Tunable negative permeability in a quantum plasmonic metamaterial

We consider the integration of quantum emitters into a negative permeability metamaterial design in order to introduce tunability as well as nonlinear behavior. The unit cell of our metamaterial is a ring of metamolecules, each consisting of a metal nanoparticle and a two-level semiconductor quantum dot (QD). Without the QDs, the ring of the unit cell is known to act as an artificial optical magnetic resonator. By adding the QDs we show that a Fano interference profile is introduced into the magnetic field scattered from the ring. This induced interference is shown to cause an appreciable effect in the collective magnetic resonance of the unit cell. We find that the interference provides a means to tune the response of the negative permeability metamaterial. The exploitation of the QD's inherent nonlinearity is proposed to modulate the metamaterial's magnetic response with a separate control field.

Figure 1

Schematic of the quantum plasmonic metamaterial. This includes a detailed sketch of one of the unit cells consisting of a MNP-QD nanoring. The unit cell has nanoscale dimensions: The radius of the MNP nanoring, QD nanoring, MNPs, and QDs are 38, 6, 16, and $\approx$1 nm, respectively. An arbitrary optical field can be injected into the metamaterial and the figure shows an example of a transverse plane wave at the center of a focused beam. However, in order to isolate the ring's magnetic dipole response, in our study we consider a quasistatic magnetic field $H$ directed along the $\widehat{\mathbf{z}}$ axis, as shown in the unit cell inset. This is a standard approach used to isolate the electric and magnetic response of a metamaterial [18, 36]. The MNPs and the QDs are excited by the induced azimuthal E field, as shown in the inset, with no net electric field, thereby isolating the unit cell's magnetic response. The interaction between the QDs and MNP fields at each site, or “metamolecule,” cause a Fano profile to appear in the scattered magnetic field of the MNP ring. This induces interference effects in the bulk magnetic response of the metamaterial which enable it to become tunable. The configuration shown is for a material with a magnetic response in the $\widehat{\mathbf{z}}$ direction only and the material is therefore anisotropic. To make an isotropic material with the same response in all directions, a cubic lattice consisting of three orthogonal arrays of nanorings should be used. This can be achieved in a face-centered cubic lattice, where up to four different nanoring orientations can be included in a single unit cell.

Figure 2

Imaginary (a) and real (b) parts of the polarizability of a MNP-QD metamolecule, whose individual dipole radii are 16 nm and $0.9$ nm, respectively, and whose separation distance is 32 nm. The MNP and QD are resonant in this example and they couple transversely ($S=-1$). The system is encased in a background medium of permittivity ${\epsilon }_b=2.2$ and is driven weakly, $E_0\mu =0.0001$ meV, where $\frac{E_0\mu }{\hslash }$ is the coupling frequency between the QD and the driving field. In (c) a sketch of the system shows that the MNP and QD are transversely coupled. In (d) we show the imaginary polarizability around the resonance frequency, highlighting the Fano dip.

Figure 3

Imaginary (a) and real (b) parts of the effective permeability of an MNP nanoring metamaterial. The ring radius $R$ and MNP radius $r$ are 38 and 16 nm, respectively. The MNPs are encased in a material with permittivity ${\epsilon }_b=2.2$ and permeability ${\mu }_b=1$. We plot the effective permeability for three values of $N$ in each nanoring, $N$ = 2 (blue line), $N$ = 3 (green dashed line), and $N$ = 4 (red dotted line), where $N_d$ = (96 nm)${}^{-3}$. The vertical dashed line corresponds to the electric resonance of a single MNP. In (c) we show a sketch of the nanoring system.

Figure 4

Imaginary (a) and real (b) parts of ${\mu }_{\mathrm{eff}}$ for a QD-MNP nanoring metamaterial. The parameters of the system are chosen to be the same as in Figs. 2 and 3, with a MNP-QD detuning $\Delta =({\omega }_0-{\omega }_x)=0.195\times {10}^{15}$ rad s${}^{-1}$. In (c) a sketch of the system is shown, where the unit cell now consists of two interacting rings, the MNP ring and the QD ring. The arrows on the MNPs show the electric field direction at each MNP and their direction also represents their adjacent QD's dipole orientation. In (d) we examine the difference between Re(${\mu }_{\mathrm{eff}}$) with (red) and without (black) QDs in the nanoring.

Figure 5

Tunability of the MNP-QD nanoring metamaterial. The real part of ${\mu }_{\mathrm{eff}}$ for different MNP-QD detunings. We plot the following: (1) $\Delta =0.195$, (2) $\Delta =0.196$, and (3) $\Delta =0.197$. All detunings are in units of ${10}^{15}$ rad s${}^{-1}$. The dashed line shows the bare MNP nanoring in the absence of QDs.

Figure 6

Nonlinear response of the MNP-QD metamolecule. A comparison of the imaginary [(a) and (c)] and real [(b) and (d)] parts of the polarizability for a weak external driving field (top row) with $E_0\mu =0.0001$ meV, and a strong driving field (bottom row) with $E_0\mu =0.1$ meV. The MNP-QD detuning is $\Delta =0.195\times {10}^{15}$rad s${}^{-1}$.