Engineering the radiative dynamics of thermalized excitons with metal interfaces

The point dipole case is particularly prevalent in quantum optics, accounting for the modification of spontaneous emission of individual atoms, molecules or other point-like quantum emitters in the vicinity of planar interfaces [20, 23], planar cavities [24], or other resonant or confining structures for light [25–27].

We consider a two-level system with an electric dipole allowed transition of frequency ω and corresponding free-space wavelength λ. We assume that its immediate vicinity is characterized by a relative permittivity ₁, but allow for the possibility of other dielectric media nearby, which might modify its emission rate. In the weak-coupling regime (to be defined shortly), the modified emission rate, Γ, normalized by the emission rate Γ₀ in an infinite medium of permittivity ₁, is given by [20, 28, 29]:

$$\begin{equation}\frac{{\Gamma}}{{{\Gamma}}_{0}}=\frac{\mathrm{I}\mathrm{m}\left[{\mathbf{p}}^{\ast }\cdot \mathbf{E}({\boldsymbol{r}}_{\mathbf{0}})\right]}{\mathrm{I}\mathrm{m}\left[{\mathbf{p}}^{\ast }\cdot {\mathbf{E}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}({\boldsymbol{r}}_{\mathbf{0}})\right]}.\end{equation} \tag{ 1 }$$

Here, p e^−iωt is a classical oscillating point dipole representing the two-level system, E( r ₀ )e^−iωt is the total field produced by the dipole in the geometry of interest at its own location r ₀ = (0, 0, z₀), and E_free( r ₀ ) is the field of the dipole in a uniform medium with a relative permittivity of ₁. The validity of this weak-coupling approach requires that the response of the electromagnetic environment itself does not appreciably vary over Γ itself. Otherwise, non-exponential or strong coupling dynamics like vacuum Rabi oscillations can occur [30].

While equation (1) is general, we now describe an efficient method to calculate the total field E for planar geometries. We outline the calculations for a point dipole with arbitrary orientation, and for convenience define the corresponding current density J( r ) = −iωδ( r − r ₀ )p. We first define the in-plane wavevector k _∥ = (k_x , k_y ), and generalizing for later discussions, the dispersion relation in medium i of ${k}_{{\Vert}}^{2}+{k}_{z,i}^{2}={{\epsilon}}_{i}{\left(\frac{\omega }{c}\right)}^{2}\equiv {k}_{i}^{2}$. The free field in an infinite medium with permittivity can be expressed as:

$$\begin{equation}{\mathbf{E}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\boldsymbol{r})=\mathrm{i}\omega {\mu }_{0}\int \mathrm{d}{\boldsymbol{r}}^{\prime }{\mathbf{G}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\boldsymbol{r},{\boldsymbol{r}}^{\prime })\cdot \mathbf{J}({\boldsymbol{r}}^{\prime }).\end{equation} \tag{ 2 }$$

For planar geometries, it is useful to express the dyadic Green's function in a plane wave representation:

$$\begin{equation}{\mathbf{G}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\boldsymbol{r},{\boldsymbol{r}}^{\prime })=\frac{\mathrm{i}}{8{\pi }^{2}}\int {\mathrm{d}}^{2}{\boldsymbol{k}}_{{\Vert}}\mathbf{M}({\boldsymbol{k}}_{{\Vert}}){\text{e}}^{\text{i}({\boldsymbol{k}}_{{\Vert}}\cdot (\boldsymbol{\rho }-{\boldsymbol{\rho }}^{\prime })+{k}_{z}\vert z-{z}^{\prime }\vert )}\end{equation} \tag{ 3 }$$

and:

$$\begin{equation}\mathbf{M}({\boldsymbol{k}}_{{\Vert}})=\left(\frac{1}{{k}_{z}{k}^{2}}\right)\left(\begin{matrix}\hfill {k}^{2}-{k}_{x}^{2}\hfill & \hfill -{k}_{x}{k}_{y}\hfill & \hfill \mp {k}_{x}{k}_{z}\hfill \\ \hfill -{k}_{x}{k}_{y}\hfill & \hfill {k}^{2}-{k}_{y}^{2}\hfill & \hfill \mp {k}_{y}{k}_{z}\hfill \\ \hfill \mp {k}_{x}{k}_{z}\hfill & \hfill \mp {k}_{y}{k}_{z}\hfill & \hfill {k}^{2}-{k}_{z}^{2}\hfill \end{matrix}\right).\end{equation} \tag{ 4 }$$

(Note that the upper sign is for z > z₀ and the lower sign for z < z₀) [29]. For the point dipole, this expression for the field simplifies to:

$$\begin{equation}{\mathbf{E}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\boldsymbol{r})=\frac{1}{4{\pi }^{2}}\int {\mathrm{d}}^{2}{\boldsymbol{k}}_{{\Vert}}\mathbf{E}({\boldsymbol{k}}_{{\Vert}}){\text{e}}^{\text{i}({\boldsymbol{k}}_{{\Vert}}\cdot \boldsymbol{\rho }+{k}_{z}\vert z-{z}_{0}\vert )}\end{equation} \tag{ 5 }$$

where

$$\begin{equation}\mathbf{E}({\boldsymbol{k}}_{{\Vert}})=\frac{\mathrm{i}{\omega }^{2}{\mu }_{0}}{2}\mathbf{M}({\boldsymbol{k}}_{{\Vert}})\cdot \mathbf{p}\end{equation} \tag{ 6 }$$

gives the polarization and amplitude of each plane wave component.

To calculate the total field, we write it as a sum of free and scattered contributions, E( r ) = E_free( r ) + E_sc( r ) and solve for the latter. For concreteness, we will focus on two cases, consisting of a single dielectric interface (figure 1(a)) and a symmetric cavity configuration (figure 1(b)). For the single interface, we consider that the point dipole is situated at a position r ₀ (with z₀ > 0) in a medium with permittivity ₁, and calculate the reflected field from an interface with a material with permittivity ₂, with the boundary between the two media situated at z = 0. For an arbitrarily polarized point dipole p, it is useful to decompose the fields into s- and p-polarizations to calculate the reflected (i.e. scattered) fields. We decompose the matrix M( k _∥ ) = M^(s)( k _∥ ) + M^(p)( k _∥ ) where:

$$\begin{equation}{\mathbf{M}}^{(s)}({\boldsymbol{k}}_{{\Vert}})=\left(\frac{1}{{k}_{z,1}{k}_{{\Vert}}^{2}}\right)\left(\begin{matrix}\hfill {k}_{y}^{2}\hfill & \hfill -{k}_{x}{k}_{y}\hfill & \hfill 0\hfill \\ \hfill -{k}_{x}{k}_{y}\hfill & \hfill {k}_{x}^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\end{equation} \tag{ 7 }$$

$$\begin{equation}{\mathbf{M}}^{(p)}({\boldsymbol{k}}_{{\Vert}})=\left(\frac{1}{{k}_{1}^{2}{k}_{{\Vert}}^{2}}\right)\left(\begin{matrix}\hfill {k}_{x}^{2}{k}_{z,1}\hfill & \hfill {k}_{x}{k}_{y}{k}_{z,1}\hfill & \hfill \mp {k}_{x}{k}_{{\Vert}}^{2}\hfill \\ \hfill {k}_{x}{k}_{y}{k}_{z,1}\hfill & \hfill {k}_{y}^{2}{k}_{z,1}\hfill & \hfill \mp {k}_{y}{k}_{{\Vert}}^{2}\hfill \\ \hfill \mp {k}_{x}{k}_{{\Vert}}^{2}\hfill & \hfill \mp {k}_{y}{k}_{{\Vert}}^{2}\hfill & \hfill \frac{{k}_{{\Vert}}^{4}}{{k}_{z,1}}\hfill \end{matrix}\right)\end{equation} \tag{ 8 }$$

gives the decomposed matrices for the incident fields and

$$\begin{equation}{\mathbf{M}}_{\mathrm{R}}^{(s)}({\boldsymbol{k}}_{{\Vert}})=\left(\frac{1}{{k}_{z,1}{k}_{{\Vert}}^{2}}\right)\left(\begin{matrix}\hfill {k}_{y}^{2}\hfill & \hfill -{k}_{x}{k}_{y}\hfill & \hfill 0\hfill \\ \hfill -{k}_{x}{k}_{y}\hfill & \hfill {k}_{x}^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\end{equation} \tag{ 9 }$$

$$\begin{equation}{\mathbf{M}}_{\mathrm{R}}^{(p,\pm )}({\boldsymbol{k}}_{{\Vert}})=\left(-\frac{1}{{k}_{1}^{2}{k}_{{\Vert}}^{2}}\right)\left(\begin{matrix}\hfill {k}_{x}^{2}{k}_{z,1}\hfill & \hfill {k}_{x}{k}_{y}{k}_{z,1}\hfill & \hfill \pm {k}_{x}{k}_{{\Vert}}^{2}\hfill \\ \hfill {k}_{x}{k}_{y}{k}_{z,1}\hfill & \hfill {k}_{y}^{2}{k}_{z,1}\hfill & \hfill \pm {k}_{y}{k}_{{\Vert}}^{2}\hfill \\ \hfill \mp {k}_{x}{k}_{{\Vert}}^{2}\hfill & \hfill \mp {k}_{y}{k}_{{\Vert}}^{2}\hfill & \hfill \mp \frac{{k}_{{\Vert}}^{4}}{{k}_{z,1}}\hfill \end{matrix}\right)\end{equation} \tag{ 10 }$$

gives the decomposed matrices for the reflected fields [29]. Note that in the p-polarized reflection matrix, the ± superscript denotes the propagation direction of the reflected field. For the geometry in figure 1(a), the reflected field propagates upward and corresponds to the choice +.

For the single interface, the reflected field at the location of the point dipole is:

$$\begin{equation}{\mathbf{E}}_{\mathrm{s}\mathrm{c}}({\boldsymbol{r}}_{\mathbf{0}})=\frac{1}{4{\pi }^{2}}\int {\mathrm{d}}^{2}{\boldsymbol{k}}_{{\Vert}}{\mathbf{E}}_{\mathbf{s}\mathbf{c}}({\boldsymbol{k}}_{{\Vert}}){\mathrm{e}}^{2\mathrm{i}{k}_{z,1}{z}_{0}}\end{equation} \tag{ 11 }$$

where

$$\begin{equation}{\mathbf{E}}_{\mathbf{s}\mathbf{c}}({\boldsymbol{k}}_{{\Vert}})=\frac{\mathrm{i}{\omega }^{2}{\mu }_{0}}{2}\left[{r}_{s}({\boldsymbol{k}}_{{\Vert}}){\mathbf{M}}_{\mathrm{R}}^{(s)}({\boldsymbol{k}}_{{\Vert}})+{r}_{p}({\boldsymbol{k}}_{{\Vert}}){\mathbf{M}}_{\mathrm{R}}^{(p,+)}({\boldsymbol{k}}_{{\Vert}})\right]\cdot \mathbf{p}.\end{equation} \tag{ 12 }$$

The reflection coefficients of a plane wave from a planar interface r_s,p ( k _∥ ) are given by [29, 31]:

$$\begin{equation}{r}_{s}({\boldsymbol{k}}_{{\Vert}})=\frac{{k}_{z,1}-{k}_{z,2}}{{k}_{z,1}+{k}_{z,2}}\quad {r}_{p}({\boldsymbol{k}}_{{\Vert}})=\frac{{{\epsilon}}_{2}{k}_{z,1}-{{\epsilon}}_{1}{k}_{z,2}}{{{\epsilon}}_{2}{k}_{z,1}+{{\epsilon}}_{1}{k}_{z,2}}.\end{equation} \tag{ 13 }$$

One can now substitute the total (free plus scattered) field into equation (1) to calculate the modified emission rate at a single interface. We consider two specific orientations of the point dipole: in-plane (p = (p_x , p_y , 0)) and out-of-plane (p = (0, 0, p)) to the interface. The total emission rate for an in-plane dipole is:

$$\begin{equation}\frac{{\Gamma}({z}_{0})}{{{\Gamma}}_{0}}=1+\frac{3}{4{k}_{1}}\enspace \mathrm{R}\mathrm{e}\left[{\int }_{0}^{+\infty }\mathrm{d}{k}_{{\Vert}}\left[\left(\frac{{k}_{{\Vert}}}{{k}_{z,1}}\right){r}_{s}({k}_{{\Vert}})-\left(\frac{{k}_{{\Vert}}{k}_{z,1}}{{k}_{1}^{2}}\right){r}_{p}({k}_{{\Vert}})\right]{\mathrm{e}}^{2\mathrm{i}{k}_{z,1}{z}_{0}}\right].\end{equation} \tag{ 14 }$$

For an out-of-plane dipole, the total emission rate is:

$$\begin{equation}\frac{{\Gamma}({z}_{0})}{{{\Gamma}}_{0}}=1+\frac{3}{2{k}_{1}^{3}}\enspace \mathrm{R}\mathrm{e}\left[{\int }_{0}^{+\infty }\mathrm{d}{k}_{{\Vert}}\left(\frac{{k}_{{\Vert}}^{3}}{{k}_{z,1}}\right){\mathrm{e}}^{2\mathrm{i}{k}_{z,1}{z}_{0}}{r}_{p}({k}_{{\Vert}})\right]\end{equation} \tag{ 15 }$$

and we note that this dipole orientation only contains p-polarized components.

For the cavity structure, we consider a dipole at z₀ = 0 in medium 1 with permittivity ₁, which is symmetrically located between two media of permittivity ₂ and where the distance between their two interfaces is given by d (figure 1(b)). The techniques described above can easily be generalized here. We again write the total field as E( r ) = E_free( r ) + E_sc( r ). The scattered field can be obtained from the reflected field of a single interface, by summing its propagation and multiple reflection to all orders. At the position of the dipole, one finds:

$$\begin{equation}{\mathbf{E}}_{\mathrm{s}\mathrm{c}}({\boldsymbol{r}}_{\mathbf{0}})=\frac{1}{4{\pi }^{2}}\int {\mathrm{d}}^{2}{\boldsymbol{k}}_{{\Vert}}\left({\boldsymbol{E}}_{\mathrm{s}\mathrm{c}}^{+}({\boldsymbol{k}}_{{\Vert}})+{\boldsymbol{E}}_{\mathrm{s}\mathrm{c}}^{-}({\boldsymbol{k}}_{{\Vert}})\right){\text{e}}^{\text{i}{k}_{z,1}d/2}.\end{equation} \tag{ 16 }$$

The scattered field components are given by

$$\begin{equation}{\boldsymbol{E}}_{\mathrm{s}\mathrm{c}}^{\pm }({\boldsymbol{k}}_{{\Vert}})=\frac{\mathrm{i}{\omega }^{2}{\mu }_{0}}{2}\left[\left(\frac{{r}_{p}\enspace {\text{e}}^{\text{i}{k}_{z,1}d/2}}{1+{r}_{p}\enspace {\text{e}}^{\text{i}{k}_{z,1}d}}\right){\mathbf{M}}_{\mathrm{R}}^{(p,\pm )}({\boldsymbol{k}}_{{\Vert}})+\left(\frac{{r}_{s}\enspace {\text{e}}^{\text{i}{k}_{z,1}d/2}}{1-{r}_{s}\enspace {\text{e}}^{\text{i}{k}_{z,1}d}}\right){\mathbf{M}}_{\mathrm{R}}^{(s)}({\boldsymbol{k}}_{{\Vert}})\right]\cdot \mathbf{p}\end{equation} \tag{ 17 }$$

for an in-plane dipole, while for an out-of-plane dipole,

$$\begin{equation}{\boldsymbol{E}}_{\mathrm{s}\mathrm{c}}^{\pm }({\boldsymbol{k}}_{{\Vert}})=\frac{\mathrm{i}{\omega }^{2}{\mu }_{0}}{2}\left[\left(\frac{{r}_{p}\enspace {\text{e}}^{\text{i}{k}_{z,1}d/2}}{1-{r}_{p}\enspace {\text{e}}^{\text{i}{k}_{z,1}d}}\right){\mathbf{M}}_{\mathrm{R}}^{(p,\pm )}({\boldsymbol{k}}_{{\Vert}})\right]\cdot \mathbf{p}.\end{equation} \tag{ 18 }$$

From equation (1), the corresponding emission rate for the in-plane dipole in the cavity is:

$$\begin{equation}\frac{{\Gamma}(d)}{{{\Gamma}}_{0}}=1+\frac{3}{2{k}_{1}}\enspace \mathrm{R}\mathrm{e}\left[{\int }_{0}^{+\infty }\mathrm{d}{k}_{{\Vert}}\left[\frac{{k}_{{\Vert}}}{{k}_{z,1}}\left(\frac{{r}_{s}}{1-{r}_{s}\enspace {\mathrm{e}}^{\mathrm{i}{k}_{z,1}d}}\right)-\frac{{k}_{{\Vert}}{k}_{z,1}}{{k}_{1}^{2}}\left(\frac{{r}_{p}}{1+{r}_{p}\enspace {\text{e}}^{\text{i}{k}_{z,1}d}}\right)\right]{\text{e}}^{\text{i}{k}_{z,1}d}\right].\end{equation} \tag{ 19 }$$

Similarly, for the out-of-plane dipole,

$$\begin{equation}\frac{{\Gamma}(d)}{{{\Gamma}}_{0}}=1+\frac{3}{{k}_{1}^{3}}\enspace \mathrm{R}\mathrm{e}\left[{\int }_{0}^{+\infty }\mathrm{d}{k}_{{\Vert}}\left(\frac{{k}_{{\Vert}}^{3}}{{k}_{z,1}}\right){\mathrm{e}}^{\mathrm{i}{k}_{z,1}d}\left(\frac{{r}_{p}}{1-{r}_{p}\enspace {\mathrm{e}}^{\mathrm{i}{k}_{z,1}d}}\right)\right].\end{equation} \tag{ 20 }$$

While the point-dipole model accurately describes a number of quantum optical emitters, it is not necessarily suitable to model an exciton with a delocalized center of mass coordinate, nor the possible dependence of emission on the center of mass temperature. Defining Q = (Q_x , Q_y ) as the in-plane, center-of-mass momentum Q of the exciton, the emitted photon will necessarily have the same in-plane momentum in addition to an out-of-plane momentum component that we denote by ${Q}_{z}^{2}={k}_{1}^{2}-{\mathbf{Q}}^{2}$. In analogy to the point dipole case, the spontaneous emission can be calculated by modeling the extended exciton as a planar dipole, with current density [7, 19]

$$\begin{equation}\mathbf{J}(\boldsymbol{r})={\mathbf{J}}_{0}\delta (z-{z}_{0}){\text{e}}^{\text{i}\mathbf{Q}\cdot \boldsymbol{\rho }}\enspace {\text{e}}^{-\text{i}\omega t}.\end{equation} \tag{ 21 }$$

Here, ρ denotes the in-plane coordinate, while z₀ is the position of the planar dipole along z. This form of the current density assumes that the exciton can only move in the monolayer. The spontaneous emission rate of the exciton of momentum Q is given by

$$\begin{equation}{\Gamma}(\mathbf{Q})=-\frac{\alpha }{\vert {J}_{0}{\vert }^{2}}\enspace \mathrm{R}\mathrm{e}\left[\int \mathrm{d}\boldsymbol{r}{\mathbf{J}}^{{\dagger}}(\boldsymbol{r})\cdot \mathbf{E}(\boldsymbol{r})\right].\end{equation} \tag{ 22 }$$

Here, α is a geometry-independent coefficient, which only depends on the microscopic properties of the exciton. For a point dipole, a natural choice to remove this coefficient is made by normalizing the emission rate relative to a uniform medium, as in equation (1). For an extended exciton, however, we will soon see that an analogous choice can be problematic, and we will present several complementary possibilities.

To illustrate how the emission of an exciton differs from a point dipole, we begin by recovering well-known results [32] regarding radiative emission of an exciton in a uniform medium of permittivity ₁. In that case, we can exploit equations (2)–(4) to find the free (and total) field,

$$\begin{equation}\mathbf{E}(\boldsymbol{r})={\mathbf{E}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\boldsymbol{r})=-\frac{\omega {\mu }_{0}}{2}\enspace {\text{e}}^{\text{i}(\mathbf{Q}\cdot \boldsymbol{\rho }+{Q}_{z}\vert z-{z}_{0}\vert )}\enspace {\text{e}}^{-\text{i}\omega t}\mathbf{M}(\mathbf{Q})\cdot {\boldsymbol{J}}_{\mathbf{0}}.\end{equation} \tag{ 23 }$$

First, we can consider an exciton with a circular, in-plane transition, corresponding to ${\boldsymbol{J}}_{\mathbf{0}}={J}_{0}{\hat{\sigma }}^{\pm }=\frac{{J}_{0}}{\sqrt{2}}(\hat{x}\pm \mathrm{i}\hat{y})$ (where the sign difference corresponds to left-handed or right-handed polarization). In that case, a natural choice to normalize emission rates is relative to the emission rate of a stationary exciton in the uniform medium, which from equation (22) yields:

$$\begin{equation}\frac{{{\Gamma}}_{0}(Q)}{{{\Gamma}}_{0}(\mathbf{Q}=0)}=\frac{1}{2}\left[\frac{{Q}_{z}^{2}+{k}_{1}^{2}}{{Q}_{z}{k}_{1}}\right]\end{equation} \tag{ 24 }$$

for Q_z < k₁, and Γ₀(Q) = 0 otherwise. Here, the subscript in Γ₀(Q) refers to the emission in a uniform medium. Notably, the above result shows that the exciton can only radiate when its momentum can match that of a propagating photon.

On the other hand, for an out-of-plane transition, ${\boldsymbol{J}}_{\mathbf{0}}={J}_{0}\hat{z}$, one finds

$$\begin{equation}{{\Gamma}}_{0}(Q)=\frac{\alpha \omega {\mu }_{0}}{2{k}_{1}}\left(\frac{{Q}^{2}}{{k}_{1}{Q}_{z}}\right)\end{equation} \tag{ 25 }$$

for Q_z < k₁, and Γ₀(Q) = 0 otherwise. Unlike the in-plane exciton, a stationary out-of-plane exciton is dark, Γ₀(Q = 0) = 0, so the previous normalization procedure cannot be applied here. One practical and experimentally meaningful way to normalize can be to consider a non-zero motional temperature for the momentum Q, and calculating the thermally averaged emission rate, which we will introduce later.

The calculation of the spontaneous emission rate of an exciton near a planar interface (figure 1(c)) proceeds in an analogous fashion to the point dipole. It can readily be shown that for an in-plane, circularly polarized transition, the total emission rate is given by

$$\begin{equation}\frac{{\Gamma}(Q,{z}_{0})}{{{\Gamma}}_{0}(Q=0)}=\frac{1}{2}\left(\frac{{Q}_{z,1}^{2}+{k}_{1}^{2}}{{Q}_{z,1}{k}_{1}}+\mathrm{R}\mathrm{e}\left[-\frac{{Q}_{z,1}}{{k}_{1}}{r}_{p}\enspace {\mathrm{e}}^{2\mathrm{i}{Q}_{z,1}\vert {z}_{0}\vert }\right]+\mathrm{R}\mathrm{e}\left[\frac{{k}_{1}}{{Q}_{z,1}}{r}_{s}\enspace {\mathrm{e}}^{2\mathrm{i}{Q}_{z,1}\vert {z}_{0}\vert }\right]\right),\end{equation} \tag{ 26 }$$

where we have normalized the emission rate by that of a stationary exciton in the uniform medium. For the out-of-plane exciton, the total emission rate is:

$$\begin{equation}{\Gamma}(Q,{z}_{0})=\frac{\alpha \omega {\mu }_{0}}{2{k}_{1}}\left(\frac{{Q}^{2}}{{k}_{1}{Q}_{z,1}}+\mathrm{R}\mathrm{e}\left[{r}_{p}\frac{{Q}^{2}}{{k}_{1}{Q}_{z,1}}\enspace {\mathrm{e}}^{2\mathrm{i}{Q}_{z,1}\vert {z}_{0}\vert }\right]\right).\end{equation} \tag{ 27 }$$

Likewise, in the cavity structure illustrated in figure 1(d), the emission rate for an in-plane, circularly polarized transition is

$$\begin{equation}\frac{{\Gamma}(Q,d)}{{{\Gamma}}_{0}(Q=0)}=\frac{1}{2}\left[\frac{{({Q}_{z,1})}^{2}+{k}_{1}^{2}}{{Q}_{z,1}{k}_{1}}+\mathrm{R}\mathrm{e}\left[\left(\frac{{k}_{1}}{{Q}_{z,1}}\right)\frac{2{r}_{s}\enspace {\mathrm{e}}^{\mathrm{i}{Q}_{z,1}d}}{1-{r}_{s}\enspace {\text{e}}^{\text{i}{Q}_{z,1}d}}\right]+\mathrm{R}\mathrm{e}\left[-\left(\frac{{Q}_{z,1}}{{k}_{1}}\right)\frac{2{r}_{p}\enspace {\mathrm{e}}^{\mathrm{i}{Q}_{z,1}d}}{1+{r}_{p}\enspace {\text{e}}^{\text{i}{Q}_{z,1}d}}\right]\right],\end{equation} \tag{ 28 }$$

while for the out-of-plane transition,

$$\begin{equation}{\Gamma}(Q,d)=\frac{\alpha \omega {\mu }_{0}}{2{k}_{1}}\left(\frac{{Q}^{2}}{{k}_{1}{Q}_{z,1}}+\mathrm{R}\mathrm{e}\left[\left(\frac{{Q}^{2}}{{k}_{1}{Q}_{z,1}}\right)\frac{2{r}_{p}\enspace {\text{e}}^{\text{i}{Q}_{z,1}d}}{1-{r}_{p}\enspace {\text{e}}^{\text{i}{Q}_{z,1}d}}\right]\right).\end{equation} \tag{ 29 }$$

With the preceeding formalism, we now consider the modified emission rates of point dipoles in the planar structures detailed in figure 1. In the calculations of the main text, we consider metal interfaces of silver, while the results for several other metals are presented in the appendix (http://stacks.iop.org/NJP/24/ 023015/mmedia). We take the permittivity of silver to be _s = −27.397 + 0.896i [33], which is accurate at the wavelength of the neutral exciton transition in MoSe₂ (λ = 755 nm) [19]. For simplicity, we consider the dielectric medium that surrounds the exciton to be vacuum (₁ = 1).

In the single-interface case, the reflection coefficient r_p (k_∥) (equation (13)) contains a pole if _s is negative, corresponding to a surface plasmon polariton (SPP) mode with in-plane wavevector component [34]

$$\begin{equation}{k}_{\mathrm{S}\mathrm{P}\mathrm{P}}=\sqrt{\frac{{{\epsilon}}_{1}{{\epsilon}}_{s}(\omega )}{{{\epsilon}}_{1}+{{\epsilon}}_{s}(\omega )}}\approx 1.019{k}_{1}\end{equation} \tag{ 30 }$$

for the silver parameters given above. For the symmetric silver cavity, the SPP wavevector is given by the solution to

$$\begin{equation}1-{r}_{p}^{2}({k}_{\mathrm{S}\mathrm{P}\mathrm{P}}){\mathrm{e}}^{2\mathrm{i}{k}_{z,1}d}=0.\end{equation} \tag{ 31 }$$

We plot the solutions for k_SPP as a function of the cavity separation d in figure 2(a).

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In figure 2(b), we plot the modified emission rates for the in-plane and out-of-plane point dipoles at a single silver interface (equations (14) and (15)), as a function of distance z₀. At large distances, the emission rate behavior is oscillatory, resulting from interference between the emitted and reflected fields. Moreover, the short distance (z₀ → 0) behavior scales as ${{\Gamma}}_{\mathrm{n}\mathrm{r}}/{{\Gamma}}_{0}\propto \left(\frac{1}{{z}_{0}^{3}}\right)\cdot \mathrm{I}\mathrm{m}\left[\frac{{{\epsilon}}_{s}-1}{{{\epsilon}}_{s}+1}\right]$. This emission is non-radiative, and emerges from the near-field component $(1/{z}_{0}^{3})$ of the dipole field. It also requires the permittivity of the silver to have an imaginary part, reflecting the energy loss produced by Ohmic dissipation of the induced currents in the silver. Mathematically, this contribution arises from the large k_∥ > k₁ tails of the integrands in equations (14) and (15). At intermediate distances, the out-of-plane dipole emits significantly into the SPP modes, due to the purely p-polarized nature of both the dipole radiation and the SPP modes. Assuming that _s is real and evaluating the pole contribution to equation (15) gives:

$$\begin{equation}\frac{{{\Gamma}}_{\mathrm{S}\mathrm{P}\mathrm{P}}}{{{\Gamma}}_{0}}\approx 1-\frac{3\pi {k}_{\mathrm{S}\mathrm{P}\mathrm{P}}^{3}}{{{\epsilon}}_{1}^{3/2}{k}_{1}^{2}\sqrt{{k}_{\mathrm{S}\mathrm{P}\mathrm{P}}^{2}-{k}_{1}^{2}}}\enspace \mathrm{R}\mathrm{e}\left[\frac{{{\epsilon}}_{s}^{3/2}}{\sqrt{1+{{\epsilon}}_{s}}(1+{{\epsilon}}_{s}^{2})}\right]{\mathrm{e}}^{-2\vert {k}_{z,\mathrm{S}\mathrm{P}\mathrm{P}}\vert {z}_{0}}.\end{equation} \tag{ 32 }$$

We plot the modified emission rate versus d for the cavity in figure 2(c). In this case, the in-plane dipole exhibits regions of enhancement and suppression with a saw-tooth behavior, in agreement with the well-known results for electric dipole transitions [24]. Coupling to SPP modes is prohibited for the in-plane dipole for this symmetric cavity geometry. For the out-of-plane orientation, a large enhancement of emission is observed at short distances d/λ ≲ 0.5, which is attributable to the SPP modes. The approximate SPP emission rate can be calculated in the near-field (d → 0) limit, where the SPP wavevector can be approximated as:

$$\begin{equation}{k}_{\mathrm{S}\mathrm{P}\mathrm{P}}\approx \mathrm{ln}\left[\frac{{{\epsilon}}_{s}-{{\epsilon}}_{1}}{{{\epsilon}}_{s}+{{\epsilon}}_{1}}\right]\frac{1}{d}.\end{equation} \tag{ 33 }$$

The corresponding tight field confinement and small mode volume at small d give rise to a significantly enhanced decay rate into SPPs:

$$\begin{equation}\frac{{{\Gamma}}_{\mathrm{S}\mathrm{P}\mathrm{P}}}{{{\Gamma}}_{0}}\approx \frac{3\pi }{{({k}_{1}d)}^{3}}\enspace \mathrm{R}\mathrm{e}\left[{\mathrm{ln}}^{2}\left[\frac{{{\epsilon}}_{s}-{{\epsilon}}_{1}}{{{\epsilon}}_{s}+{{\epsilon}}_{1}}\right]\right].\end{equation} \tag{ 34 }$$

Similar to the single-interface analysis, at short distances d → 0 both dipole orientations also experience a non-radiative emission rate that scales like Γ_nr ∝ (Im _s )/d³ due to Ohmic dissipation. However, this contribution always remains smaller than the SPP emission for the out-of-plane dipole, which also has a 1/d³ scaling.

In this section, we present results for the momentum-resolved emission rate of an extended exciton for the single interface and cavity structures. Figure 3(a) shows the normalized emission rate Γ(Q, z₀)/Γ₀(Q = 0) for a single interface as a function of the center-of-mass momentum Q and the distance z₀ from the surface for an in-plane transition as given by equation (26). In figure 3(b), we plot an analogous quantity ${\Gamma}(Q,{z}_{0})/{{\Gamma}}_{0}(Q={k}_{1}/\sqrt{2})$ for the out-of-plane case, equation (27). Here, we have chosen the (arbitrary) normalization of an exciton in a uniform medium with momentum $Q={k}_{1}/\sqrt{2}$, as Q = 0 yields a zero emission rate. In both the in-plane and out-of-plane cases, the emission rate for a fixed momentum in the radiative region (Q < k₁) is oscillatory as the exciton position z₀ is varied, much like the oscillations seen in the point dipole case. We note, however, that for the extended exciton, the visibility of the oscillations does not decrease with increasing z₀. In the non-radiative region (Q > k₁), one observes an emission into SPP's when Q = k_SPP, and lossy emission for other Q, provided that the exciton is close enough to couple evanescently, z₀ ≲ 1/|Q_z |.

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One particular limit is the case of zero motional temperature, Q = 0. In figure 2(b), we compare the normalized emission Γ(Q = 0, z₀)/Γ₀(Q = 0) for an in-plane transition of an extended exciton at zero temperature with that Γ(z₀)/Γ₀ of a point dipole. The extended exciton curve is notably different in its undamped oscillations with z₀, and the absence of a near-field non-radiative emission for small z₀. This is because the non-radiative emission comes from the contribution of evanescent waves, which have momenta Q > k₁, the wave vector of a radiative photon in medium 1. Thus, at T = 0, an extended exciton with only Q = 0 has no non-radiative emission. In the out-of-plane case, the extended exciton has a zero-temperature emission rate Γ(Q = 0, z₀) = 0, unlike the point dipole case, as seen from equation (25) for Q = 0. Physically, an oscillating, spatially uniform (Q = 0) planar current density must emit electromagnetic radiation in a direction orthogonal to the plane, with a resulting electric field polarization that is parallel to the plane. As there can be no preferred direction of this polarization due to rotational symmetry, the field and thus the emission rate must be zero.

In figures 3(c) and (d), we repeat the calculations of the momentum and distance dependent emission rates, Γ(Q, d)/Γ₀(Q = 0) and ${\Gamma}(Q,d)/{{\Gamma}}_{0}(Q={k}_{1}/\sqrt{2})$, for in-plane and out-of-plane transitions, respectively, this time for the cavity geometry. Now in the radiative region Q < k₁, the emission is sharply enhanced for cavity separation distances d that yield resonances at the exciton emission frequency. These resonance conditions are determined by the poles of the terms in equations (28) and (29). In the case of a nearly perfect conductor _s → −∞, the resonance condition at normal incidence Q = 0 would occur at d/λ = m + 1/2 for non-negative integer m. The splitting of the resonance condition for Q ≠ 0, observed in the in-plane case, arises due to the slight phase difference between the s- and p-reflection coefficients for the finite permittivity of silver. Similar to the single interface, the emission rate is periodic in d, but now with a Purcell-enhanced maximum on resonance that scales with the cavity finesse, $\sim 1/(1-\vert r{\vert }^{2})$. For Q > k₁, the out-of-plane transition also exhibits a sharp feature following the SPP dispersion relation given in equation (33), Q ≈ k_SPP, while the in-plane transition does not couple to SPP's for the symmetric geometry considered here.

Here, we study the radiative emission of thermalized in-plane polarized excitons near a single metal interface using the same material parameters as in section 3. We take the exciton mass to be m = m_e + m_h where m_e = 0.49m₀ and m_h = 0.52m₀ for MoSe₂ are obtained from DFT calculations (m₀ is the electron mass) [45, 46]. For the MB distribution (equation (37)), we use the Q-dependent emission rates given in equations (24) and (26) to calculate the normalized, average emission rate, ⟨Γ⟩/⟨Γ₀⟩, for the in-plane, circularly polarized plane dipole at non-zero temperatures. We plot the thermally averaged emission rate as a function of temperature and distance z₀, in figure 4(a) and provide the corresponding value of T/T_c on the upper axis. The characteristic energy scale defined by ${k}_{\mathrm{B}}{T}_{\mathrm{c}}={(\hslash {k}_{1})}^{2}/2m$ corresponds to when the exciton has the same momentum as the radiative photon (T_c ∼ 30 mK for our parameters). We find that for z₀ > 0.5λ, the emission rates for different temperatures behave similarly, oscillating interferometrically as a function of z₀ with a visibility that increases for T ≲ T_c. However, at small distances, when the 2D TMDC is placed close to the metal interface (z₀ < 0.03λ), the differences in their temperature dependence become apparent. This is evident in figure 4(c), where we plot the emission rate as a function of z₀ for various temperatures. Interestingly, as the temperature of the system increases, the functional form of ⟨Γ⟩/⟨Γ₀⟩ approaches that of a point dipole, Γ(z₀)/Γ₀, as given by equation (14) and plotted with the dashed black curve. At sufficiently low temperatures (T ∼ T_c), the exciton momentum distribution contains a negligible contribution from non-radiative components Q > k₁, which results in a significant suppression of non-radiative emission at short distances, as compared to the high temperature and point-dipole cases where non-radiative emission is dominant.

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We now calculate the average emission rate associated with a BE distribution, using equations (24), (26) and (38). Here, we choose a fixed total density of n = 10¹² cm⁻² typical of interlayer exciton systems [4] and use the same exciton mass m as before. In figure 4(b), we plot the emission rate ⟨Γ⟩/⟨Γ₀⟩ as a function of distance z₀ and temperature T. For convenience, we also indicate the corresponding value of $n{\lambda }_{\mathrm{d}B}^{2}$ on the upper x-axis. In particular, one sees that a qualitative change in behavior occurs around $n{\lambda }_{\mathrm{d}B}^{2}\sim 1$, with values $n{\lambda }_{\mathrm{d}B}^{2}\lesssim 1$ approaching the MB emission properties discussed earlier. On the other hand, for values $n{\lambda }_{\mathrm{d}B}^{2}\hspace{2pt}\gtrsim \hspace{2pt}1$, bosonic enhancement of low-energy (Q ∼ 0) states yields more prominent oscillations in the emission at large distances z₀, reminiscent of that of a planar dipole with center-of-mass momentum Q = 0 (see figure 3(a)). This similarity can be better seen in figure 4(d), where we plot ⟨Γ⟩/⟨Γ₀⟩ as a function of distance, for several different temperatures. In particular, we compare the results obtained from the MB and BE distributions (red and blue curves, respectively), with the emission calculated at T = 0 (black curve). It is seen that for $n{\lambda }_{\mathrm{d}B}^{2}\gg 1$, the BE curve essentially approaches the T = 0 result, even though the temperature T ≫ T_c is still much larger than that to narrow the single-particle MB momentum distribution to only radiative components. Conversely, for $n{\lambda }_{\mathrm{d}B}^{2}\ll 1$, the emission curves for the MB and BE distributions coincide. Finally, in figure 4(e), we plot the emission rate ⟨Γ⟩/⟨Γ₀⟩ obtained by the BE distribution, as a function of temperature T and total density n, for a fixed distance of z₀ = 0.011λ nm. The dashed contours indicate the values of $n{\lambda }_{\mathrm{d}B}^{2}$. The transition around $n{\lambda }_{\mathrm{d}B}^{2}\sim 1$ from the MB result to essentially the T = 0 result (for $n{\lambda }_{\mathrm{d}B}^{2}\gg 1$) is also evident here.

We can repeat the calculations of the previous subsection, but now for an out-of-plane transition. In particular, in figure 5(a), we plot the normalized average emission rate ⟨Γ⟩/⟨Γ₀⟩ as a function of distance and temperature, assuming an MB distribution, while in figure 5(c), we provide line cuts showing the distance dependence for several specific temperatures. In figure 5(c), we also plot the normalized emission rate Γ(z₀)/Γ₀ for a point dipole (dashed black curve). Just as in the case of an in-plane transition, the out-of-plane transition results converge to the point dipole one at sufficiently large temperatures and distances. Also as before, for small distances and low temperature, suppression of non-radiative emission is responsible for the divergence of the planar and point dipole results from one another.

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In figure 5(b), we plot the average emission rate versus distance and temperature for the BE distribution, taking the same total density n = 10¹² cm⁻² as in section 5.1. Again, one observes a qualitative transition in behavior around $n{\lambda }_{\mathrm{d}B}^{2}\sim 1$.

We now use equations (24), (28) and (38) to calculate the normalized average emission rate for the cavity configuration. In figures 6(a) and (b), we plot the temperature and distance (d) dependent rates for the MB and BE distributions, respectively. In the latter case, we again take a total density n = 10¹² cm⁻². In figure 6(c), we plot linecuts of the MB results at select temperatures at small distances (left), and at larger distances (right), where the latter shows the strong sawtooth shaped Purcell enhancement associated with the cavity. Similar to the single-interface case, we find that as the temperature increases, the plane-dipole model begins to behave like the point dipole model (dashed black curves of figure 6(c)) as the high-Q states that contribute to non-radiative emission become populated. For T ⩾ 5 K, we find that the minimum emission rate is ⟨Γ⟩/⟨Γ₀⟩ ≈ 0.01 for a cavity separation of d ∼ 0.27λ and is sharply enhanced for smaller separations, d < 0.27λ, due to non-radiative decay. In comparison, this non-radiative decay becomes negligible at short distances once T ∼ T_c. In the Purcell-enhanced region, it can be seen that low temperatures T ≲ T_c can result in a greater level of spontaneous emission enhancement, as one would expect as the Q distribution narrows toward Q = 0 (compare with figure 3(c) at Q = 0).

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For BE distributions with $n{\lambda }_{\mathrm{d}B}^{2}\hspace{2pt}\gtrsim \hspace{2pt}1$, bosonic enhancement allows for the normalized emission rate to come close to the T = 0 case, similar to what was observed at a single interface. This is evident in figure 6(d), where we plot the distance-dependent emission rates at a fixed temperature of T = 5 K $(n{\lambda }_{\mathrm{d}B}^{2}=11.01)$, using the MB and BE distributions (red and blue, respectively), and also the T = 0 result (black dashed). Notably, for example, a maximum Purcell enhancement of ⟨Γ⟩/⟨Γ₀⟩ ∼ 10², as allowed for T = 0, is also observed for the BE distribution. In figures 6(e) and (f), we now consider the emission rate versus temperature and total density, at a close distance of d = 0.011λ and large distance of d = 0.439λ, respectively. The transition from MB to BE behavior around $n{\lambda }_{\mathrm{d}B}^{2}\sim 1$ is evident. Notably, at the distance of d = 0.439λ, Purcell enhancements on the order of $\sim 1{0}^{2}$ are observed when $n{\lambda }_{\mathrm{d}B}^{2}\gg 1$.

An important caveat to these discussions of orders of magnitude of enhancement is the possibility of strong coupling between the excitons and the metal cavity, in which case the calculations are no longer valid. For a metal cavity with $\sim 5$ THz linewidth, interlayer excitons with emission rates in the 10–100 MHz regime would remain weakly coupled even with the $\sim 100$ times Purcell enhancement shown in figure 6(d) [3]. However, for individual monolayers where excitons can have THz radiative decay rates, the system would enter the strong coupling regime and the calculations are no longer valid [46].

We now consider an out-of-plane transition in the cavity structure. Figure 7(a) plots the emission rate dependence on temperature and cavity separation d for the MB distribution. At low temperatures and small separations d, the large non-radiative emission present at high temperatures becomes significantly suppressed. Figure 7(b) shows the emission rate calculated using the BE distribution. As before, a large phase space density allows the behavior associated with the MB distribution and low temperatures T < T_c to be observed at much higher temperatures. Figure 7(c) plots linecuts of the emission rate for the MB distribution at select temperatures. We find that at T = T_c, the non-radiative emission is suppressed by several orders of magnitude compared to higher temperatures and does not monotonically grow as d → 0, unlike the point dipole case.

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