Ultra-subwavelength resonators for high temperature high performance quantum detectors

In this article we have investigated two important properties of metallic nano-resonators which can substantially improve the temperature performances of infrared quantum detectors. The first is the antenna effect that increases the effective surface of photon collection and the second is the subwavelength metallic confinement that compresses radiation into very small volumes of interaction. To quantify our analysis we have defined and discussed two figures of merit, the collection area Acoll and the focusing factor F. Both quantities depend solely on the geometrical parameters of the structure and can be applied to improve the performance of any detector active region. In the last part, we describe three-dimensional electronic nano-resonators that provide highly subwavelength confinement of the electromagnetic energy, beyond the microcavity limits and illustrate that these device architectures have a tremendous potential to increase the temperature of operation of infrared quantum detectors.

nanostructures and absorbing regions. The detector optimization, following the guidelines of our figures of merit, will lead us to a resonator architectures where the electric field is confined in nano-metric volumes, with strong potential for high temperature, high performance THz quantum detectors.
In figure 1(a) we have presented a general scheme for a detector built with a metallic resonator of geometrical cross section σ. For our discussion, we shall consider the specific case of unipolar intersubband detectors such as QWIP. We suppose that the absorbing region of the detector ('resonant absorber' as indicated in figure 1(a)) is made, for instance, of a two-level semiconductor quantum well with subband energies E 1 and E 2 , as depicted in figure 1(b). A typical example of photonic structure of such detector is a double-metal patch resonator, which acts both as a microcavity and an antenna [26,31]. Let Φ be the number of photons incident per unit surface and unit time on the structure. We define an area A coll , larger than the geometrical cross section σ, such that the number of photons absorbed by the detector per unit time are exactly equal to A coll Φ. The total current produced in the detectors then comprises a photocurrent I photo , and a thermally activated dark current I dark : Here R is the internal responsivity of the detector, J 0 is a constant that depends solely on the properties of the absorbing region [26], E act is an activation energy that in most cases can be identified with the energy of the quantum transition E 21 =E 2 − E 1 , and k B is the Boltzmann constant. In the following, we will assume that the electromagnetic mode of the metallic nano-structure has a resonant frequency f res that is always matched with the frequency of the quantum transition, f res =E 21 /ћ. When the photon flux arises from the thermal emission of a 300 K background, we can define an important figure of merit, which is the background limited performance (BLIP) temperature of the detector, T BLIP [26]. This quantity is inferred from the temperature dependence of the dark current equation (2) through the condition I photo =I dark (T BLIP ). The value of T BLIP sets the temperature above which the detectivity is limited by the dark current. Below T BLIP the noise properties are imposed by the photocurrent from the 300 K background and no detectivity improvement can be obtained by lowering the temperature of operation. Typical values of T BLIP for intersubband detectors are 70 K for resonant wavelength around 9 μm [33,34] and much lower (10 K-17 K) for detectors operating at THz frequencies [35]. This decrease of the performance can be easily deduced from equation (2) where E act ∼E 12 scales with the energy separation between levels 1 and 2 [36]. The benefit from the antenna effect is apparent immediately from equation (1) as the photo-current I photo is proportional to the photon collection area A coll , and the dark current to the geometrical cross section σ. In that sense, the areas A coll and σ can be identified as the 'photonic' and 'electric' area of the detector [27]. Therefore, in the case where A coll ?σ the photocurrent signal is enhanced with respect to the dark current, leading to a Figure 1. (a) A general scheme of a detector which comprises a resonant absorber embedded in a metallic nanoresonator. Thanks to the antenna function of the resonator, photons are gathered on a collection area A coll that can be much larger than the detector cross section σ. The quantity σ indicates the number of photons incident on the structure per unit surface and unit time. (b) A resonant absorber based on the confined subbands in semiconductor quantum well of a thickness L qw , typical for a QWIP. For simplicity, the in-plane parabolic dispersion is not indicated. Electrons are promoted from the first level E 1 to the second E 2 either by photoabsorption (photocurrent I phot ) or thermal excitation (dark current I dark ). (c) A standard geometry for the characterizations of absorbing regions based on quantum wells. The photon flux is coupled into the absorber through the angle-polished facet of the semiconductor substrate. In that case, the photon absorption area A coll coincides with the geometrical cross section σ.
Using the model from reference [15], the internal responsivity R of the structure, which is defined in equation (1) can be expressed as: The term 'internal' is used in the sense that the photocurrent has been normalized on the number of photons coupled into the device, in accordance with the definition of A coll from equation (1). Here Q is the quality factor of the resonant structure and the coefficient B isb describes analogously the fraction of the electromagnetic field absorbed by the quantum transition, and averaged per cycle of oscillation, normalized on the total number of photons stored in the cavity. The quality factor Q thus has three contributions: 1/Q=B isb +1/Q rad +1/Q nr , with Q rad describing the radiation loss of the resonator, owe to its coupling with the free space radiation, and Q nr describes all non-radiative loss channels (i.e. loss in the metal) other than the resonant QWIP absorption described by B isb . The parameter g is the photo-absorption gain (number of electrons generated per photon absorbed), and e is the electron charge. Using the results from [15], the coefficient B isb (E) can be expressed as a function of the energy E of the incident photons: Here G  is the linewidth of the quantum transition, E P is the plasma frequency that depends on the number of available carriers for photo-absorption [15], and f w is the geometrical overlap between the electric field of the nano-structure and the quantum absorbing region in the direction perpendicular to the surface of the device. In the case of a double-metal structure, as described in figure 1(a), it can be expressed as f w =N qw L qw /L, with L the thickness of the double-metal structure (figure 1(a)), L qw the thickness of the quantum well and N qw the total number of quantum wells inserted the metallic structure.
In order to evaluate the impact of the geometrical layout from figure 1(a) we compare it to a standard substrate-coupled geometry, also referred as 'mesa', depicted in figure 1(c). The mesa configuration is used to characterize the detector absorbing region alone [38]. In this configuration the incident photons are coupled through the angle-polished facet of the semiconductor substrate. To recover the standard definition of the photocurrent in this case [30], we assume that the quantity A coll coincides with physical detector dimensions, A coll =σ. The photocurrent is then expressed as I 0 photo =tR 0 σΦE 21 , where t is the transmission coefficient of the facet, and h = ( ) R eg N E qw 0 2 1 is the intrinsic responsivity of the QWIP. Here η is the absorption quantum efficiency of the transition [36]. The BLIP temperature of the mesa T 0 BLIP is defined as I 0 photo =I dark (T 0 BLIP ), and depends solely on the quantum design of the absorbing region and the properties of the semiconductor material, but not on the device cross section σ. Using the expression of the quantum efficiency η from [36] we obtain the following relation: Here, θ is the incident angle in figure 1(c) that shall be considered θ=45°as usually found in experiments, and l = c E 21 21 is the resonant wavelength. Combining all definitions stated above, we arrive at the following link between the BLIP temperature T BLIP of the antenna-coupled structure as a function of T 0 BLIP of the reference sample described in figure 1(c): l n l n l n In this formula we have defined two quantities F and K such as: where V=σL is the volume of the nano-structure, as shown in figure 1( Equation (6) indicates that the BLIP temperature of the detector can be increased if the factor F is increased by judicious of the geometry of the structure. This quantity, which is an important figure of merit for any detector architecture, has a straightforward interpretation in terms of the local electric field enhancement of the photonic resonator [39]. Indeed, let E out be the electric field amplitude of the incident wave. Then the power flow density of the incident wave can be expressed as ε 0 c|E out | 2 /2. If E in is the amplitude of the electric field stored inside the resonator, then the total energy density can be expressed as ε 0 ε|E in | 2 V/2 with ε=n 2 (the factor ½ arises from cosine variation of the electric field along the patch [7]). We used the fact that the electric energy density ε 0 ε|E in | 2 V/4 is equal to the magnetic energy density at resonance. Then the energy conservation dictates that A coll ε 0 c|E out | 2 /2=ε 0 ε|E in | 2 V/2×(Q/ω res ) with ω res the angular frequency of the resonant mode, which leads to the equality: ε|E in | 2 /|E out | 2 =F/2π. Because of this interpretation, the factor F shall be referred in the following as 'focusing factor', as it expresses quantitatively how the density energy of free space photons is compressed into the cavity volume V. It is remarkable that if we consider the case of a single elastically bound electron, we have A coll =3λ 21 2 /4π, and then the factor F/2π ultimately becomes the Purcell factor of the system.
These considerations indicate that the definition (7) has a more general bearing than the particular case of intersubband detectors that we used to establish the results (6) and (7). Indeed, we expect that for other type of detectors the numerical constant lnK will be different, yet of the order of unity, and the equation (6) will still hold.
Since, according to equation (6), the BLIP temperature T BLIP has a logarithmic dependence on the focusing factor F, a substantial increase in the temperature performance can be achieved when F is varied through several orders of magnitude. From equation (7), the design parameters at hand are the collection area A coll , the volume of the absorbing region V and the quality factor Q. This last term can be analyzed by comparing the intrinsic quality factor of the resonator Q cav , defined from 1/Q cav =1/Q rad +1/Q nr and the coefficient B isb . Typically, in double-metal resonators with strongly subwavelength thickness we have 1/Q rad =1/Q nr [7]. In the case of very low Q cav cavities, a system too lossy with 1/Q cav ?B isb , the detector is inefficient as most of the incident photons are absorbed by the metal rather than the quantum transition. In the opposite regime, where B isb ?1/ Q cav , all photons are absorbed by the quantum wells. This situation occurs in the case of very high Q cav resonators (which could be, for example, photonic crystal resonators [30]). Notice that in this limit the responsivity R does not depend anymore on B isb and the focusing factor becomes inversionally proportional to B isb . There is no interest therefore in further increasing the rate of absorption as it will simply increase the noise due to the increase of photogenerated carries. In this counter-intuitive limit the high Q cav cavity acts as a photon storage that 'slows down' the absorption in the active region, thus increasing the T BLIP . Moreover, increasing the absorption would imply the use of heavily doped QWIPs [40,41], which have a strong dark current and low BLIP temperature and detectivity. Optimum detector performance has been already established by relatively low doping levels: on the order of 10 10 cm −2 -10 11 cm −2 depending on the spectral region of operation [36]. The only degrees of freedom that are left to improve the photonic resonator that contains the absorbing region are the geometrical properties contained in the factor F defined in equation (7).
It has been experimentally and theoretically shown that arrays of resonators can feature a strong ability to absorb the incident radiation on a large cross-section [7,42,44]. Therefore, following the ideas developed in previous works [8,15,16,45], we have analyzed diluted arrays of double-metal square patch-cavity resonators as a model system to illustrate the roles of the parameters A coll and F. In that case B isb =0 and Q=Q cav . For our analysis, we started by varying the filling factor of a set of arrays with identical patches as described in figure 2(a). The resonators are made of a 2 μm thick GaAs layer sandwiched between two metal plates. The patches have a lateral width of s=10 μm, thus resonating at a frequency f res =c/(2n eff s) around 4.5 THz, where n eff ∼3.3 the effective index of the confined mode [7]. The arrays are periodic with a square unit cell area p 2 =Σ. As shown in figure 2(a) the semiconductor material is dry-etched by inductively coupled plasma everywhere except under the square metal patch. While the area of the square patch is kept constant at 10×10 μm 2 , we studied arrays with periods p=15 μm, 25 μm, 35 μm, 55 μm and 80 μm (respective unit cells: Σ=225 μm 2 , 625 μm 2 , 1225 μm 2 , 3025 μm 2 and 6400 μm 2 ). In figure 2(b) we show the reflectivity spectra of the arrays, obtained with the Globar source of a Fourier Transform Interferometer at almost normal incidence (15°with respect to the array normal). The mode of the double-metal patch is revealed as a dip in the reflectivity spectra at the expected value f res =4.5 THz. Apart the quality factor Q and the resonant frequency f res , the dip is characterized by its 'contrast' R = -C 1 min , with R min the reflectivity minimum. The values of both C and Q, extracted from Lorentzian fits of the data are plotted as function of Σ in figure 2(c). Note that both C and Q depend strongly on the array periodicity, as previously observed with split-ring resonators [43] and patch wire cavities [8]. According to energy conservation, the contrast C provides the fraction of photons absorbed by the array at resonance. Since the number of incident photons per unit time on each element of the array is ΦΣ, the collection element for each patch in the array is provided by the equation: The reflectivity spectra therefore allow a direct determination of the collection area A coll per element. This result has been plotted in figure 2(d) (dots). We can already notice that the typical values of A coll are much larger than the cross section of the square pad (10×10 μm 2 ), up to 15 times in the measured values. The remarkable behavior that is evident from the results in figure 2 is that, while the contrast C is optimum for dense arrays and decreases with the unit cell area Σ, the collection area A coll increases monotonically with Σ and seems to saturate. We thus observe a rather counter-intuitive result that, when put together, the absorbing elements tend to decrease their individual absorption area. On the other side, the system as a whole has an optimum ability to absorb the incident radiation, when the contrast C reaches unity for a particular value of Σ (critical coupling point, [7]). To capture this behavior and to estimate the maximum value of A coll in our system we rely on the analytical model described in [8], where we provided an explicit expression of peak contrast C as a function of Σ and the characteristics of the individual resonators: In this model, the effect of the radiation loss latter is evaluated according to the expression Q rad =λ 21 πn eff 2 / (4LD rad )=73 [8] with n eff =λ 21 /2 s=3.3, and thus Q rad is typically one order of magnitude greater than Q nr .
Using the values of the measured quality factor Q as a function of the unit cell area Σ ( figure 2(c)) and our analytical model from equation (10) we obtained C as a function of Σ, as shown as a continuous curve in figure 2(c) (upper panel). The agreement between our model and the experiment is excellent, without any fit parameters, except for the denser array (Σ=225 μm 2 ). The discrepancy at that point can be explained by nearfield coupling between the resonators [45], which introduced additional linewidth broadening. Using our model, we can now provide an analytical expression for the collection area A coll of the patch antenna array as defined in equation (9):  is the collection area obtained by letting S  ¥ for a given quality factor Q nr and as such it represents the collection area of a single element of the array. Note that the area A coll 1 is 4 times bigger than the area of the unit cell Σ at the critical coupling, i.e. when C=1 and S = A 4 .
coll 1 As seen from the data in figure 2(c) the quality factor has a strong dependence on Σ and therefore the asymptotic limit that provides the collection area of a single element is obtained as: with the ¥ Q nr the asymptotic limit of the non-radiative contribution in the quality factor for very diluted arrays, obtained from the data in figure 2(c). Equation (11) accounts well for the monotonic increase of the collection area observed in experiments, as shown in figure 2(d). It expresses the fact that the collection area per element is always smaller when elements are arranged in an array. Using our model, we can extrapolate the 'intrinsic' value of ¥ A coll to be 1717 μm 2 =41×41 μm 2 , almost 17 times larger than the cross section of the square pad. With the analytical expression of ¥ A coll we evaluate the maximum local field enhancement for the patch antenna geometry, for a volume V=Ls 2 : coll nr 21 2 Here, = ¥ e Q Q nr rad is the extraction efficiency of the resonator. It is remarkable that the limiting value ¥ F is completely independent from the aspect ratio of the resonator, and namely the resonator volume, but depends only on the resonator loss. This limit is intimately related to the fact that the patch antenna resonance relies on propagation effects, i.e. the lateral dimensions of the cavity are commensurable with the wavelength in the material. Indeed, the modes of these structures can be described as a standing-wave Fabri-Perot like resonances [7]. As the wavelength of the first order mode is provided by λ 21 =2n eff s, the resonator dimensions cancel by combining equations (7) and (12). The same considerations can be easily shown to be valid for patch antenna resonators with an arbitrary cross sections. In particular, taking we obtain from equation (13) a limiting value F ∞ =1.1×10 4 . Let us considering the THz detector from [16], which operates at 5.4 THz and had a BLIP temperature T 0 BLIP =17 K. An activation energy E act =17.7 meV was obtained by fitting the dark current as a function of the temperature with equation (2) from the data reported in [16]. These figures lead to a maximum achievable BLIP performance of = ¥ T K 38 BLIP from equation (6). As ¥ F is independent from the shape of the microcavity, and hence the resonant wavelength, we can use the above results to make predictions for infrared detectors operating in other ranges. For instance, our results can be extrapolated to the Mid-Infrared (MIR) range (λ=9 μm), which is practically important for thermal imaging [36]. Typically, for a λ=9 μm we have T BLIP 0 =70 K [34], and the quality factors are twice lower with respective to THz cavities, because of the increased metal loss [46]. Extrapolating these numbers to the MIR domain we obtain =¥ F 3 10 3 which provides = ¥ T 100 K BLIP for a cavity-embedded QWIP detector at 9 μm.
From a practical point of view, it is important to compare the collection efficiency of the detector, seen as a single pixel, with the performance of the optical system, used, for instance, in an imaging array. Provided a wavelength of operation λ 21 , the ultimate diffraction-limited spot has an area of the order of l . 21 2 This value is generally superior to the collection area of a single detector. Indeed, in the present geometry, using equation (12) the ratio l ¥ A coll 21 2 is readily estimated at 0.4. Since it is practically difficult to achieve diffraction-limited spots, a natural solution is to use an array of elements that covers wide illuminated area. However, as shown from figure 2(d), this leads to a reduction of the individual collection area A coll and hence a lower temperature performance. These considerations show that there is always some trade-off between the maximum temperature T BLIP required and the overall collection ability of the detector array. This is further illustrated in figure 3, where we have shown both the BLIP temperature and the reflectivity contrast C as a function of the array unit cell surface Σ, assuming the same absorbing region as in [16]. This graph shows that the maximum temperature T ∞ BLIP =38 K is obtained for Σ=3015 μm (array period p=55 μm), where the photon rejection rate is 1 − C=57%, however the temperature drops only by less than one degree if the array is operated with 50% rejection rate. On the other hand, T BLIP for a dense array operating at the critical coupling with C=1 is equal to 25 K, still about 8 K higher than the intrinsic value T 0 BLIP =17 K. A possible method to optimize A coll is to use planar antennas, such as bow-tie or spiral have the ability to collect efficiently the incident radiation on areas larger than the diffraction limit [47,48], however these structures are not naturally compatible with the intersubband selection rule. Alternative solutions have been proposed, such as the combination of planar antennas with patch microcavities [23], or exploring 'monopolar' resonators combined with loop antennas [49][50][51]. In the following, we propose another approach to increase the focusing factor of the array F, and hence the value of T BLIP even for dense arrays, which adds degrees of freedom for the optimization of the system.
Since the fundamental limitation (equation (13)) in microcavities arises from propagation effects, a possible alternative is to explore electromagnetic resonators that are not subject to this limitations. A natural choice for such systems is the circuit resonators, which are used as building blocks of metamaterials in the high frequency part of the microwave spectrum [52]. Indeed, such resonators operate well below the propagation limit [53], and could therefore provide much higher focusing factors F. Such structures have been recently demonstrated specifically for intersubband THz devices [51,[54][55][56][57][58][59][60]. Typically, such circuit-like structure have a double-metal part, that plays the role of the capacitance, and an inductive part, that confines the magnetic field. In that case the volume V in the expression of the focusing factor (equation (7)) corresponds to the volume of the capacitance [61]. Note that this type of electromagnetic resonators could be implemented also at higher frequencies.
However their fabrication could be already very challenging in the mid-infrared spectral region and extremely difficult in the near-ir/visible.
We recently proposed a three-dimensional architecture, which allows for an increased confinement of the THz electric field and is compatible with the selection rule of intersubband transitions [59]. Our design is recalled in figure 4(a). It consists of a thin metal strip instead of a continuous ground plate, a dielectric slab and a top metal loop. In figures 4(b) and (c) we present simulations of the electric E z and magnetic H z field of the structure in a plane that crosses the middle of the dielectric slab, with the z-axis is perpendicular to the plane of the slab. These simulations reveal that the electric field E z is strongly confined in the regions of overlap between the top and bottom metals, while the magnetic field H z is rather localized around the loop. The analysis presented in [59] allows to assign a capacitance C db and an inductance L loop as a function of the geometrical parameters of the structure. The resonant frequency is then obtained as res db loop This design allows therefore the reduction of the capacitive part by expending of the inductive one, while the resonant frequency f res is kept constant [59]. In that case the capacitance volume V can be made much smaller than the resonant wavelength, resulting in a high focusing factor F, according to our definition equation (7).
To probe this concept experimentally, we realized structures with progressively shrinking capacitances while the inductive loops have been adjusted so that all structures resonate around 3 THz. For these proof of principle studies we used SiO 2 dielectric layers which simplified the fabrication process. Scanning electron microscope Figure 3. Plot of the peak contrast C and the BLIP temperature T BLIP as a function of Σ, assuming an absorbing region resonant at 5.4 THz, as the one described in [16]. The dashed line indicates C=0.5, which corresponds to a 50% of photon rejection rate. For the 0.5×0.5 μm 2 structure ( figure 5(b)) we measured arrays with two different Σ as indicated in figure 5(e). The collection area for these systems that have both transmission and reflection ports is now defined as coll min min in a straightforward generalization of equation (9). In our structures, we observed no features in the reflectivity port (R min =0) and therefore T = S -( ) A 1 coll min . In order to estimate the effective volumes of our structures, we simulated the electromagnetic field by using finite difference domain software. The effective volume of the electric field was estimated as where w e is the time-averaged electric energy density. To take into account the intersubband selection rule, this quantity was corrected with a factor 1/Ψ, defined as The values obtained for the effective volume V eff /Ψ are thus used for the estimation of the focusing factor in equation (7). The numerical results are summarized in table 1, in comparison with the geometrical volume V=2σL.
The focusing factors F for both patch cavities and LC-resonators is presented in figure 6(a) as a function of the array cross section Σ. Even if the microcavities and LC resonators studied for this plot operate at different THz frequencies, we expect that the orders of magnitude of the focusing factors will be the same. For the patch  cavities we have indicated the single resonator limit from equation (13), that is independent from the operating wavelength, as the quality factors in such systems do not change significantly from 3 THz to 5 THz. We observe that the LC structures have systematically higher focusing factors F than the patch cavities, even for dense arrays with small Σ. According to our results for the evolution of A coll as a function of Σ (figure 2(d)), even higher focusing factors should be expected in low density arrays. In figure 6(b) we show the expected elevation of the BLIP temperature as a function of the focusing factor F (equation (5)) for the THz detector from [16]. The continuous curve is a numerical result obtained from equation (6). The stars correspond to the experimental data from [16]. The projected T BLIP for the LC circuits, even in dense arrays is above 70 K, while the limit for the patch-cavity systems is = ¥ T 38 K.

BLIP
Capacitive parts with widths as small as 100 nm are feasible by electrical lithography. This resonator architecture therefore could allow reaching operating temperatures close to the liquid Nitrogen temperature. Clearly, this concept will be beneficial also for Mid-IR detectors where the T BLIP temperature could reach 130 K from = T 70 K BLIP 0 . Another important figure of merit that can be impacted by the architecture of the resonators is the specific detectivity, which should be redefined appropriately. Since we expect the effect of the dark current to be reduced in our architectures, we are going to consider the background limited specific detectivity D * BLIP . We use the following definition: Here, differently from the standard definition in [32,36] the specific detectivity has been normalized on the collection area A coll instead of the area of the device σ. This choice is justified as follows: at low temperature, the dark current is negligible, and the main source of noise in the detector is the photocurrent I photo . Then, using equation (1), we can express the specific detectivity at low temperature: with Φ 300K the number of photons per unit surface and per unit time radiated by a black-body at the peak absorption energy E 21 of the detector. In equation (15), the specific detectivity becomes independent of the collection area A coll , as required for the standard definition. The responsivity entering this formula is the internal responsivity defined in equation (3). In figure 7, we have plotted the predictions from equation (14) for the different THz devices envisioned in this work and [16]. The maximum detectivities at low temperature have been computed from the experimental data reported in [16]. The first feature that is clearly visible is the increase of the temperature performance of the detector through the higher T BLIP as discussed above. In accordance with Figure 6. (a) A summary of the focusing factors F from arrays of nano-resonators with different geometries discussed in this work. The squares are the rectangular patches and the other symbols indicate the LC resonators described in figure 5, with the corresponding size of the square capacitance. This figure combines data with structures that operate from 3 to 5 THz, assuming that the dependence of F on the frequency is weak in the spectral range. The dashed line is the limit for patch antennas from equation (13). (b) T BLIP as a function of the focusing factor F (continuous line). The line+symbols curves correspond to the different resonators discussed in this work, and the dashed line to the patch antenna limit. The stars corresponds to experimental values reported in [16].
equations (1), (2) and (14) the T BLIP corresponds to the temperature where the detectivity is less than a factor of 2 1/2 than its maximal value (dashed lines in figure 6). While the T BLIP is increased, the decrease of the specific detectivity with the temperature is also slowed down, meaning that the detector can operate efficiently at temperature higher than T BLIP . For these plots, we have assumed that the gain g and the absorption coefficient B isb of the detector are constant as a function of the temperature. Note that this hypothesis is a simplification since the gain can exhibit temperature dependence [62] and the absorption coefficient decreases with the temperature due to thermal redistribution of carriers [63].
Another impact of the cavity geometry that is visible in figure 6 is the increase of the maximum detectivity with respect to the mesa device. This increase can be explained by considering equation (5), which links the absorption coefficient in the cavity geometry (B isb ) with that the quantum efficiency in the mesa geometry (λ). The absorption coefficient is enhanced due to the ratio λ 21 /L, which expresses the vertical confinement in the cavity. Indeed, this factor is higher for structures with highly sub-wavelength thickness (L=λ 21 ), which results to higher responsivities for the cavity geometry with respect to the mesa [16].
It is interesting to note that our results provide a theoretical limit for the maximum achievable background limited detectivity in a microcavity-coupled system. Indeed, using the expression (3) for the case, B isb ?1/Q cav , and replacing the resulting expression for the responsivity in equation (15), we obtain: This results states that the maximum BLIP specific detectivity is optimized in detectors which contain a single quantum well, N qw =1 which absorbs most of the photons fed in the cavity. In that case, the photon flux from the thermal background coupled in the detector F 300K depends only on the energy (Planck's radiation law) and the spectral bandwith of the detector. The maximum BLIP detectivity is then provided solely by the absorption energy of the detector E 21 and the spectral width of the responsivity curve, as the only source of noise in the system are the carriers generated by the photons from the 300 K background. A typical value estimated from equation (16) for a THz detector operating at 5.4 THz with a single quantum well and 10% spectral bandwidth is D * BLIP =6×10 12 cmHz 1/2 /W. Note that this value does not limit the dark current-limited detectivity D * signal , which is expected to increase with the ratio (A coll /σ) 1/2 [27]. This corresponds to a situation where the detector is shielded from the thermal radiation and is illuminated only by the source to be measured.
In summary, we have studied the impact of the lateral and vertical photonic confinement for quantum detectors of infrared radiation. We have pointed out the importance of two figures of merit, the collection area of the detector A coll and a dimensionless local field enhancement factor F. We have shown how the collection area A coll can be inferred directly from reflectivity measurements on arrays of nano-resonators. We also commented the effect of such array configuration on the detector performance, taking into account the finite size of the incident beam. Namely, there is always a trade-off between the collection efficiency of the detector array and its temperature performance. We linked the maximum background limited operating temperature T BLIP to the local field enhancement factor F, which has been expressed in equation (7) through A coll , the quality factor of the structure Q and the volume V. We have found that in systems such as patch microcavities, which rely on Figure 7. Background limited specific detectivity D * as a function of the temperature, for the case of different structures presented in this work. The maximum value of D * depends only on the vertical confinement of the structure, expressed as the ratio λ 21 /L ('cavity enhancement effect'). An increased focusing factor F leads to an increase in the temperature range of the detector operation through an increased T BLIP .
propagation effects, F is ultimately limited by the resonator loss. This limit can be exceeded in electromagnetic resonators based on quasi-static effects, such as LC circuits. In particular, our theoretical predictions indicate that THz detectors based on such resonators would operate close to the liquid nitrogen temperature. The figures of merit A coll and F introduced here can also be applied to other systems which provide highly subwavelength confinement, such as, for instance, the localized plasmon modes in metallic nano-particles [64]. As another example, we expect very strong high order non-linear effects proportional to F 2 or F 3 , due to the very tight field confinement in these nanostructures [65].