Intersubband surface plasmon polaritons in all-semiconductor planar plasmonic resonators

We theoretically discuss properties of intersubband surface plasmon polaritons (ISPPs) supported by the system consisting of a multiple quantum well (MQW) slab embedded into planar resonator with highly doped semiconducting claddings playing the role of cavity mirrors. Symmetric structures, where the MQW slab occupies the whole space between the claddings and asymmetric structures, where the MQW occupy only half of the space between mirrors, are considered. We focus mainly on the nearly degenerate structures where intersubband frequency is close to frequency of the surface plasmon of the mirrors. The ISPP characteristics are calculated numerically using a semiclassical approach based on the transfer matrix formalism and the effective-medium approximation. The claddings are described by the lossless Drude model. The possibility of engineering the dispersion of the ISPP branches is demonstrated. In particular, for certain parameters of the asymmetric structures we observe the formation of the multimode ISPP branches with two zero group velocity points. We show that the properties of the ISPP branches are reasonably well interpreted employing quasiparticle picture provided that the concept of the mode overlap factor is generalized, taking into account the dispersive character of the mirrors. In addition to this, we demonstrate that the lossless dispersion characteristics of the ISPP branches obtained in the paper are consistent with the angle-resolved reflection-absorption spectra of the GaAlAs-based realistic plasmonic resonators.

Figure 1

Diagram illustrating schematically the geometry of the structures discussed in the paper: (a) the three-layer structure and (b) the four-layer structure.

Figure 2

The normalized frequencies of the modes $S_{\mathrm{SPP}},A_{\mathrm{SPP}},I_{\mathrm{SPP}}$ and ${\mathrm{TM}}_2$ as a function of ${\overline{k}}_x$ (solid curves). For comparison, we also present results predicted by simplified Eq. (42) (black dotted line). The short-dotted (dashed-dotted) line corresponds to the spacer light line (the normalized surface plasma frequency ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}$). The area located above the boundary curve ${\overline{\alpha }}_{\mathrm{mirr}}=0$ is shaded. The inset displays ${\overline{k}}_x$ dependence of the parameters ${\overline{\alpha }}_{\mathrm{mirr}/\mathrm{spac},N}$ corresponding to the modes $S_{\mathrm{SPP}}$ ($N=S),A_{\mathrm{SPP}}$ ($N=A)$ and $I_{\mathrm{SPP}}(N=I)$. The dotted curves in the inset correspond to Im $\left({\overline{\alpha }}_{\mathrm{spac},A}\right),{\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.5$.

Figure 3

Same as in Fig. 2, except that now ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.25$.

Figure 4

The ${\overline{\omega }}_{p,\mathrm{mirr}}$ dependence of the normalized cutoff frequency ${\overline{\omega }}_A^{\mathrm{cutoff}}$ (thick solid line) and the normalized crossing frequency ${\overline{\omega }}_A^{\mathrm{cross}}$ (thin solid line). The dashed line corresponds to the normalized surface plasmon frequency ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}$.

Figure 5

The normalized frequencies of the $A_{\mathrm{SPP}}$ mode and the $A_{\mathrm{ISPP}}^U$ polariton branch supported by the three-layer structures, with different values of ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}$, as a function of ${\overline{k}}_x$. The short-dotted (dashed-dotted) line corresponds to the light line (the normalized frequency ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}$). ${\overline{\varpi }}_{p,\mathrm{MQW}}=0.3\times {\overline{\omega }}_{\mathrm{IT}}$.

Figure 6

The thick (thin) solid curves present the ${\overline{k}}_x$ dependence of the overlap factors corresponding to the modes $S_{\mathrm{SPP}}$ and $A_{\mathrm{SPP}}$ supported by resonator with ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.5$ (${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.25$). For comparison, we present also the ${\overline{k}}_x$ dependence of the overlap factors corresponding to the modes ${\mathrm{TM}}_0\equiv \mathrm{TEM}$ and ${\mathrm{TM}}_1$ supported by the resonators with perfect metallic mirrors (the dotted curves).

Figure 7

The normalized frequencies of the ISPP branches supported by the three-layer structure as a function of ${\overline{k}}_x$ (the thick solid curves). For illustration, the normalized frequencies of the plasmonic modes supported by the passive structure are also displayed (the thin solid curves). The short-dotted (dashed-dotted) line corresponds to the light line (the normalized intersubband frequency ${\overline{\omega }}_{\mathrm{IT}}$ and surface plasma frequency ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}$). The interpretation of the black and gray horizontal arrows is given in the main text. The area where in-gap intersubband polariton branches are located is hatched. The regions ${\mathcal{S}}_{\mathrm{MQW}}^U$ and ${\mathcal{S}}_{\mathrm{MQW}}^L$ where the intersubband polariton branches have surface character are light gray shaded.${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}=0.5$ and ${\overline{\varpi }}_{p,\mathrm{MQW}}=0.3\times {\overline{\omega }}_{\mathrm{IT}}$.

Figure 8

Same as in Fig. 7, except that now ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}=0.25$. Moreover, for illustration we present additionally the dispersion of the branches $A_{\mathrm{ISPP}}^U$ and $A_{\mathrm{ISPP}}^L$ obtained employing Eq. (26) (open circles).

Figure 9

The normalized frequencies of the branches $S_{\mathrm{ISPP}}^U$ and $S_{\mathrm{ISPP}}^L$ supported by the nondegenerate structures as a function of ${\overline{k}}_x$. Thick solid curves represent exact results predicted by Eq. (12). Open circles are obtained employing the generalized secular equation (26) based on the single-mode cavity approximation. Thin solid (dashed) curves represent ${\overline{k}}_x$ dependence of the $S_{\mathrm{SPP}}$ (TEM) mode. Calculations have been performed taking (a) ${\overline{\omega }}_{\mathrm{IT}}=0.1$ and (b) ${\overline{\omega }}_{\mathrm{IT}}=0.4$. ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.5$ and ${\overline{\varpi }}_{p,\mathrm{MQW}}=0.3\times {\overline{\omega }}_{\mathrm{IT}}.$

Figure 10

The normalized frequencies of the hybrid ISPP branches supported by the four-layer structure as a function of ${\overline{k}}_x$ (the thick solid curves). For illustration, the normalized frequency of the modes $A_{\mathrm{SPP}}$ and $S_{\mathrm{SPP}}$, as well as the normalized frequencies of the branches $A_{\mathrm{ISPP}}^U$ and $A_{\mathrm{ISPP}}^L$ supported by the three-layer structure with ${\overline{\varpi }}_{p,\mathrm{MQW}}=0.3\times {\overline{\omega }}_{\mathrm{IT}}$ are also displayed (the thin solid curves). The short-dotted (dashed-dotted) line corresponds to the spacer light line (the normalized intersubband frequency ${\overline{\omega }}_{\mathrm{IT}}$ and surface plasma frequency ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}$). The interpretation of the scan line (dashed line) is given in Appendix pp4. The hatched area indicates the $\overline{\omega }$ region where the in-gap polariton branches are located. ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}=0.5$ and ${\overline{\varpi }}_{p,\mathrm{MQW}}=\sqrt{2}\times (0.3\times {\overline{\omega }}_{\mathrm{IT}}).$

Figure 11

Same as in Fig. 10, except that now ${\overline{\omega }}_{p,\mathrm{mirr}}={\overline{\omega }}_{\mathrm{IT}}=0.25.$

Figure 12

The ${\overline{k}}_x$ dependence of reciprocal of the normalized effective mode length corresponding to the modes $S_{\mathrm{SPP}}$ (black solid curves), $A_{\mathrm{SPP}}$ (gray solid curves), and $I_{\mathrm{SPP}}$ (dotted curves) calculated for different values of the normalized distance ${\underline{z}}_{\mathrm{QW}}=(L_{\mathrm{MC}}/2-z)/L_{\mathrm{MC}}$. (a) The structure with ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.5$ and (b) the structure with ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}=0.25$.

Figure 13

The spectral dependence of the absorptance of the passive structure (open circles), the three-layer (black solid curves) and four-layer (gray solid and dashed curves) structures described in the text. The gray dashed (solid) curve corresponds to the four-layer structure with the MQW located near the coupling (back) mirror. The calculations have been performed taking $\hslash {\gamma }_{\mathrm{IT}}=0.5$ meV and $\hslash {\gamma }_{\mathrm{IT}}=0.005$ meV (see the inset). ${\omega }_{p,\mathrm{mirr}}^{\mathrm{surf}}={\omega }_{\mathrm{IT}}=69.3$ meV and $L_{\mathrm{MC}}=13.468\mu \mathrm{m}.$ It corresponds to ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}=0.5$.

Figure 14

Same as in Fig. 12, except that now $L_{\mathrm{MC}}=6.734\mu \mathrm{m}$. It corresponds to ${\overline{\omega }}_{p,\mathrm{mirr}}^{\mathrm{surf}}={\overline{\omega }}_{\mathrm{IT}}=0.25$.