Semiconductor Quantum Plasmonics

We investigate the frontier between classical and quantum plasmonics in highly doped semiconductor layers. The choice of a semiconductor platform instead of metals for our study permits an accurate description of the quantum nature of the electrons constituting the plasmonic response, which is a crucial requirement for quantum plasmonics. Our quantum model allows us to calculate the collective plasmonic resonances from the electronic states determined by an arbitrary one-dimensional potential. Our approach is corroborated with experimental spectra, realized on a single quantum well, in which higher order longitudinal plasmonic modes are present. We demonstrate that their energy depends on the plasma energy, as is also the case for metals, but also on the size confinement of the constituent electrons. This work opens the way toward the applicability of quantum engineering techniques for semiconductor plasmonics.

Figure 1

(a) Sketch of a doped semiconductor layer embedded between two undoped layers. If the thickness $L$ is smaller than the plasma wavelength in the material ${\lambda }_P=2\pi c/{\mathrm{\Omega }}_P$, it is possible to optically excite a Berreman mode at $E_P=\hslash {\mathrm{\Omega }}_P$. (b) Dispersion of a volume plasmon in a classical model. (c) Effect of size confinement in the optical response of a semiconductor QW with three occupied electronic states. The optical response of the system is a collective resonance at a greater energy than $E_P$ due to size confinement contribution. (d) Longitudinal dispersion of a volume plasmon taking into account the material band structure and the Fermi distribution of electrons.

Figure 2

Measured (red lines) and simulated (black lines) emission spectra under thermal excitation of the plasmons for different values of the internal angle of light propagation. The thermal emission spectra have been simulated by solving quantum Langevin equations in the input-output formalism [34]. The plasmon eigenmodes have been calculated from a full numerical diagonalization accounting for the finite barrier height as well as for nonparabolicity effects [35].

Figure 3

(a),(b) Normalized microcurrents ${\xi }_{i\to i+j}(z)$ calculated for an infinite barrier QW for $j=1$ (a) and $j=3$ (b) and $i=2$ (black), $i=6$ (red), $i=12$ (green) and in the limit of $i\gg 1$ (blue). (c) Schematic representation of a square QW for the plasmon modes (dashed line). We have plotted in this effective potential the plasmon microcurrents (black continuous lines) of the $j=1$, $j=3$, and $j=5$ modes, with $j$ the quantum number of the confined longitudinal wave vector. The charge density distributions are also plotted for each longitudinal mode.

Figure 4

(a) Squared energy of the plasmon modes extracted from the measured thermal emission spectra (red squares) plotted as a function of the quantum number $j$ of the corresponding confined plasmon. The line shows the result of Eq. (6). (b),(c) Squared plasmon frequencies numerically calculated with our full quantum model (bullets) compared to the results of Eq. (6) (lines). In panel (b), the electronic density has been set to $N_v=7.5\times {10}^{18}{\mathrm{cm}}^{-3}$, while the QW thickness is varied and in panel (c) $L=100\mathrm{nm}$ and the electronic density is varied. The experimental results are also reported for comparison (squares).

Figure 5

Upper panel: Calculated absorptivity for a square GaInAs highly doped semiconductor layer with thickness 54 nm and electronic density $N_v=2\times {10}^{19}{\mathrm{cm}}^{-3}$. Lower panel: Calculated absorptivity for a system of asymmetric tunnel coupled GaInAs/AlInAs quantum wells with the same electronic density as the 54 nm structure. The conduction band profile and the square moduli of the wave functions for the two structures are presented as insets. The black horizontal lines indicate the Fermi energy.