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Abstract
We investigate THz metamaterial resonators which provide highly subwavelength confinement of the electromagnetic field. In these structures the capacitive part is reduced into nano-scale dimensions and allows strong local electric field enhancement, useful for infrared detectors and sensors. A crucial issue for these structures is the fraction of the incident THz radiation that couples into the capacitive part of the metamaterial. To this end we develop a strategy to enhance the harvesting of the incoming free space radiation into volumes as small as 5x10−8λ3 for a free space wavelength λ. We also provide an analytical model in order to evaluate propagation effects in the inductive part of the structures.
© 2017 Optical Society of America
1. Introduction
The unit cell of metamaterials [1] consists of metal nanostructures, such as split-ring resonators [2] that realize the high-frequency counterpart of inductance-capacitance circuit resonator [3]. As already underlined in a seminal work by Pendry et al. [3], a key feature in such resonators is the ability to achieve strong local electric field enhancement in the capacitive element. This concept has ever since been experimentally validated in many works from the microwave, through the far and to the near infrared part of the electromagnetic spectrum [4–10]. For THz and mid-infrared frequencies the local field enhancement is particularly useful for fundamental studies of the light-matter interaction [9,11–17], as well as for detector applications [5,6,18–25]. In these spectral ranges, it is possible to employ semiconductor quantum wells, where far-infrared photons are absorbed or generated by intersubband electronic transitions. In this case it is desirable that the resonator architecture satisfies the intersubband selection rule, which requires that the electric field is perpendicular to the epitaxial layers [26]. This is not the case for the majority of planar metamaterial resonators, which provide a strong in-plane electric field [1, 2, 13, 17]. On the other hand, metal-dielectric-metal resonators, which essentially exploit the absence of cut-off of the TM0 mode, naturally satisfy the intersubband selection rule, and furthermore provide a very strong overlap with the quantum electronic system [11, 14, 15, 27, 28].
We recently proposed meta-atom architecture which benefits from this concept and can be described in a lumped element framework [29]. This architecture is schematically recalled in Fig. 1(a), together with an indication of the electric and magnetic field lines. This design consists of two metal parts, which are separated by a dielectric layer of thickness T. The regions where the two parts overlap play the role of capacitors. Here the electric field is mainly vertical, i.e. perpendicular to the metal plates. Owe to charge conservation in each metallic element the dynamic charges induced on the edges of each element are of opposite signs, meaning that the two capacitors may be considered in series. The ensemble of two metallic elements that are dubbed “upper loop” and “lower ground plate” as further illustrated in Fig. 1(b) can be considered of inductive elements. As the geometrical shapes of inductances and capacitors are now completely independent, the resonant frequency of the structure can be tuned solely by the size of the upper loop while maintaining the strong electric field confinement in the capacitors [25,29].
Fig. 1 (a). Schematics a three dimensional meta-atom architecture, indicating the capacitive and inductive elements, the dynamical distribution of charges and currents in the resonators as well as the corresponding electric and magnetic field lines. (b) A top view of a resonator realized with 1µm thick Si3N4 dielectric layer.
In the present work we make an in depth investigation of this concept by studying resonators with different shapes and geometries. Our two objectives are i) optimize the coupling of the THz radiation from the free space into the resonator and ii) explore structures with nanometer sized capacitors in all three dimension of space to realize extremely high confinement factors.
2. Coupling to free space
Our approach to understand and optimize the coupling of THz radiation from the free-space into the capacitors is inspired by the coupling mechanism in double-metal patch antennas [30, 31]. These antennas are aperture antennas, were the radiating apertures correspond to the lateral edges the patch [30]. In that case one considers the stray electric field lines around the patch antenna, which must have a non-zero total projection on the planar component of the incoming electric field in order to couple to the cavity mode. To investigate this point, we prepared four different designs illustrated in Figs. 2(a)-2(d). In all these designs the metal strips are 1µm wide and the dielectric is a 1µm thick Si3N4 layer. The resonators were fabricated onto GaAs substrates which have negligible absorption in the THz range.
Fig. 2 Four sample geometries investigated in this work (optical microscope pictures), together with schematics of the electric field lines: (a) initial design, (b) initial design with extended lower ground plate (c) “top hat” upper loop and (d) split-ring upper loop. In (b), A1 and A2 indicate the radiating apertures of a capacitor, with n1 and n2 the corresponding normal vectors.
Figure 2(a) is the initial design described in the introduction, which will be our reference in this investigation. Next to the optical microscope picture, we illustrate schematically the electric field lines which connect the two metal parts. We always shall consider the case where the fundamental resonance is excited in the structure. In that case the charges induced in the two capacitive regions are of opposite signs, as indicated in Fig. 2. The projection of the stray fields on the edges is indicated with thick arrows, which point either leftwards (blue line) or rightwards (red line). For the case of Fig. 2(a), we would expect the leftward and rightward projections to almost compensate each other, leading to zero or very weak coupling strengths with an incident wave. However, as the metamaterial realization is not perfectly symmetric, the electric lines are also not perfectly symmetric around each metal pad and therefore the coupling efficiency never vanishes. This is no longer the case for the design of Fig. 2(b), where we extend the lower ground plate around the top pads. The stray electric field lines are now less dependent on the fabrication and become very symmetric, thus implying a complete cancelation between the rightward and leftward arrows. In terms of antenna theory [30], each capacitor has two radiation apertures A1 and A2 as indicated in Fig. 2(b). The only radiating currents in that case is the magnetic current density, provided by the vector product Ez1,2xn1,2 where n1 and n2 are the outward normal to the apertures [Fig. 2(b)], and Ez1 and Ez2 are the vertical electric field in the each aperture. Because of the high symetry, Ez1 = Ez2 = Ez the total magnetic current density is Ezx (n1 + n2) and goes identically to zero.
This discussion indicates that to optimize the in-coupling of the resonators we need to break the symmetry of the stray electric field lines so to reduce one of the contributions as it illustrated in Figs. 2(c) and 2(d), where the leftward arrows are practically suppressed, as the top loop covers completely the edges of the ground strip. Due to the shape of the inductive loops, the design from Fig. 2(c) will be referred as “top hat” resonator, “THR”, while that of Fig. 2(d) is dubbed “split-ring” resonator, “SRR”.
To evaluate experimentally the performance of these designs, we prepared arrays of resonators which were tested in transmission experiments under normal incidence with linearly polarized light. Each array consisted of 2x2mm2 panels where the distance between the edges of the resonators was kept at 10µm. The transmission experiments were performed with Fourier Transform Infrared Spectrometer, where the THz source is a Globar lamp and the detector is a helium cooled bolometer. The spectrometer is vacuum-pumped in order to avoid water absorption. The radiation of the Globar lamp was polarized with a grid polarizer either parallel or orthogonal to the ground strip. All spectra were normalized at the Globar reference spectra obtained for each polarization without any sample. As it has been explained in Fig. 2, the coupling to the resonators occurs through the projection of the incoming electric field on the planar component of the fringing fields of the capacitor and therefore the active polarization is when the electric field of the incident wave is parallel to the ground strip.
The experimental results are presented in Fig. 3 for the four designs. A weak absorption feature is observable for the initial design, while no absorption is detected for the resonator with the extended ground plate [Figs. 3(a) and 3(b) respectively]. Conversely, strong absorption resonances are visible for the case of THR [Fig. 3(c)] and SRR [Fig. 3(d)]. The structures are not optically active for a polarization perpendicular to the ground strip, as expected. Furthermore, the much stronger absorption of the THR and SRR confirms the coupling mechanisms described above. We underline the fact that the coupling mechanism of our resonators, based on the electric field component, is qualitatively very different from the magnetic coupling described in [28]. Moreover it provides more degrees of freedom, as many different planar geometries of the inductive upper loop can be envisioned.
Fig. 3 Transmission spectra with polarized light for each of the four designs indicated in Fig. 2: (a) initial design, (b) initial design with extended lower ground plate (c) “top hat” upper loop (THR) and (d) split-ring upper loop (SRR). For each case, the dotted curve corresponds to light polarized perpendicular to the ground strip (Eperp), while the continuous curve to a polarization along the ground strip (E//). In 3(a) the absorption resonance is indicated by an arrow.
In order to remove the baseline from the experimental curves, we have normalized the transmission for the active polarization [electric field E//, as indicated in the insets of Fig. 3] on the transmission for the inactive one (electric field Eperp). To take into account the air to GaAs substrate transmission this ratio has been multiplied by T0 = 4ns/(1 + ns)2 = 0.7 where ns = 3.5 is the refractive index of the GaAs substrate. The resulting curves are reported in Fig. 4, except for the design with extended ground strip [Fig. 2(b)] where no clear resonance was visible. The baseline correction permits to render the absorption resonance of the initial design [indicated by an arrow in Fig. 3(a)] clearly visible.
Fig. 4 Normalized transmission spectra obtained from the data in Fig. 3 by dividing spectra for the active on the inactive polarization. The spectra have been multiplied by T0 = 4ns/(1 + ns)2 to take into account the transmission of the GaAs substrate with index ns = 3.5. The dashed curves are fit using Eq. (1).
The transmission spectra from Fig. 4 can be modelled analytically by using an approach similar to the one described in [31]. In this approach, the in-plane components of the incident, reflected and transmitted wave are projected on the fringing fields of the resonators [thick arrows in Fig. (2)] in order to relate the inner and outer electric fields. The problem is solved completely by using the Poynting’s theorem, and we obtain for the following expression for the transmission as a function of the frequency f:
Here L(f) = 1/(1 + 4Q2(f/fres-1)2) is a normalized Lorentzian that describes a frequency-dependent absorption and η is the absorption strength. The experimental spectra are very well fitted by Eq. (1) (dashed curves in Fig. 4) indicating that the resonances are homogeneously broadened. From the fits we extract the absorption amplitude η and the quality factor Q for each structure: η = 0.32 (initial design), η = 1.06 (SRR) and η = 1.6 (THR), which quantifies the coupling mechanism described above. The respective quality factor are Q = 12 (initial design), Q = 7.5 (SRR) and Q = 12 (THR). The THR and the SRR have therefore similar free-space coupling, while the THR has a better quality factor. In our experiments we did not observe clearly any resonant features in the reflectivity, however Eq. (1) takes automatically into account the reflectivity port. Indeed, because of the high refractive index of the substrate the amplitude of the reflectivity features is decreased with respect to the transmission dip.
In the following, for simplicity we shall provide directly the normalized transmission spectra as described above and the vertical axis shall be labelled as “transmission”.
3. Geometrical scaling of the resonant wavelengths
In our structures the resonant frequencies/wavelengths can be tuned by changing the geometry of both capacitive and inductive parts [29]. For fixed capacitor geometry, the wavelengths can be tuned across the THz range by changing the perimeter of the inductive loop. On the other side, the capacitor sizes can be decreased in order to increase the electric field confinement, which is important for the reduction of dark current in infrared detectors [25] or for fundamental studies of ultra-strong light-matter coupling regime. However, for a fixed resonant frequency, the reduction of the capacitive parts must be compensated by increased size of the inductive loop, which increases the propagation effects in the structures, as it will be discussed in Section 4.
We have first investigated the scaling of T = 1µm SRR thick resonances as a function of the form of the inductive loop, as reported in Fig. 5. All measurements are performed in dense array of resonators with 10µm separation between the resonators, as described in the previous paragraph. The geometry of the loop as a function of the dimensions X, Y, W, is schematized in the inset of Fig. 5(c). In Fig. 5(a) we report the transmission spectra where the internal length Y of the loop is varied, while its internal width X is kept constant. In Fig. 5(b), spectra with X + Y = const are indicated. Figure 5(c) resumes the observed resonant wavelengths as a function of the loop perimeter of the upper metallic part. The measured wavelengths (full circles) are compared with simulations of the electromagnetic eigenmodes of the structure (open stars) which were obtained with commercial finite difference domain (FDD) software. All the investigated SRR structures appear in Figs. 5(a) and 5(b), with the numbers WXY indicated in microns below each picture. This convention is kept for all structures that are commented further in Figs. 6 and 7.
Fig. 5 (a) Normalized transmission spectra for a series of SRR resonators with variable inductive loop length Y. Each resonator is labelled with the triplet WXY defined in the inset of (c), and provided in microns below each picture. (b) Normalized transmission spectra for a series with constant X + Y, where X is the width of the inductive loop. (c) Resonant wavelength versus the medium perimeter X + 2Y + 8W of the upper loop that connects the two capacitors. The open circles are obtained from experimental data, while the open stars are results from FDD simulations. The inset indicates the labels for the geometrical dimensions (W, X and Y) of the inductive loop.
Fig. 6 (a) Normalized transmission spectra for a series of THR resonators with variable inductive loop length Y and width X. Each resonator is labelled with the triplet WXY defined in the inset of panel (b). (b) Resonant wavelength versus the length of the path X + 2Y + 4W on the upper loop that connects the two capacitors. The full dots are experimental data, while the full stars are results from FDD simulations. The inset shows the geometry of the top inductive loop with definition of the dimensions X, Y, W.
Fig. 7 (a) Normalized transmission spectra for THR structures with T = 0.5µm thick Si3N4 layer and W = 0.5µm.(b) Normalized transmission spectra for THR structures with T = 0.25µm thick Si3N4 layer and W = 0.35µm. The THR structures are named according to the nomenclature introduced in Fig. 6(b). (c) Resonant wavelengths as a function of the path length X + 2Y + 4W connecting the two capacitors.
In a fully quasi-static vision, we would expect that the resonance wavelength will depend only on the total perimeter 2(X + Y) of the inductive loop of the structure. The data from Fig. 5(b) shows clearly that this is not the case. Furthermore, as displayed in Fig. 5(c), the resonant wavelength is proportional to the length of the path on the top metal that connects the two capacitors. This is an indication that in the case of SRR the resonant wavelengths cannot be inferred from a purely quasi-static picture, and propagation effects should be taken into account. Very similar results are obtained for 1µm thick THRs, as shown in Fig. 6(a) and 6(b). In that case the length of the path that connects the two capacitive parts is provided by X + 2Y + 4W, following the schematics indicated in Fig. 6(b).
In order to increase the confinement of the electric field in all three dimensions of space we have fabricated THR structures where both the thickness T of the Si3N4 layer and the width W of the metal stripes have been reduced. Correspondingly, the inductive loops of the structures have been increased in order to keep the resonances in the same wavelength range as the structures with thicker dielectric layer. Transmission measurements of these structures are shown in Fig. 7. In Fig. 7(a) we report on structures with T = 0.5µm thick Si3N4 layer and W = 0.5µm width of the metal parts. Here the dimension Y of the inductive loop is varied, while X is kept at a fixed value, X = 4µm. In Fig. 7 (b) we report on structures with T = 0.25µm thick Si3N4 layer, and W = 0.35µm for the metal wire width, whereas the length X of inductive loop is X = 6µm. For both these measurements, the distance between resonator edges were 5µm, as denser arrays provide higher contrast of the resonances [25,31].
The geometrical volumes of the double-metal parts of the THR shown in Fig. 6(a), Figs. 7(a) and 7(b) are respectively W2T = 1µm3, 0.125µm3 and 0.031µm3. The resonances from Figs. 7(a) and 7(b) have typical quality factors Q~8-9, which are slightly smaller than the typical values Q~11-13 observed for the thicker structures from Fig. 6(a). These values are confirmed by finite difference domain simulations. The lowering of the quality factors for thinner THRs seems plausible as the thinner structures have larger inductive loops. One would therefore expect an increase of the resistance and radiation loss in such designs. At lower frequency, the ohmic loss could be improved by employing superconductor material such as YBCO [32]. The radiation loss can be decreased by altering the form of the inductive loop [33].
Following the previous discussion, the resonant wavelengths are plotted as a function of the medium perimeter X + 2Y + 4W in Fig. 7(c). Also in this case we observe a linear dependence, which indicates the presence of propagation effects. This behavior is somewhat expected as larger dimensions of the inductive loop favor the appearance of standing wave phenomena.
4. Transmission line model
We now seek a model of the structures which would allow understanding more precisely the nature of the electromagnetic resonances observed. Given the linear dependences observed in the plots from Figs. 5-7, it is tempting to model the resonant wavelength λres by the expression λres = 2neffP with neff an effective index and P the propagation length between the two capacitive parts. For instance, data in Fig. 5(c) is well fitted with neff = 2.03, the data from Fig. 6(b) with neff = 2.13, and the data from Fig. 7(c) with neff = 2.45. Similar picture has been suggested for planar split-ring resonators [34]. However, this picture would imply that the metal stripe supports a guided mode, similar to a planar Goubau line, which forms a standing wave between the two capacitors. Such lines have been recently investigated by time-domain THz spectroscopy, and it has been found that the guided modes are strongly dependent on the substrate thickness [35]. In particular, the guided modes show very little confinement in the case of thick substrates as considered in the present work, and this picture cannot explain the values of the effective index that we observe.
We therefore propose another model for the observed standing waves inspired from dual planar transmission lines [36]. To precise this model, we first examine the field distributions provided by the FDD solver. In Figs. 8(a) and 8(b) we have plotted respectively the magnetic energy density and the electric energy density distributions for the eigenmode of the THR structure with W = 1µm, X = 4µm and Y = 6µm. The magnetic energy density is essentially localized next to the backside of the upper magnetic loop, where, as indicated by FDD simulations, the current density is also maximal (not shown). On the other hand, the electric energy density shown in Fig. 8(b) is located around the capacitive double-metal parts, as also shown quantitatively in Fig. 1(a). In both Figs. 8(a) and 8(b), one can notice a leakage of the magnetic/electric energy along the parallel sections of the upper inductive loop. These parallel sections can then be assimilated to a coplanar stripline transmission line (TL) of length Y, connecting the inductive and capacitive elements, as described by the equivalent circuit model in Fig. 8(c). The inductance L is modelled from the formula of a thin wire of length X and rectangular cross section of side δ [37]:
Fig. 8 (a) Distribution of the magnetic energy density in the plane containing the upper inductive loop, obtained by FDD simulations. (b) Distribution of the electric energy density at the middle plane of the resonator. (c) Circuit model of the structures. (d) Effective index as a function of the TL length Y for all structures, with the thickness T of the Si3N4 layer indicated.
Here δ is the skin depth for gold, which is numerically provided in microns by the expression δ = 0.0756/f0.5 where f is the frequency expressed in THz [38] and µ0 = 4π10−7 H/m is the magnetic constant. Typical values for the inductance are thus on the order of 10 pH.
In the FDD simulations, negligible magnetic energy is observed around the ground strip, which is then treated as a wire without inductance. In this case, the region comprising the double-metal parts and the lower ground strip is described as a pure capacitance Ceq, resulting from two identical capacitances in series, as indicated in Fig. 8(c). To infer the exact values of Ceq, we performed electrostatic FDD simulations of this region of the structure, that provided the values Ceq = 70aF for the T = 1µm thick THR and SRR (W = 1µm), Ceq = 35aF for the T = 0.5µm thick THR (W = 0.5µm) and Ceff = 29aF for the T = 0.25µm thick THR (W = 0.35µm). The parallel plate formula provides Ceq = 0.5ε0εSiNW2/T, with values respectively 17.7aF, 8.8aF and 8.6aF for the three thicknesses, T = 1µm, 0.5µm and 0.25µm. As expected, the actual values are increased with respect to the parallel plate formula, owe to the important stray field contributions [29]. Note that, contrary to the structures in [29], there are not known analytical expressions for the capacitances, owe to extensions of the capacitive parts above the metal stripe [see Fig. 2].
The coplanar stripline TL is characterized by its impedance Z and its effective index neff that is related to the propagation constant β through the formula β = neffω/c with c the speed of light in vacuum and ω = 2πf. The impedance Z is provided by the expression [36]:
Here Z0 = 377Ω is the free space impedance, k’ = X/(X + 2W), k = (1-k’2)0.5, and K(k) is the complete elliptical integral of the first kind [36]. The equation providing the eigenfrequency ω of the circuit in Fig. 8(c) is obtained by zeroing its total impedance of the circuit in Fig. 8(b) and can be cast in the form:
The propagation effects are contained in the r.h.s of Eq. (4), where the contribution of the tangent function comes from the standing wave pattern on the transmission line [37]. Indeed, in the case Y = 0 we recover an LC circuit with a frequency ω = (1/LCeff)1/2, whereas for very short lines, neffωY/c<<1, it adds an effective inductance proportional to the length Y of the TL.
We use Eq. (4) to infer the values of the effective index neff of the transmission line mode from the measured values of the frequency ω. The resulting values are reported in Fig. 8(d), for all structures, as a function of the TL length Y. We observe a universal behavior, where the values of the effective index cluster around the value neff = 2.2 similar to the index of the Si3N4 layer (n = 2.0). This value is close and yet smaller than the value neff = 2.7 predicted by the theory from [39] for an infinite TL, which has been plotted for comparison. This indicates that the field is more localized in the Si3N4 slab, than for an infinite TL line, where the field leakage into high refractive index (n = 3.5) GaAs substrate increases the value of neff. The value of neff increases strongly for shorter lines. Indeed, in this case the shortening of the Y dimension is compensated by an increase in the X dimension, which leads to propagating effects in the x direction as well. Such effects are difficult to capture in quasi-static approaches such as lumped element and transmission lines, and only complete resolution of the full Maxwell equations provide satisfactory answer. However, the model of Eq. (4) works well for the case of thin structures such as the THR from Fig. 7(a), which provide the strongest confinement of the electric field. Furthermore, the model explains the linear trend of the resonant wavelength with the perimeter of the inductive loop. Indeed, because of propagation effects the wavelength is roughly proportional to the length of the transmission line Y, and also increases with the dimension X as the inductance in Eq. (2) is roughly proportional to X.
5. Nanoscale electric field confinement
The electric field confinement in our structures has been quantified through systematic FDD simulations of the electric density energy distribution at resonance. In Figs. 9(a) and 9(b) we show the typical results for 1µm thick SRR and THR respectively. The electric energy is located around the double-metal parts, with hotspots around the metallic edges. To quantify the effective volume of the electric energy, we use the following definition of the effective volume:
Here We is the time-averaged electric energy density, We = ε(r)ε0|E(r)|2/4, E(r) is the total electric field and Ez is the vertical component of the electric field. In the second factor, in the r.h.s. of Eq. (5) the integration runs only on the Si3N4 layer. This factor has been added to take into account the selection rule for intersubband transitions [40]. The effective volumes for all structures have been plotted in Figs. 9(c) and 9(d). The 1µm thick SRR and THR have similar effective volumes, despite the different geometry of the inductive loop. The typical values are Veff~1µm3, lower than the geometrical volumes of the double-metal parts, 2µm3. This is attributed to the strongly inhomogeneous distribution of the electric energy density, as seen from the images in Figs. 9(a) and 9(b). As expected, the thinnest structures from Fig. 7(a) and 7(b) have the smallest effective volumes. In the case of the 0.25µm thick THR the effective volume Veff decreases with frequency, reaching a minimal value of Veff ~0.024µm3 = 5x10−8λ3 for the structure resonating at 3.9 THz (λ = 77µm). The trend observed in Fig. 9(d) can be explained through the electric energy leakage discussed in the previous section. Structures that resonate at lower frequencies have longer TL section, and therefore the electric energy is more delocalized.Fig. 9 (a) Distribution of the electric energy density in a 1µm thick SRR obtained by FDD. The upper part is the distribution obtained in the middle plane of the Si3N4 layer, while the lower part is through a plane crossing the middle of the two capacitors. In the latter the range of the color code has been expanded by a factor of two. (b) Similar plots for the case of a 1µm thick THR. (c) Effective volumes in SRRs as a function of the resonant frequencies. (d) Effective volumes in THRs as a function of the resonant frequencies, and for different capacitor sizes: circles: 1µm thick THRs [Fig. 6(b)], squares: 0.5µm thick THRs [Fig. 7(a)] triangles: 0.25µm thick THRs [Fig. 7(b)].
While the effective volume is a measure of the electric field confinement of the structure, its overall performance must also include its ability to harvest free-space photons. We recently introduced a single figure of merit for the context of quantum detectors, which quantifies both the antenna effect of the structure and the electric field confinement [25]:
Here Acoll is the collection area of the structure [25], defined such as all incident photons impinging on the area Acoll are absorbed, considering a plane wave under normal incidence, and Q is the resonator quality factor. The square root of F in Eq. (6) then quantifies the local electric field enhancement in the resonator in comparison with the electric field of the incident wave [25,41]. For the present structures, using the approach from [25] and taking into account the transmission port the collection area can be evaluated as Acoll = 4ηΣ/(1 + nS + η)2, where Σ is the array unit cell and η obtained by fitting the spectra from Fig. 5(a), 5(b), 6(a), 7(a), 7(b) with Eq. (1). For THR and SRR with W = 1µm and T = 1µm there is 10µm distance between the resonator edges, and the array unit cell is computed as ΣTHR = (10µm + 6W + X)(10µm + Y + 2W) and ΣSRR = (10µm + 2W + X)(10µm + 2W + Y). For the THR with W = 0.5µm (T = 0.5µm) and W = 0.35µm (T = 0.25µm) the distance is 5µm and therefore the collection area is provided by ΣTHR = (5µm + 6W + X)(5µm + Y + 2W).
The values of F as a function of the resonant frequency have been plotted for all structures in Figs. 10(a) and 10(b), using the effective volumes obtained by FDD simulations according to Eq. (4). From experiments we estimate typical values Acoll = 40µm2 and 11µm2 for respectively the 1µm thick and 0.5µm and 0.25µm thick THR structures, and Acoll = 20µm2 for the SRR structures, that are almost independent from the resonant frequencies. The general trends in Fig. 10 are then determined essentially by the Q/Veff ratio that is the tradeoff between resonator loss and electric field confinement. We generally observe that THR have higher focusing factors as SRR of the same thickness, mainly because of better quality factors. The T = 1µm THR and T = 0.5µm THR have similar focusing factors, as the increased electric field confinement is compensated by a lower quality factors. As expected, highest focusing factors are obtained for the T = 0.25µm thick THR. In all cases, the focusing factors are on the order of 104-105, which makes these architectures very interesting for intersubband detectors with suppressed dark current, according to the theory presented in [25]. Furthermore, these double-metal structures seem to be advantageous with respect to other planar designs in the THz range, where enhancement factors of the order of 102 have been reported [7].
Fig. 10 (a) Focusing factors for SRRs as a function of the resonant frequency. (b) Focusing factors for THRs as a function of the resonant frequency, for and for different capacitor sizes: circles: 1µm thick THRs [Fig. 6(b)], squares: 0.5µm thick THRs [Fig. 7(a)] triangles: 0.25µm thick THRs [Fig. 7(b)].
6. Conclusion
We have explored both experimentally and numerically three dimensional THz metamaterial resonators that provide very strong electric field confinement in nanoscale volumes. The increased confinement is achieved thanks to a double-metal architecture, which reduces the leakage of the electric field lines observed in planar structures. We have provided optimized designs for the free space coupling based on the engineering of the fringing field distribution around the capacitive parts. Furthermore, we have discussed propagation effects in these structures in terms of a transmission line model that depends on a single fitting parameter for a large variety of resonator geometries. While these structures have been developed specifically for intersubband devices, we believe that owe to their ability to couple in free space photons in highly subwavelength dimensions, they can be useful for integration with a large variety of nano-emitters/receivers of infrared radiation.
Funding
H2020 European Research Council (ERC grant “ADEQUATE”), Agence Nationale de la Recherche (ANR-16-CE24-0020 Project “hoUDINi”).
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