Three-dimensional THz lumped-circuit resonators

Yanko Todorov, Pascal Desfonds, Cherif Belacel, Loic Becerra, Carlo Sirtori

To cite this version:

Yanko Todorov, Pascal Desfonds, Cherif Belacel, Loic Becerra, Carlo Sirtori. Three-dimensional THz lumped-circuit resonators. Optics Express, 2015, 23 (13), pp.16838-16845. 10.1364/OE.23.016838 . hal-01275503

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Distributed under a Creative Commons Attribution 4.0 International License
Three-dimensional THz lumped-circuit
resonators

Yanko Todorov,¹,* Pascal Desfonds,¹ Cherif Belacel,¹ Loïc Becerra,² and Carlo Sirtori¹

¹Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, Sorbonne Paris Cité, CNRS-UMR 7162, 75013 Paris, France

²Institut des Nanosciences de Paris, Université Pierre et Marie Curie, UMR7588, 4 place Jussieu, 75252 Paris,
France

*yanko.todorov@univ-paris-diderot.fr

**Abstract:** Our work describes a novel three dimensional meta-material resonator design for optoelectronic applications in the THz spectral range. In our resonant circuits, the capacitors are formed by double-metal regions cladding a dielectric core. Unlike conventional planar metamaterials, the electric field is perpendicular to the surface and totally confined in the dielectric core. Furthermore, the magnetic field, confined in the inductive part, is parallel to the electric field, ruling out coupling through propagation effects. Our geometry thus combines the benefit of double-metal structures that provide parallel plate capacitors, while maintaining the ability of meta-material resonators to adjust independently the capacitive and inductive parts. Furthermore, in our geometry, a constant bias can be applied across the dielectric, making these resonators very suitable for applications such as ultra-low dark current THz quantum detectors and amplifiers based on quantum cascade gain medium.

©2015 Optical Society of America

**OCIS codes:** (160.3918) Metamaterials; (230.5750) Resonators; (260.3090) Infrared, far;
(310.4165) Multilayer design.

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# 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 Helmholtz 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 $T = 2\mu m$ thick dielectric $SiO_2$ 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 $2\mu m$. 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 $SiO_2$ 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 field 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 $f_{res} = 1/2\pi(CL)^{1/2}$ 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, $E = (hf_{res}/\varepsilon\varepsilon_0V_{capa})^{1/2}$.
<figure><img src="image_2.png" /><figcaption>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.</figcaption></figure>

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 $T = 2$µm. 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: ($W = 2$µm, $L_x$, $L_y$ = 2µm), ($W = 2$µm, $L_x$ = 2µm, $L_y$) and ($W = 4$µm, $L_x$ = $L_y$), for light incident at $\theta = 10^\circ$ with respect to the normal of the array. In Fig. 2(d) we also report measurements for a dielectric thickness $T = 1$µm. Reflectivity spectra with $\theta = 45^\circ$ and $\theta = 60^\circ$ 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].

<figure><img src="image_3.png" /><figcaption>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 $T = 2\mu m$. The triplet of numbers indicates the parameters ($W, L_x, L_y$) defined in Fig. 1(c), for instance 264 indicates nominal values $W = 2\mu m$, $L_x = 6\mu m$ and $L_y = 4\mu m$. (d) Spectra of the resonators 222 and 242 for two different thicknesses $T = 2\mu m$ (dotted curves) and $T = 1\mu m$ (continuous lines). The grey area corresponds to the phonon absorption of the GaAs substrate.</figcaption></figure>

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 $E = 2\mu m$ the thickness of the Π-ring. The definition of $p$ is motivated by the formula of the resonant frequency $f_{res} = 1/2\pi(CL)^{1/2}$ 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.
<figure><img src="image_4.png" /><figcaption>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.</figcaption></figure>

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 $L = L_1 + L_2 + L_3 + L_4 - 2M_{13} - 2M_{24}$, 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 depth 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.

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]:

$$C = \varepsilon \varepsilon_0 \frac{WE}{T} \left\{ 1 + \frac{T}{\pi E} \left( 1 + \ln \left[ \frac{2\pi E}{T} \right] \right) \right\} \left\{ 1 + \frac{T}{\pi W} \left( 1 + \ln \left[ \frac{2\pi W}{T} \right] \right) \right\}. \quad (1)$$

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 $C = \varepsilon\varepsilon_0 WE/T$.

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 $f_{LC} = 1/2\pi[(C/2 + C_{inter})L]^{1/2}$ (blue open squares).

<figure><img src="image_5.png" /><figcaption>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 $T = 2\mu m$, $1\mu m$ or $0.5\mu m$ of the dielectric layer. The red line is the parallel plate capacitor value $C/2 = 0.5\epsilon_0 WE/T$. (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 $f_{LC} = 1/2\pi[(C/2 + C_{inter})L]^{1/2}$.</figcaption></figure>

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 $\lambda \ll \lambda_{res}$ with $\lambda$ 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 $\lambda$, we use the maximum optical path $L_{opt}$, proportional to $n\lambda$, ($n$ 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 $\lambda_{res}$ such as $L_{opt} = K\lambda_{res}$ where $K$ is an integer. The ratio $L_{opt}/\lambda_{res}$ then quantifies the limit between propagation effects, where $L_{opt}/\lambda_{res} = 1, 2, 3..., $ and the quasi-static regime, where $L_{opt}/\lambda_{res} \ll 1$. In Fig. 5 we have applied this criterion to the fundamental mode of several types of resonators: wires [30] ($L_{opt} = 2n\lambda$), 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 $\lambda_{res}/4$ 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 $L_{opt} = 2n((W + L_y)^2 + L_x^2)^{1/2}$ with $n = 2$ (SiO$_2$ 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 $L_{opt}/\lambda_{res}$ 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, $\lambda_{res} > 100\mu m$, whereas our resonator operates well within the quasi-static limit
for the entire THz range.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Ratio between the minimal photon optical path $L_{opt}$ and the resonant wavelength $\lambda_{res} = c/f_{res}$ 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 $L_{opt}/\lambda_{res} = 1$ corresponds to the first standing wave mode in the structure. Higher order modes are provided by the condition $L_{opt}/\lambda_{res} = K$, where K is positive integer. The case $L_{opt}/\lambda_{res} < 1$ 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; $L_{opt}/\lambda_{res} > 1$ signifies that the effective index of the guided mode is higher than the bulk refractive index.</figcaption></figure>

**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*<sub>x</sub> = *L*<sub>y</sub> ~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.

**Acknowledgments**

The authors acknowledge financial support by ERC Project “Adequate”, ANR-11-IDEX-0005-02 (Labex SEAM Project “Capture”) and the network “Salles Blanches Paris Centre”.