Quantized fluctuational electrodynamics for three-dimensional plasmonic structures

We recently introduced a quantized fluctuational electrodynamics (QFED) formalism that provides a physically insightful definition of an effective position-dependent photon-number operator and the associated ladder operators. However, this far the formalism has been applicable only for the normal incidence of the electromagnetic field in planar structures. In this work, we overcome the main limitation of the one-dimensional QFED formalism by extending the model to three dimensions, allowing us to use the QFED method to study, e.g., plasmonic structures. To demonstrate the benefits of the developed formalism, we apply it to study the local steady-state photon numbers and field temperatures in a light-emitting near-surface InGaN quantum-well structure with a metallic coating supporting surface plasmons.

Figure 1

The studied structure formed by a $\mathrm{Ag}/\mathrm{GaN}/{\mathrm{In}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}/\mathrm{GaN}/{\mathrm{Al}}_2{\mathrm{O}}_3$ heterostructure. The background temperature is $T=300$ K, the band gap of the light emitting ${\mathrm{In}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ QW is $E_g=2.76$ eV, and the QW excitation corresponds to an applied voltage of $U=2.6$ V. Note that the figure is not to scale.

Figure 2

(a) The base-10 logarithm of the total EM LDOS, (b) the effective temperature of the total EM field in the case of a thermally excited QW, and (c) the effective-field temperature in the case of an electrically or optically excited QW corresponding to the bias voltage $U=2.6$ V for photon energy $\hslash \omega =E_g+k_{\mathrm{B}}T=2.786$ eV as a function of position and the in-plane component of the wave vector. The position $z=0$ is fixed to the Ag/air interface. The white dashed lines represent the light cones of GaN, sapphire, and air.

Figure 3

The base-10 logarithm of the total EM LDOS as a function of photon energy and the in-plane component of the wave vector (a) in the QW, (b) in air at 1 nm above the surface, and (c) in air at $1\mu \mathrm{m}$ above the surface. (d), (e), and (f) The effective temperature of the total EM field at the corresponding positions in the case of a thermally excited QW. (g), (h), and (i) The effective-field temperature at the corresponding positions in the case of an electrically or optically excited QW corresponding to the bias voltage $U=2.6$ V.