_b\omega \mathcal{V}}}.$$

$$\delta (\hslash cq/\sqrt{{\varkappa }_b}-\hslash {\omega }_0)=\delta (\hslash c|q_z|/\sqrt{{\varkappa }_b}-\hslash {\omega }_0),$$

$${\Gamma }_0({\varkappa }_b)=\frac{2\pi q_b}{{\varkappa }_b\hslash }{\left(\frac{e{p}_{cv}\phi (0)}{m_0{\omega }_0}\right)}^2.$$

$${\Gamma }_0({\varkappa }_b)=\frac{{\Gamma }_0^{\mathrm{vac}}}{\sqrt{{\varkappa }_b}}.$$

$$r_{b0}=\frac{\sqrt{{\varkappa }_b}-1}{\sqrt{{\varkappa }_b}+1},r_{0b}=-r_{b0},$$

$$r_{+}=r_0+\frac{t_0^2{r}_{0b}}{1-r_0{r}_{0b}}=\frac{r_{0b}[{\omega }_0-\omega -\mathrm{i}(-{\Gamma }_0^{\mathrm{vac}}+\Gamma )]+\mathrm{i}{\Gamma }_0^{\mathrm{vac}}}{{\omega }_0-\omega -\mathrm{i}[{\Gamma }_0^{\mathrm{vac}}(1+r_{0b})+\Gamma ]},$$

$$r_{tot}=r_{b0}+\frac{t_{b0}{t}_{0b}{r}_{+}}{1-r_{0b}{r}_{+}}.$$

$$r_{tot}=\frac{\mathrm{i}{\Gamma }_0^{\mathrm{vac}}\frac{1+r_{0b}}{1-r_{0b}}}{{\omega }_0-\omega -\mathrm{i}\left({\Gamma }_0^{\mathrm{vac}}\frac{1+r_{0b}}{1-r_{0b}}+\Gamma \right)}.$$

Full text

1. SI Experimental Methods

2. SII Measurement of the cavity thickness using AFM and reflectivity spectra

3. SIII Bi-exponential fit of the time-resolved photoluminescence

4. SIV Cw photoluminescence spectra and estimation of the radiative lifetime in sample I

5. SV Time-resolved photoluminescence of the neutral exciton at 90 K

6. SVI Possible role of non-radiative channels in the PL dynamics

7. SVII Reflection contrast

8. SVIII Theory of Purcell effect in van der Waals heterostructures

9. SVIII.1 General approach using the transfer matrix method

10. SVIII.2 Fermi golden rule approach to emission into homogeneous medium

11. SVIII.3 Comparison with electrodynamical approach

12. SVIII.4 Emission in the semi-infinite dielectric structure

13. SIX Analytical model applied to our structure

14. SIX.1 Effects of the in-plane wavevector of exciton

SI Experimental Methods

We fabricate the van der Waals heterostructures by mechanical exfoliation of high quality hBN crystals and bulk MoSe2 (similar results have been obtained with commercial MoSe2 provided by 2D Semiconductors or HQ Graphene). A first layer of hBN is mechanically exfoliated and deposited onto a 80 nm SiO2/Si substrate using a dry-stamping technique. The deposition of the subsequent MoSe2 ML and the second hBN capping layer is obtained by repeating this procedure. The in-plane size of the investigated MoSe2 MLs is typically $\sim 10\times 10~{}\mu$m2. The samples are held on a cold finger at a temperature $T=7$ K in a closed-cycle He cryostat. Attocube X-Y-Z piezo-motors allow for positioning with nm resolution of the ML with respect to the microscope objective (numerical aperture NA$=0.82$) used for the excitation and collection of luminescence. The cw PL experiments are performed with a He-Ne laser (633 nm) for excitation focused onto a spot diameter of 1 $\mu$m. The PL signal is dispersed in a spectrometer and detected with a Si-CCD cooled camera. For time-resolved photoluminescence experiments presented in the main text, the flakes are excited by $\sim 1.5$ ps pulses generated by a tunable mode-locked Ti:Sa laser with a repetition rate of 80 MHz and wavelength of 712 nm, i.e. about 100 meV above the neutral exciton transition energy. Similar results have been obtained for laser excitation wavelength in the range 710…753 nm (laser detuning from exciton resonance ranging from 7 to 104 meV), see Fig. S2. The PL signal is dispersed by a simple spectrometer and detected by a synchro-scan Hamamatsu streak camera Lagarde et al. (2014); Robert et al. (2016). By measuring the backscattered laser pulse from the sample surface, we obtain the overall instrumental response of the time-resolved setup, Fig. 2(a) in the main text.

SII Measurement of the cavity thickness using AFM and reflectivity spectra

The thicknesses of the bottom hBN layer $d$ is roughly determined by the color obtained in reflectivity Fig. S1(a). The reflectivity spectrum on each terrace is fitted using the calculated reflectivity obtained with the transfer matrix method, Fig. S1(b). AFM measurements confirm this preliminary determination and lead to more precise values, Fig. S1(c) and (d).

SIII Bi-exponential fit of the time-resolved photoluminescence

The exciton kinetics are fitted with simple bi-exponential fits based on a simple two-level model assuming that the system can be described by the population of the “reservoir” of photogenerated hot carriers $n_{hot}$ and of the cold excitons $n_{X}$ (inset of Fig. 3(b) in the main text):

[TABLE]

where $\tau_{relax}$ is the relaxation rate of the hot excitons into the emitting states and $\tau_{rad}$ is the radiative decay rate of excitons. In Eqs. (S1) we disregarded non-radiative recombination processes. Then the calculated exciton PL intensity simply writes:

[TABLE]

Here $n_{hot}^{(0)}$ is the initial photogenerated population of the hot carriers. Note that depending on the relation between the relaxation and radiative times the rise time of PL and its decay time are controlled by different parameters. Figure S3 displays the corresponding calculated intensity for two cases: (a) $\tau_{rad}>\tau_{relax}$ and (b) $\tau_{relax}>\tau_{rad}$. The rise-time does correspond to the radiative recombination time if $\tau_{relax}>\tau_{rad}$.

SIV Cw photoluminescence spectra and estimation of the radiative lifetime in sample I

The time-resolved photoluminescence measurements show that the radiative lifetime in sample I is limited by the temporal resolution (in the main text we infer $\tau_{X}<1.5$ ps) We can tentatively estimate the radiative lifetime by combining the cw and time-resolved PL results. The exciton homogeneous linewidth writes: $\Gamma=\Gamma_{rad}+\Gamma^{\prime}$, where $\Gamma_{rad}$ is the radiative linewidth and $\Gamma^{\prime}$ includes both non-radiative and pure-dephasing processes. The linewidth contributions from inhomogeneous broadening, light scattering, non-radiative processes, and radiative decay cannot be, unfortunately, disentangled by linear techniques such as photoluminescence or reflectivity spectroscopy used here. Nevertheless a rough estimate can be obtained assuming that the linewidth in sample II is mainly determined by pure dephasing processes and neglecting inhomogeneous broadening. We find $\Gamma^{\prime}\approx 1.1$ meV (FWHM); the measured radiative lifetime ($\approx 10$ ps) gives a negligible radiative linewidth contribution : $\Gamma_{rad}\approx 0.06$ meV. Assuming that non-radiative and pure dephasing processes are identical in samples I and II, we can deduce a radiative linewidth of $\approx 0.9$ meV in sample I corresponding to a radiative lifetime of $\approx 740$ fs. Although this analysis is rather approximate, the results are consistent with the recent FWM measurements in similar samples demonstrating that the homogenous broadening is not fully controlled by the radiative broadening Martin et al. (2018), for example, disorder-induced broadening still plays a significant role. The trend observed in Samples I and II is confirmed by the photoluminescence spectra measured in sample IV (same monolayer embedded in a cavity with different thicknesses), see Fig. S4.

SV Time-resolved photoluminescence of the neutral exciton at 90 K

Figure S5 displays the neutral exciton PL dynamics at $T=90$ K of sample III for three different bottom hBN thicknesses $d$. Two regimes are observed. In the first one, the excitons are in a non-thermal regime and their dynamic is dominated by a competition between radiative rate and escape of the light cone through exciton-phonon interactions. The second regime corresponds to the long decay time of 1 ns related to the lifetime of thermalized excitons in agreement with Ref. Robert et al. (2016) . This decay time does not depend on the bottom hBN thickness showing that the decay rate of excitons at this temperature is no more dominated by radiative recombination, in contrast to the results presented in the main text at $T=7$ K.

SVI Possible role of non-radiative channels in the PL dynamics

Our attribution of the decay time of neutral exciton to the relaxation of hot excitons in the main text assumes for simplicity the absence of non-radiative channels. Although this work demonstrates that the lifetime of an exciton in the radiative cone is dominated by its radiative recombination (tunable by cavity effects), we cannot claim that all hot excitons necessarily relax to the light cone. Indeed, any non-radiative channel such as defects trapping, relaxation to a dark state or formation of a trion at a timescale shorter than the relaxation to the light cone would lead exactly to the same dynamics but with smaller neutral exciton PL yield. In this subsection we briefly analyze this scenario depicted in Fig. S6 where these non-radiative decay processes of excitons are considered. The occupancies of the hot exciton reservoir and of the excitons in the light cone are given in this case by the following set of rate equations:

[TABLE]

Here the non-radiative decay rate of excitons includes contributions of trapping to defects, trion formation as well as possible formation of dark states. For excitons within the light cone we neglect non-radiative processes since

[TABLE]

The calculation shows that the exciton PL dynamics under the condition

[TABLE]

is given by the exponential law $\propto\exp{(-t/\tau_{decay})}$. We emphasize that if the trion formation from the hot exciton reservoir is the dominant process, then it is not surprising to observe the same rise time for the trion than the decay time for the neutral exciton.

SVII Reflection contrast

In order to provide an additional evidence for the variation of the exciton radiative lifetime as a result of the cavity-like effect we present in Fig. S7 the measured reflectivity contrast

[TABLE]

where $R_{ML}$ is the intensity reflection coefficient for the sample with the MoSe2 monolayer and $R$ is the intensity reflection coefficient of the same sample but without the monolayer (measured in a different spot). Panel (a) shows the data on the sample I where the exciton radiative recombination is enhanced and panel (b) shows the data on the sample II where the exciton radiative recombination is inhibited. Despite certain spread of the $DR/R$ values, one can see that the reflection contrast in the sample I is systematically much larger than in the sample II ; the exciton linewidth in sample I is also substantially larger than the one in sample II. We stress that these reflectivity data are fully in line with the cw PL and time-resolved PL measurements presented in the main text. We abstain from the detailed multi-parameter fit of the reflectivity data which requires also careful analysis of the light scattering and inhomogeneous broadening (as well as possible effects of finite NA of our optical setup).

SVIII Theory of Purcell effect in van der Waals heterostructures

In this section we outline the general electrodynamical method to calculate the radiative decay rate based on the transfer matrix formalism, present its justification on the basis of quantum mechanical approach.

SVIII.1 General approach using the transfer matrix method

We use the transfer matrix method in order to calculate the elementary response functions (reflectivity, transmission and absorbance) of our structure “cap hBN layer/TMD ML/bottom hBN layer/SiO2/Si” Robert et al. (2018)

[TABLE]

Here $\hat{T}_{i\to j}$ is the transfer matrix through the interface between the layers $i$ to $j$, $\hat{T}_{i}$ is the transfer matrix through the layer $i$, the prime denotes the bottom hBN layer. For the TMD ML transfer matrix we consider the situation of the monolayer embedded into the infinite hBN:

[TABLE]

[TABLE]

Here $r$ and $t$ are the reflection and transmission coefficients of the TMD MLs in the infinite homogeneous hBN, $\omega_{0}$ is the exciton resonance frequency, $\Gamma_{0}^{\rm hBN}$ is the exciton radiative decay rate (into the hBN, $\Gamma_{0}^{\rm hBN}=\Gamma_{0}^{\rm vac}/n_{\rm hBN}$, where $n_{\rm hBN}$ is the hBN refractive index and $\Gamma_{0}^{\rm vac}$ is the radiative decay rate into free space, see Eq. (S21) and discussion below) and $\Gamma$ is the exciton non-radiative decay rate. The Si layer is assumed to be thicker than the absorption length, hence, the reflection of light at the interface Si and air is disregarded. The transfer matrix provides the following relation between $r_{tot}$ and $t_{tot}$ the amplitude reflection and transmission coefficients through the structure:

[TABLE]

Equation (S9) allows us to obtain $r_{tot}(\omega)$ and $t_{tot}(\omega)$ from the transfer matrix and the absorbance of the monolayer can be expressed as

[TABLE]

where $n_{\rm Si}$ is the refraction index of Si.

According to the general theory Landau and Lifshitz (2000); Ivchenko (2005); Glazov et al. (2014) the poles of the response functions $r_{tot}(\omega)$, $t_{tot}(\omega)$ and, correspondingly, the feature in $\mathcal{A}(\omega)$, correspond to the eigenmodes of the system. In the vicinity of exciton resonance frequency the functions $r_{tot}(\omega)$ and $t_{tot}(\omega)$ can be recast in the form

[TABLE]

Thus, in order to calculate the radiative decay rate $\Gamma_{0}^{\rm eff}$ (HWHM)in our structure, we found numerically the absorbance, $\mathcal{A}(\omega)$ at $\Gamma\to 0$,111In this way we disregarded possible weak asymmetry of the optical spectra due to exciton-phonon interaction which could be taken into account by using $\Gamma$ as a function of frequency Christiansen et al. (2017); Shree et al. (2018). fitted it by Eq. (S11c) and extracted $\Gamma_{0}^{\rm eff}$ as a function of the bottom hBN thickness $d$. These data are plotted by solid blue line in Fig. 2(b) of the main text.

SVIII.2 Fermi golden rule approach to emission into homogeneous medium

Here and below we provide quantum mechanical approach to emission of excitons into homogeneous dielectric media. Let us start with the calculation of the emission rate of the exciton in a TMD ML into a free space. We neglect dielectric contrast between the ML and the surrounding. Hereafter we consider only $s$-polarization of light and fix the direction of the electric and magnetic fields.

Normalization of electric field per photon reads

[TABLE]

[TABLE]

where $\mathcal{V}$ is the normalization volume, $E_{0}$ and $B_{0}$ are the complex amplidutes of the electric and magnetic field. It is convenient to use the vector potential $\bm{A}$ such that $\bm{E}=-c^{-1}\partial\bm{A}/\partial t$. Hence $\bm{E}_{0}=(\mathrm{i}\omega/c)\bm{A}_{0}$ and the normalization for the vector potential reads

[TABLE]

The light-matter coupling operator can be expressed as

[TABLE]

where $\bm{j}$ is the current associated with exciton transition. In the two-dimensional layer the matrix element of the perturbation (S14) related to the formation of exciton per one photon with the center of mass wavevector $\bm{K}=(K_{x},K_{y})=0$ reads

[TABLE]

where $\mathcal{S}$ is the normalization area, $\varphi(0)$ is the exciton envelope wavefunction at the coinciding electron and hole coordinates, $p_{cv}$ is the interband momentum matrix element, $\bm{q}_{\parallel}=(q_{x},q_{y})$ is the in-plane wavevector of light. Using the Fermi golden rule we obtain the exciton radiative decay rate

[TABLE]

Here $\omega_{0}$ is the exciton resonance frequency, and the light wavevector $q=\sqrt{q_{\parallel}^{2}+q_{z}^{2}}$ where $q_{\parallel}=0$ and $q_{z}\neq 0$ are the in-plane and normal components of the wavevector. We transform the sum over $\bm{q}$ into the integral over $q_{z}$ as

[TABLE]

The energy conservation law provides two values of $q_{z}=\pm\omega_{0}/c$. As a result, for the radiative decay rate $\Gamma_{0}$ we have in agreement with Refs. Ivchenko (2005); Glazov et al. (2014)

[TABLE]

Here $q_{0}=\omega_{0}/c$ is the wavevector of emitted radiation.

Let us now analyze the modifications of Eq. (S18) for emission into a dielectric medium with the background dielectric constant $\varkappa_{b}$. Normalization of the field and vector potential reads (note that the relation between $\bm{E}$ and $\bm{A}$ does not depend on $\varkappa_{b}$ and in the homogeneous medium $B_{0}=\sqrt{\varkappa_{b}}E_{0}$ so that electric and magnetic energies are the same):

[TABLE]

Additional modification in the Fermi golden rule approach comes from Eq. (S16) where the energy conservation $\delta$-function reads now

[TABLE]

therefore, the removal of the $\delta$-function yields $\sqrt{\varkappa_{b}}$ in the numerator. As a result Ivchenko (2005); Glazov et al. (2014):

[TABLE]

Here $q_{b}=\omega_{0}\sqrt{\varkappa_{b}}/c$ is the wavevector of emitted radiation in the medium. The comparison of Eqs. (S18) and (S20) shows that

[TABLE]

In this approach we disregard, for simplicity, the modification of exciton wavefunction at the coinciding coordinates $\varphi(0)$, interband momentum matrix element $p_{cv}$ and the exciton resonance frequency $\omega_{0}$ due to the dielectric environment effects, including the screening of Coulomb potential of electron-hole interaction. This approach allows us to clarify the electrodynamical effects which can be interpreted here as: (i) change of the photon density of states and (ii) change of the field amplitude per photon.

SVIII.3 Comparison with electrodynamical approach

We demonstrate now that this result, Eq. (S21), simply follows from the electrodynamical approach presented above. We consider a monolayer surrounded by the homogeneous hBN semi-infinite layers. We introduce

[TABLE]

the reflection coefficients from the dielectric to vacuum ($r_{b0}$) and from vacuum to the dielectric ($r_{0b}$). First, we calculate the reflection coefficient $r_{+}$ from the half space filled with a dielectric with the ML on its left side, Fig. S8(a). Summing up all the reflections between the ML and the dielectric we arrive at

[TABLE]

where $r_{0}$ and $t_{0}=1+r_{0}$ are the reflection transmission coefficient through the ML for light incident from the vacuum. These quantities differ from $r$ and $t$ in Eq. (S8) by the replacement $\Gamma_{0}^{\rm hBN}\to\Gamma_{0}^{\rm vac}$ in Eq. (S18), see Ref. Ivchenko (2005). Next, summing up all the reflections between the left-hand side dielectric in Fig. S8(b) and the structure “ML+semi-infinite dielectric” depicted in Fig. S8(a) we arrive at

[TABLE]

Here $t_{0b}=1+r_{0b}$ and $t_{b0}=1+r_{b0}$ are the corresponding transmission coefficients. Substitution of Eq. (S23) into Eq. (S24) gives

[TABLE]

Equation (S25) has the same form as Eqs. (S8) and provides the explicit connection between the renormalized and non-renormalized exciton radiative decay rate

[TABLE]

in full agreement with Eq. (S21) confirming the electrodynamical approach.

SVIII.4 Emission in the semi-infinite dielectric structure

Now we turn to the structure shown in Fig. S8(a) where the semi-infinite dielectric is placed on the right of the ML. The reflection coefficient is given by Eq. (S23). In order to find the decay rate of exciton within the electodynamical approach it is sufficient to analyze the pole of this expression, cf. Eqs. (S11). Thus, for the radiative decay rate we have Chance et al. (1978); Glazov et al. (2011)

[TABLE]

This result was derived by the electrodynamical approach. Note, that this equality is general and can be applicable for any system with complex reflection coefficient, e.g., if the barrier is finite or the absorption is involved or if the barrier consists of several layers. In the particular case of a barrier made of a homogeneous dielectric with the real susceptibility $\varkappa_{b}$ we have

[TABLE]

It is instructive to derive Eq. (S27) by means of the Fermi golden rule. For convenience and in order to avoid problems with fact that the dispersion of waves in the left-hand side and in the right-hand side of the structure is different we assume that at $z\to+\infty$ the background dielectric constant $\varkappa_{b}$ goes smoothly to $1$. Asymptotic behavior of the modes of electromagnetic field at $z\to\pm\infty$ can be written as:

[TABLE]

Here $|E_{0}|$ is given by Eq. (S12), the $t_{b}$ is the transmission coefficient through the dielectric barrier from the left to the right, $r_{b}^{\prime}$ is the reflection coefficient from the barrier for the radiation incident from the right, $t_{b}^{\prime}$ is the transmission coefficient through the barrier from the right to the left, see Fig. S8(a). Functions (S29) are properly normalized which can be checked by means of the following relations De Martini et al. (1991)

[TABLE]

which hold true for non-absorbing media only, where, strictly speaking, quantum mechanical approach is applicable. The dispersion relation remains the same as for the waves in the empty space.

Now we apply the Fermi golden rule [cf. Eq. (S16)]

[TABLE]

where $V^{(i)}$ is the perturbation for interaction with the waves (S29), $i=1,2$. The summation over $q_{z}$ is understood as follows [cf. Eq. (S17)]

[TABLE]

Taking into account that the amplitude of the electromagnetic field at the TMD ML for the mode $1$ is given by $E_{0}(1+r_{0b})$ and for the wave $2$ is given by $E_{0}t_{b}^{\prime}$, the ratio

[TABLE]

Equation (S33) can be recast in the form (S27) making use of the following relations $|t_{b}^{\prime}|^{2}=|t_{b}|^{2}$, which follows from the time-reversal symmetry, and Eq. (S30):

[TABLE]

Thus, Fermi golden rule, Eq. (S33) and the electrodynamical approach, Eq. (S27), gives the same result for non-absorbing dielectric media. It is worth stressing that Eq. (S27) is more general in a sense that it can be applied for absorbing structures as well where the quantum mechanical approach fails Glazov et al. (2011).

SIX Analytical model applied to our structure

In this section we provide simple but accurate analytical approximation for the Purcell effect in our structure. We also consider the effects related to emission at oblique angles and demonstrate that they are not particularly important in the studied system.

The main simplification comes from the fact that the thickness of the top hBN layer (typically within 10 nm range) does not affect electrodynamical properties of the structure because it is much smaller than the light wavelength at the exciton resonance frequency. That is why we can consider a simplified structure in our analytical model shown in Fig. S9, which differs from the real sample [Fig. 1(a) of the main text] by the absence of the top hBN layer. Thus, in accordance with Eq. (S27) in order to find the exciton radiative decay rate we need to calculate the reflection coefficient $r_{bg}$ for the three-layer structure (bottom) hBN/SiO2/Si for the light incident from the left.

We introduce the following notations for the reflection coefficients:

[TABLE]

we use $r_{j}^{\prime}=-r_{j}$ for the reflection through the same interface but in the backwards direction, and we use $t_{j}=1+r_{j}$, $t_{j}^{\prime}=1+r_{j}^{\prime}$ for the transmission coefficients, $j=1,\ldots,3$. We use $q_{j}$ and $L_{j}$ to denote corresponding wavevector and thickness, i.e., $q_{1}$ is the wavevector in hBN, $L_{1}\equiv d$ (in the notations of the main text) is its thickness. For the SiO2/Si part of the structure (light is incident from hBN) the reflection coefficient reads

[TABLE]

Analogously, for the whole structure hBN/SiO2/Si

[TABLE]

The Purcell factor, i.e., the ratio of the emission rate in our structure and in vacuum, in agreement with written above is given by

[TABLE]

We stress that we analyze here the impact of the electrodynamical environment only. The effects related to possible variation of the exciton binding energy and oscillator strength due to modification of the Coulomb interaction dielectric screening are disregarded. The calculation of the exciton radiative lifetime after Eq. (S38) and Eq. (1) of the main text is presented in Fig. S10 together with the experimental data and the full numerical model. The difference between the full model and the analytical approximation is almost negligible and results from the non-zero thickness of the top hBN layer.

SIX.1 Effects of the in-plane wavevector of exciton

The aim of this section is to determine the influence of the emission angle because in the experiments, a high NA objective is used which collects photons emitted by excitons with non-zero momentum. To that end, let us now consider the emission of exciton with an in-plane wavevector $\bm{k}$. In this case, the polarization of radiation becomes important. As before, we start from the emission into free space and present the side-by-side comparison of the Fermi golden rule approach and electrodynamical treatment of the problem. The light-exciton coupling matrix elements depend on the light polarization [cf. Eq. (S15)] and read:

[TABLE]

where $\cos{\vartheta}=q_{z}/q=\sqrt{k^{2}-q^{2}}/q$. Additionally, in the Fermi golden rule the energy conservation $\delta$-function should be transformed as

[TABLE]

which produces extra factor $1/\cos{\vartheta}$ in the Fermi golden rule. This factor arises from van Hove-like singularity of photon density of states at the edge of the light cone. As a result, we have the following relations

[TABLE]

This result can be also obtained from the electrodynamical approach Ivchenko (2005); Glazov et al. (2014). Similar relations (with the replacement $\rm vac\to\rm hBN$) hold for emission in the homogeneous hBN. The enhancement of $\Gamma_{0,s}$ for the grazing incidence ($|\vartheta|\to\pi/2$, $\cos{\vartheta}\to 0$) results from the simple fact that for light propagating in the ML plane the light-matter length in $s$-polarization becomes infinite.

The average decay rate for exciton population characterized by the distribution function $f(k)$ can be written as:

[TABLE]

In the averaging we take into account the population of states within the light cone and assume that the distribution of exciton is independent of their polarization. If $f(k)$ is constant within the light cone, which is reasonable due to the fact that the light cone is very “narrow” in the energy space, the integrals can be readily calculated via the substitution $\cos{\vartheta}=\mu$ with the result

[TABLE]

Equation (S42) can be easily extended for our system depicted in Fig. S9. To that end we introduce $r_{bg}^{(s)}$ and $r_{bg}^{(p)}$ for the reflection coefficients of our structure in $s$- and $p$-polarizations, respectively. It follows from Eqs. (S38) and (S42) that

[TABLE]

For completeness we present the expressions for reflection coefficients in $s-$ and $p-$polarizations at the boundaries of homogeneous dielectrics. We use the following notations for the wavevectors

[TABLE]

$j=0$ corresponds to vacuum, $n_{0}=1$, $j=1$ is the hBN, $n_{1}=n_{\rm hBN}$ (note that we disregard optical anisotropy of hBN, which can lead to quantitative changes), $n_{2}=n_{\rm SiO_{2}}$, and $n_{3}=n_{\rm Si}$. We present the reflection coefficients for light incident from the layer $j$ to $j+1$ as

[TABLE]

The transmission coefficients read $t_{j,j+1}^{(s)}=1+r_{j,j+1}^{(s)}$ and $t_{j,j+1}^{(p)}=1+r_{j,j+1}^{(p)}$. Note that we use extra minus sign in the definition of $r_{p}$ in order to have $r_{s}=r_{p}$ at the normal incidence.

We also note that in actual calculations is it convenient to change from the integration over $\bm{k}$ to the integration over the angle of incidence $\vartheta$. Assuming that $f(k)$ is constant in the relevant wavevector range, by virtue of $kdk=q_{0}^{2}\sin{\vartheta}d\sin{\vartheta}=q_{0}^{2}\cos{\vartheta}\sin{\vartheta}d\vartheta$ the integral in Eq. (S44) can be recast as ($\mu=\cos{\vartheta}$):

[TABLE]

In experiments a finite collection angle is used, $\vartheta_{c}<\pi/2$. In order to describe such a situation we suggest to use corresponding partial averaging and obtain the Purcell factor in the form

[TABLE]

[TABLE]

where the prefactor comes from the normalization condition for emission into a homogeneous environment:

[TABLE]

Particularly, at $\vartheta_{c}\to 0$ we obtain $F_{p}=1+\mathop{\rm Re}\{r_{bg}\}$ in accordance with Eq. (S38), at $\vartheta_{c}\to\pi/2$ we obtain Eq. (S47) [up to a coefficient $4/3$ which follows from Eq. (S43)].

Figure S11 demonstrates the results of calculations. The analysis shows that accounting for the in-plane wavevectors of excitons does not strongly affect the results of previous analysis.