1. Introduction

Metamaterials rely on arrays of miniature circuit-resonators, such as split-rings, that can confine the electromagnetic field into sub-wavelength volumes [1], down to nanometer sizes [2–4]. This property allows highly efficient surface sensing [5–7], very low-dark current quantum detectors [8–10], modulators [11] and new schemes for exploring the ultra-strong light-matter interaction [3, 12–14]. The majority of meta-materials, regardless their operation frequency are obtained by using a two-dimensional planar technology [3, 14, 15]. The interaction between the electric field and the active material hosted in the substrate is therefore not optimized as it relies only on the fringing fields that leak out from the circuit capacitor [16, 17]. In the THz region, other designs for metamaterial-like LC-resonators based on double-metal geometries have been proposed [18, 19], with the advantage to optimize the field confinement within the dielectric core. Our design introduces a substantial difference with the previously reported structures, where the electric and magnetic field are orthogonal when calculated in the same plane. In our configuration the dominant electric and magnetic field components (E_z and H_z) are parallel to each other in a plane crossing half a way the capacitor. This parallelism between the magnetic and electric field is essential to exclude propagation effects, since such configuration with parallel electric and magnetic fields cannot satisfy the Helhmoltz wave equation [20]. As a result, the inductive and the capacitive parts can be adjusted independently for a fixed THz frequency of operation, similarly to the case of lumped element electronic circuits [21]. Three-dimensional resonators based on this design have been demonstrated across the whole THz range (1-20 THz). This architecture allows for ultra-sub-wavelength electric field confinement in a dielectric core, opening new possibilities for advanced opto-electronic devices, such as emitters [18, 22] and detectors of infrared radiation, based on a single quantum object [23].

2. Resonator design, characterization and modelling

The resonator design investigated in this work is depicted in Fig. 1(a). It consists of a ground strip (rectangular metallic patch), deposited on substrate, a dielectric layer of thickness T (transparent in Fig. 1(a)) that covers the ground strip and a Π−shaped half ring deposited just above the strip. Figures 1(b) and 1(c) show realization of the structure in an array of resonators on a GaAs substrate, with the metallic parts made of gold (150nm thick) and T = 2µm thick dielectric SiO₂ layer. In Fig. 1(c) we also define the geometrical parameters W, L_x and L_y that have been varied in this work, while the width E of the Π−ring is kept fixed at 2µm. The key feature of this structure is that the geometrical overlaps between the upper ring and the ground plate are acting as the plates of two capacitors. The capacitors localize the electric field into the SiO₂ dielectric, within a strongly sub-wavelength volume. This feature is confirmed by finite difference simulations of the electric E_z [Fig. 1(e)] and magnetic H_z [Fig. 1(f)] fields. In the bottom part of Figs. 1(e) and 1(f) the total electric and magnetic energy densities are also shown. While the electric filed is confined mostly in the capacitive parts [Fig. 1(e)], the magnetic field is localized around the loop formed by the upper ring and the ground plate [Fig. 1(f)]. This spatial separation of the electric and magnetic energy is reminiscent of the quasi-static nature of our resonator, in which the energy exchange is driven by the real current in the metallic parts, rather than propagative coupling in the bulk of the resonator [20, 24, 25]. This is a crucial point, as it allows to adjust independently the capacitance C and the inductance L of the structure, while the resonant frequency f_res = 1/2π(CL)^1/2 is kept fixed. Furthermore, as seen from the simulations from Figs. 1(e) and 1(f) the electric and magnetic fields components in the plane of the capacitors (“xy-plane”) are parallel to each other, ruling out any possible coupling mediated by propagation effects. The reduction of the capacitive part can then lead to highly subwavelength volumes V_capa for the electric energy density, possibly down to nanoscale range, with corresponding high electric field amplitudes, E = (hf_res /εε₀V_capa)^1/2.

 

Fig. 1 (a) Perspective drawing of the resonator, indicating the subwavelength capacitor parts. (b,c) Microscope pictures of arrays realized by laser lithography on GaAs substrate and with SiO₂ dielectric layer. The main geometrical parameters are indicated in (c). (d) Coordinate system and a set of planes chosen for the illustration of the electromagnetic field distribution. The surfaces A, B and C are contained by the dashed lines and the z-direction. (e) Upper panel: electric field component E_z in the (x,y)-plane from (d). Lower panel: electric energy density plotted in the planes A and B. (f) Upper panel: magnetic field component H_z in the (x,y)-plane. Lower panel: magnetic energy density in the planes (x,y) and C.

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In order to demonstrate that the resonant frequency can be tuned separately through the capacitive and inductive parts, we prepared arrays of resonators where the dimensions of the parameters W, L_x and L_y were varied from 2µm to 8µm, keeping a dielectric thickness T = 2µm. This allowed us to vary independently the capacitor size W, and the perimeter of the inductive loop L_x + L_y. In Figs. 2(a), 2(b), 2(c) we report the reflectivity spectra for three different families of parameters: (W = 2µm, L_x , L_y = 2µm), (W = 2µm, L_x = 2µm, L_y) and (W = 4µm, L_x = L_y), for light incident at θ = 10° with respect to the normal of the array. In Fig. 2(d) we also report measurements for a dielectric thickness T = 1µm. Reflectivity spectra with θ = 45° and θ = 60° yield essentially the same results and are not shown. We also performed transmission measurements, however the resonances were in that case less pronounced than in the reflectivity spectra. The measurements were performed with light polarized perpendicular to the two capacitors (along the x-direction). The resonator modes of the structure are clearly seen as the reflectivity minima of the spectra. The shaded band corresponds to the phonon absorption band of the GaAs substrate. In Fig. 2(a) we observe sharp reflectivity dips, while in Figs. 2(b) and 2(c) we rather observe Fano-type features with a reflectivity maximum immediately following the less pronounced resonant dips. These features probably arise from the increased radiation loss of the structure due to the inductive loop [19, 26]. It is remarkable that, apart from the quasi-static mode illustrated in Fig. 1, we do not observe any higher order modes (in the range 1-20 THz) that are typically present in other types of resonators [19, 27–30].

 

Fig. 2 (a,b,c) Experimental reflectivity spectra of resonator arrays such as the one depicted in Fig. 1(c), with light polarized along the two capacitors, and under almost normal incidence on the array surface. The thickness is T = 2µm. The triplet of numbers indicates the parameters (W L_x L_y) defined in Fig. 1(c), for instance 264 indicates nominal values W = 2µm, L_x = 6µm and L_y = 4µm. (d) Spectra of the resonators 222 and 242 for two different thicknesses T = 2µm (dotted curves) and T = 1µm (continuous lines). The grey area corresponds to the phonon absorption of the GaAs substrate.

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In Fig. 3, we plot the resonant frequencies extracted from the spectra in Fig. 2 as a function of a “scale parameter” p, defined as $p=(1/\sqrt{EW/T})(1/\sqrt{L_x+L_y})$ with E = 2µm the thickness of the Π-ring. The definition of p is motivated by the formula of the resonant frequency f_res = 1/2π(CL)^1/2 as a function of the capacitance C and the inductance L of the structure. Indeed, we expect that the capacitance is roughly proportional to EW/T, while the inductance L is proportional to the perimeter 2(L_x + L_y). From Fig. 3 we observe that indeed f_res has an essentially linear dependence on p. Note that from this plot we used the parameters E, W, L_x, L_y measured for the real structures shown in the left panel of Fig. 3. From Fig. 3, we can note that the pairs of structures (242, 224), (244, 262) and (282, 264) that have similar loop perimeter L_x + L_y also have identical frequencies, as expected. Furthermore, almost the same frequency is recovered by shrinking the loop and increasing the capacitor, as seen for the pair (444, 284) and the triplet (422, 224, 242). This indicates clearly that, provided the resonant frequency f_res, the inductive part and the capacitive part can be adjusted separately so that very sub-wavelength values of the electric field confinement volumes V_capa are achieved.

 

Fig. 3 Left: plot of the resonator frequencies measured from the data in Fig. 3 as a function of the “scale parameter” p, that is roughly proportional to the square root of the inverse of the product of the capacitance and inductance of the structure, as explained in the main text. Right: pictures of the different family of resonators.

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In order to understand fully the dependence of the resonant frequency on the shape of the structures, we developed a lumped element model that is described in Fig. 4. Figure 4(a) presents the equivalent circuit with the correspondence to the geometrical parts that play role to inductances and capacitors. The total inductance is defined as L = L₁ + L₂ + L₃ + L₄-2M₁₃-2M₂₄, where L_i are the self-inductances of the rectangular blocks indicated by numbers in Fig. 4(a) and the M_ij are the corresponding mutual inductances. The values of L_i and M_ij are computed with the exact formulas of Hoer and Love for thin tapes [31], as the skin dept of the metal is much smaller than the lateral size of the blocks at this frequency range [32]. The computed values of L are plotted in Fig. 4(b) as a function of the half-perimeter L_x + L_y. The total inductance is monotonic function of L_x + L_y as anticipated, but more than linear because of the logarithmic factors in the exact expressions [31]. The mutual inductance accounts typically for 20% of the total inductance.

 

Fig. 4 Circuit model of the structure. (a) Scheme of the equivalent circuit, with indication of the inductive and capacitive elements. (b) Total inductance as a function of the half-perimeter L_x + L_y of the magnetic loop (circles). The dotted line is a quadratic fit. (c) Total capacitance as a function of the parameter EW/T (diamonds). The dotted lines are extrapolation of the circuit model for decreasing capacitor surface EW, for a thickness T = 2µm, 1µm or 0.5µm of the dielectric layer. The red line is the parallel plate capacitor value C/2 = 0.5εε₀WE/T. (d) Ratio f_LC/f_res between the modelled (f_LC) and measured frequencies (f_res) as a function f_res. The full squares are the values obtained with the full circuit model from Fig. 4(a), while the open squares are the values of the simplified formula f_LC = 1/2π[(C/2 + C_inter)L]^1/2.

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The full capacitance of the system consists of the double metal parts with capacitances C, highlighted in Fig. 4(a), and also of the mutual capacitance C_inter of the ring gap. To compute C_inter, we have used the formulas for slotline [20]. For the double-metal capacitors, we use the two dimensional version of the Palmer formula [33, 34]:

Besides the contribution of the double-metal region (factor WE/T), this formula contains corrections due to the fringing fields [33, 34]. Typical values for C from Eq. (1) are hundreds of aF, while for C_inter we obtain tens of aF, showing that indeed the majority of the electric energy of the resonator is located inside the dielectric core. The total capacitance C/2 + C_inter is plotted in Fig. 4(c). Moreover, by decreasing the thicknesses T below 0.5μm C_inter becomes completely negligible. However, for capacitors with aspect ratio close to 1 the Palmer expression predict values that are considerably higher than the parallel plate formula C = εε₀WE/T.

With the inductances and capacitances of the system determined, we have computed the resonant frequencies by solving the electrical circuit in Fig. 4(a). In Fig. 4(d) we compare the results with experiments (red squares). Very good match is observed for all structures, while the small discrepancies can be attributed to the uncertainty in the measured values of the geometrical parameters. Furthermore, as seen in Fig. 4(d) the resonant frequencies are also well accounted by the simplified formula f_LC = 1/2π[(C/2 + C_inter)L]^1/2 (blue open squares).

To ensure whether a resonator operates in the limit of propagation or in the quasi-static limit it is very convenient to use the well-known size vs. wavelength criterion, which sets that the quasi-static regime is the case where λ << λ_res with λ a typical resonator size [20, 24, 25]. This limit is easily attained in all high frequency electronic devices up to tens of GHz, but has to be refined for the case of THz resonators that operate typically at the cross-over between optics and electronics [19, 28]. For this purpose, rather than the physical length λ, we use the maximum optical path L_opt, proportional to nλ, (n is the refractive index of the resonator) defined as the distance for a photon to propagate back and forth within the resonator. It is easy to understand that in the propagation regime a resonator sustains standing waves with resonant wavelengths λ_res such as L_opt = Kλ_res where K is an integer. The ratio L_opt/λ_res then quantifies the limit between propagation effects, where L_opt/λ_res = 1, 2, 3…, and the quasi-static regime, where L_opt/λ_res<<1. In Fig. 5 we have applied this criterion to the fundamental mode of several types of resonators: wires [30] (L_opt = 2nλ), the “LC” resonators from Ref [18, 35, 36], where L_opt = 2n(2r + l) (r is the radius of the “capacitive” part and l the length of the “inductive” part), with n = 3.5 (GaAs dielectric core). We have also included the “monopolar” disk resonators from Refs [19, 27, 28]. where L_opt = 4nD (D is the diameter of the disk resonators) for Ref [27], and L_opt = 4(nD + d) for Ref [19], where the inductive part of length d is standing in the air. The factor 4 comes from the fact that this resonator can be interpreted as λ_res/4 standing wave with a maximum of the electric field in the capacitor and zero in the inductive loop. For the structures reported here, we use L_opt = 2n((W + L_y)² + L_x²)^1/2 with n = 2 (SiO₂ dielectric core). As another criteria for our structure, we can take an optical path L_opt = 2[2nW + L_x + L_y], which yields similar, but slightly higher values for L_opt/λ_res compared to those reported in Fig. 5. The results summarized in Fig. 5 show that the structures from Ref [19]. indeed satisfy the quasi-static condition, in particular for very large wavelengths, λ_res > 100µm, whereas our resonator operates well within the quasi-static limit for the entire THz range.

 

Fig. 5 Ratio between the minimal photon optical path L_opt and the resonant wavelength λ_res = c/f_res for different resonators reported in the literature (open dotted symbols) as comparison to our work (full symbols, with the same code as in Fig. 3). The case L_opt/λ_res = 1 corresponds to the first standing wave mode in the structure. Higher order modes are provided by the condition L_opt/λ_res = K, where K is positive integer. The case L_opt/ λ_res < 1 corresponds to resonators operating in the quasi-static regime. For simplicity, in this diagram we used the bulk refractive index of the dielectric filling the resonators; L_opt/ λ_res >1 signifies that the effective index of the guided mode is higher than the bulk refractive index.

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3. Conclusion

In summary, we have reported three dimensional THz resonator architectures with a very sub-wavelength effective volume of the capacitive parts. An extrapolation of our circuit model shows that, by reducing the widths E and W and the thickness T volumes as small as 0.02µm³ can be achieved in the 1-10 THz range while the inductive loop is kept at sub-wavelength dimensions L_x = L_y ~5µm. In this limit the electric field is well localized within the capacitor and almost perfectly polarized in the z-direction. Such structures are therefore very suitable for studying the ultra-strong coupling regime with intersubband transitions, in the few electron limit [13]. Clearly, in such small volumes light harvesting could become problematic, as the impedance of our resonator Z = (L/C)^1/2 strongly mismatches the free space value 377 Ω. However this issue can be overcome by employing side antenna coupling [26]. Another advantage of our architectures is that they naturally allow applying a bias between the ground plate and the Π−ring, as required for optoelectronic applications.