Paul Henri Allé

Patricia Garcia-Muñoz

Kopoin Adouby

Nicolas Keller

Didier Robert

Efficient photocatalytic mineralization of polymethylmethacrylate and polystyrene nanoplastics by TiO2 /β-SiC alveolar foams

Keywords: nanoplastic, polymethylmethacrylate, Polystyrene, water treatment, photocatalysis, β-SiC

Many household wastewaters contain Microplastics (MPs) and nanoplastics (NPs). Most of this type of pollution is not stopped by current wastewater treatment plants provoking their entrance into the global aquatic systems. This is an emerging issue and potential threat to marine life and human health. However, advanced technologies for efficient MPs and NPs control and elimination remain largely underdeveloped. Heterogeneous photocatalysis represents an attractive and efficient decomposition technique for the nanoplastic particle degradation. The first objectives of this work were i) to study the feasibility of nanoplastic degradation by photocatalysis and ii) to determine the influence of certain reaction parameters i.e. the flowrate, the pH0 and the light intensity on the photocatalytic degradation of calibrated polymethylmethacrylate (PMMA) and polystyrene (PS) nanoparticles with TiO2-P25/β-SiC foams under UV-A. The first results are very encouraging, because to our knowledge, we are the first to show that it is possible to mineralize PMMA and PS nanobeads by UV-A photocatalysis. We were able to convert in 7 hours, 50% of an aqueous suspension loaded with PMMA nanobeads (TOC = 12 mg/l) by working at an irradiance of 112 W/m 2 , with a flowrate of 10 mL/min and at initial pH value of 6.3, using TiO2-P25/β-SiC foams in a flow-through mode. We also showed that the photocatalytic treatment can be applied to various polymers, such as polystyrene and PMMA.

Introduction

Nowadays, we can find micro-and nano-sized plastics (MPs and NPs) worldwide in coastal regions and aquatic ecosystems transported by wind and ocean currents. In a recent report of the European Union (European Commission 2017), microplastics most commonly defined as man-made plastic items smaller than 5 mm, are of particular concern for the marine and inland aquatic environment.

The scientific community has not given a precise definition for nanoplastics but we can consider that they represent plastic particles less than ten micrometers. The potential impacts of micro and nanoplastics present in products on the (aquatic) environment and possibly on the human health have generated concerns in Member States of the European Union and worldwide, with several implemented or proposed bans/restrictions taking various different forms.

In the past decade, there has been a major change in the pollution potential for nanoplastic pollution [START_REF] Moore | Synthetic polymers in the marine environment: a rapidly increasing, long-term threat[END_REF] in the oceans, with the shift from natural exfoliators to nanoplastics in skin cleansers.

Consequently, certain NPs such as nanobeads derived from toothpaste and facial cleanser are directly introduced into wastewater by human activities. In the 1990s, these nanoplastics were mainly present in hand cleaners, in the form of soaps plastic-sand liquids, but recognized as a minor source of plastic pollution [START_REF] Gregory | Plastic 'scrubbers' in hand-cleansers: a further (and minor) source for marine pollution identified[END_REF]. However, as nanoplastics have now replaced natural exfoliating materials (pumpkin seeds, apricot kernels...) in facial cleansers, many consumers now have a product containing nanoplastics (nanobeads) at home and uses it daily. Even if some countries have started to ban personal care products containing plastic nanobeads (SAM 2019), due to their persistence, the threat will still be here for long time to the environment (Nemls et al. 2019).

Then the nanobeads travel through the wastewater systems of the cities, but because of their small size a large part pass through the filters of the wastewater plants (typically coarse, >6 mm, and fine filters, 1.5-6 mm Vesilend, 2003) and enter the oceans [START_REF] Browne | Microplastic an emerging contaminant of potential concern?[END_REF]. Thus, wastewater treatment plants (WWTPs) can be also considered as one of the most important source points of MPs and NPs release [START_REF] Cole | Microplastics as contaminants in the marine environment: a review[END_REF][START_REF] Talvitie | Do wastewater treatment plants act as a potential point source of microplastics? Preliminary study in the coastal Gulf of Finland, Baltic Sea[END_REF] in the aquatic and marine environment. While the larger plastic particles are efficiently removed in WWTPs, this is not the case for the NPs that are entering and accumulating in the aquatic environments [START_REF] Murphy | Wastewater Treatment Works (WwTW) as a Source of Microplastics in the Aquatic Environment[END_REF]. MPs and NPs can be considered as emerging pollutants of growing particular concern for human health and environment.

NPs are strong adsorbents for hydrophobic toxic pollutants and may affect their fate and toxicity in the environment. For example, photosynthesis disruption was reported in studies investigating effects on phytoplankton. Current literature suggests that the plastic particle toxicity depends on the concentration, the particle size, the exposure time as well as the shape and the type of the polymers.

In a recent review, authors describe nanoplastic removal methods in wastewater that include sorption and filtration, removal based on chemical phenomena, biological ingestion treatments and advanced oxidation technologies [START_REF] Padervand | Removal of microplastics from the environment. A review[END_REF]. Among the different advanced oxidation technologies (AOTs) able to remove nanoplastics from wastewater, heterogeneous photocatalysis represents an attractive and efficient technique for the degradation of these particular pollutants.

Heterogeneous photocatalysis is the most efficient and economically most feasible method from the mineralization point of view, compared to ozonation or chlorination. Moreover, with chlorination, there is the risk of forming organochlorine intermediates which are carcinogenic. Nanofiltration could be considered, but it is an expensive technique, which does not eliminate plastic nanoparticles but displace this pollutant. There is also a great risk of fouling the filters.

Very few works has been done on the elimination of plastics particles by heterogeneous photocatalysis and most of them concern polymeric films such as Polyvinylchloride PVC [START_REF] Horikoshi | Photocatalyzed Degradation of Polymers in Aqueous Semiconductor Suspensions. 3. Photooxidation of a Solid Polymer: TiO2-Blended Poly(vinyl chloride) Film[END_REF], Low Density PolyEthylene LDPE [START_REF] Ali | Photocatalytic degradation of low density polyethylene (LDPE) films using titania nanotubes[END_REF][START_REF] Tofa | Visible light photocatalytic degradation of microplastic residues with zinc oxide nanorods[END_REF] and Polystyrene PS [START_REF] Shang | Photocatalytic degradation of polystyrene plastic under fluorescent light[END_REF]. To our knowledge, in the literature only one example concerns the photocatalytic removal of polymer nanoparticles [START_REF] Wang | Photocatalytic TiO2 Micromotors for Removal of Microplastics and Suspended Matter[END_REF]. The authors used a photocatalytic TiO2-based micromotor (Au@mag@TiO2, mag = Ni, Fe) for the elimination of nanoplastics contained and extracted from washing powders, toothpastes and face cleansing creams.

Although TiO2 in the powder form demonstrated high photocatalytic activity under UV-A light in the degradation of a wide range of molecules in water, the filtration process required after the water treatment makes its application on an industrial scale complex and costly. To solve this problem, we used open-cell β-SiC alveolar foams as photocatalyst support. The integration of these materials in a photoreactor allows operating in a continuous mode [START_REF] Kouamé | β-SiC foams as a promising structured photocatalytic support for water and air detoxification[END_REF], M'Bra et al 2019).

In this manuscript, we use a b-SiC foam supported TiO2-P25 photocatalyst integrated into a flowthrough photoreactor for nanoplastics removal in wastewater. We took calibrated polymethylmethacrylate (PMMA) and polystyrene (PS) nanobead particles as model systems for microplastic pollutants. The photocatalytic performances have been evaluated by analyzing the timeevolution of the total organic carbon (TOC) upon UV-A irradiation. The influence of some important operating parameters (ie. pH, flowrate, incident irradiance in the photoreactor, particles size…) on the photocatalytic removal of PMMA and PS nanobeads has been studied in order to optimize the process.

Experimental part

Photocatalytic materials and calibrated polymer nanobeads

Open-cell β-Silicon carbide alveolar foams were employed as structured support for the Aeroxide© TiO2-P25 photocatalyst (Evonik, Germany) due to their high chemical and thermal stability, as well as their large open porosity that maintain ultra-low pressure drop and favors the light transmission into the reactor core. In order to obtain a homogeneous TiO2 coating inside the 3D structure of the foam, the dip-coating technique followed by thermal annealing has been employed to obtain a high adhesion of the TiO2-P25 coating on the substrate. For a detailed preparation protocol, please refer to Marien et al 2019. Finally, these foams were integrated in a photoreactor especially designed for this study (see figure 1).

In order to measure the efficiency of the process, we used commercial monodisperse suspensions of polymers supplied by Microparticles GmbH. The calibrated polymer nanoparticles have a high monodispersity (CV < 3.8 %), spherical shape, high optical transparency, high mechanical, and chemical stability as well as low density (1,05 g/cm 3 ). Polystyrene and polymethylmethacrylate products in the nanometer size range are as 5 % (w/v) aqueous suspensions. For this study, we used three monodisperse suspensions of nanobeads: one with polymethylmethacrylate nanobeads (NBs-PMMA (C5O2H8)n) particles size: 105 nm with a standard deviation of 0.005 μm) and two with polystyrene nanobeads (C8H9)n) (NBs-PS1 particles size: 140 nm, Standard Deviation : 0.005 μm and NBs-PS2 particles size: 508 nm Standard Deviation : 0.015 μm).

Photocatalytic reactor and analysis

Photocatalytic experiments were performed in a recirculation mode photoreactor. The photocatalytic system is presented in detail in figure 1. Three TiO2-P25/β-SiC foams (diameter 38 mm; length 30 mm) were inserted in the tubular quartz reactor (diameter 40 mm; length 200 mm, volume 250ml) surrounded by 2 to 4 UV-A lamps (1 cm between the quartz tube and the lamps).

Each foam sample (10 g) contains 10 ± 1 wt% of TiO2. Effluent inlet and outlet diameters are 5 mm.

Each lamp (Philips T5 15W 10 Actinic BL) have a maximum emission wavelength at 354 nm. A volume of 600 ml of aqueous nanoplastic suspension to be treated is introduced into the system. These suspensions are prepared by mixing 300 mg of commercial monodisperse solution in 1000 mL of distilled water. This corresponds to an initial Total Organic Carbon, (TOC0) around 12 mg/L. The reactor was connected to a peristaltic pump (Masterflex L/S, Cole Parmer) to control the flow rate of the nanobeads suspension in the photoreactor. The lamps are switched on after 30 min under recirculation in the dark for allowing the establishment of the equilibrium of the system. Except for the experiments with pH control, all the tests were carried out at free pH0. During the process under irradiation, 10 mL of solution were taken at regular time-intervals to evaluate the Total Organic Carbon without filtration. The TOC measurement is not directly related to the concentration of polymer nanobeads during irradiation. However, it can demonstrate the transformation of polymer nanoparticles under the action of photocatalysis. TOC measurements were determined by a Shimadzu TOC-L apparatus.

Results and discussion

The main objective of this work is to demonstrate the feasibility to degrade polymer nanobeads in aqueous solution by heterogeneous photocatalysis with TiO2-P25/β-SiC foams under UV-A irradiation. The tests were carried out on diluted suspensions of NBs-PMMA and NBs-PS corresponding to approximately 12 mg/L of total organic carbon (TOC0). We followed the TOC concentration of unfiltered solutions during the UV-A irradiation time. The residual concentration of organic carbon in the solution during irradiation process is a good indicator of the polymer degradation, although the TOC is not directly related to the concentration of nanoplastic during irradiation. Indeed, the value of the TOC during the irradiation represents the intermediates of degradation of the polymer nanoparticles which have been photo-oxidized by TiO2 under irradiation, but also the polymer nanoparticles which remains in suspension.

The first preliminary results showed that it is possible to oxidize PMMA and PS nanobeads in the presence of TiO2-P25/β-SiC foams. As it can be observed in figure 2a, TOC evolution without irradiation and photolysis (without TiO2) of NBs-PMMA are very low, while after 7 h of irradiation with TiO2-P25/β-SiC, the half of the organic carbon contained in the PMMA nanobeads is transformed into carbon dioxide, and consequently mineralized. The reaction follows a zero-order reaction kinetics (limitation by mass transfer) with a kinetic rate constant of 0.0141 mg/L.min which is conventional in supported catalysis [START_REF] Ollis | Kinetics of Photocatalyzed Reactions: Five Lessons Learned[END_REF]. The oxidation rates and the efficiency of the photocatalytic system are highly dependent on a number of operation parameters that govern the kinetics of the photomineralization. Therefore, we have determined the influence of some parameters like flow rate, pH0 and light intensity on the photocatalytic degradation of the nanoplastics.

Flow rate influence

As shown in Figure 2b, a significant effect of the flow rate on the nanoplastic photocatalytic degradation is observed, which means that an external transfer limitation intervenes, at least at low flowrate values. The higher the flowrate, the lower the kinetic rate constant and consequently the lower the TOC conversion achieved. After 7 h of irradiation at a flowrate of 50 mL/min, only 14% of PMMA has been mineralized with a kinetic rate constant of 0.0032 mg/L.min, while this percentage increases to 26% and 50% by working at 30 mL/min and 10 mL/min, respectively, with kinetic rate constants of 0.0066 mg/L.min and 0.01410 mg/L.min, respectively. The residence time of a polymer nanobeads in the photoreactor for a single pass corresponds to 25 min for a flow rate of 10 ml mL/min, 8.4 min for 30 mL/min and 5 min for 50 mL/min. Indeed, this can be explained by the size of the nanobeads which are much larger than a simple molecule of pesticide or dye [START_REF] M'bra | Heterogeneous photodegradation of Pyrimethanil and its commercial formulation with TiO2 immobilized on SiC foams[END_REF].

Most importantly, the aggregates of TiO2 on the foams have sizes of comparable range to those of the nanobeads (a few hundred nanometers). Consequently, there is little point of contact between the solid (NBs-PMMA) and the TiO2 surface, for this reason the kinetics seem to be long compared to organic molecules.

Initial pH effect

pH is one of the most important operating parameters that affect particularly the surface charge of the TiO2 photocatalyst crystallites. For Degussa TiO2-P25, the point of zero charge (pzc) occurs at pH ~6.3 [START_REF] Carabin | Photo-degradation of carbamazepine using TiO2 suspended photocatalysts[END_REF]. At free pH 6.3, we are working under optimum conditions for the photooxidation which shows mineralization rate values of 0.0141 mg/L.min (figure 3a). On the contrary, at pH 9, the TiO2 surface is negatively charged and a part of the ester functions of the nanobeads surface can be hydrolyzed into negative carboxylate groups, causing electrostatic repulsion forces and thus showing a drastic reduction of the TOC conversion (17% in 7 h) .Under these conditions, these repulsion forces between TiO2 and NBs-PMMA lead to hinder the photocatalytic reactions as can be inferred from the decrease of the kinetic constant rate down to k = 0.0116 mg/L.min. At acidic pH, there will be competition between the chloride ions (coming from the HCl solution) and NBs-PMMA towards the active sites of the TiO2 surface.

Light intensity influence

At high light intensity, the reaction rate is found to be proportional to the square root of light intensity. However, at the low-intensity level, the photodegradation rate is directly proportional to the light intensity [START_REF] Chen | Photo-degradation kinetics of 4-nitrophenol in TiO2 suspension[END_REF]. Light intensity effect has been studied by varying the number of UV-A lamps irradiating the TiO2-P25/β-SiC foams (Figure 3b), in this case also the results are consistent because the process is more efficient with an irradiance value of 112 W/m 2 ) rather than 56 W/m 2 . In our type of reactors with integrated photocatalytic 3D materials (TiO2/β-SiC foams in our case), it is necessary to achieve efficient exposure of the catalyst to light irradiation. Without photons of appropriate energy, the catalyst shows no activity. In fact, in a photocatalytic reactor, besides conventional reactor complications such as mixing, mass transfer, reaction kinetics, catalyst installation, etc., an additional engineering factor related to the illumination of catalyst becomes relevant. The illumination factor is of utmost importance since the amount of catalyst that can be activated determines the water treatment capacity of the reactor. Figure 3b shows the influence of the irradiance on the removal degree of NBs during the photocatalytic process. The reaction rates and so the TOC conversions increase when increasing the light irradiance from 56 W/m 2 to 112 W/m 2 , with kinetic rate constants of 0.0039 mg/L.min and 0.0148 mg/L.min.

Chemical structure and particles size effects

In order to show that the photocatalytic process can be effective for polymers with different molecular structures, we have applied it for the removal of polystyrene nanobeads of two different average sizes: NBs-PS1 (particles size: 140 nm) and NBs-PS2 (particles size: 508 nm). As it can be observed in figure 4, for a similar particle size, PMMA mineralization appears to be faster (kNBs-PPMA = 0.0148 mg/L.min while kNBs-PS1 = 0.0065 mg/L.min). This is probably due to the C=O and C-O bonds which are more "breakables" than the C-C, C=C and C-H bonds of polystyrene. Indeed recently, Dinood et al (2019), have highlighted the photocatalytic degradation mechanism of polystyrene films by FTIR. According to these authors, PS photodegradation starts with the phenolic rings of PS absorbing UV radiation and getting excited into singlet states and then various photochemical reactions could be originated from excited triplet from of benzene rings with CC and/or C-H bond scissions at various sites could be initiated. FTIR spectra have shown the formation of CO, -OOH, -OH… groups. But this implies, that the transformation of organic carbon into CO2 (mineralization) is a step which takes place at the end of the mechanism. On the contrary in the case of PMMA, one of the primary steps is the abstraction of the ester side group -OCH3 [START_REF] Vinu | Photocatalytic degradation of methyl methacrylate copolymers[END_REF]. The polymer radical fragment thus produced can form the formaldehyde, which finally forms carbon dioxide, which explains the faster mineralization of PPMA. Regarding the influence of particles size (PS) on the photomineralization, the effect is less visible, because there is probably a degradation of the PS nanobeads surface. Nevertheless, the results of figure 4 show that the kinetics are faster with the 105 nm PS nanobeads than those of 540 nm. This result seems logical because smaller particles lead to larger contact area with TiO2 photocatalyst.

Conclusion

We have shown that it it possible to mineralize nanobeads of PMMA and PS under irradiation of TiO2 in aqueous solution. Working at an irradiance of 112 W/m 2 , with a flowrate of 10 mL/min and at initial pH value of 6.3, using TiO2-P25/β-SiC foams in a flow-through mode allowed to reach 50% of TOC conversion after 7 h of treatment in the removal of PMMA nanoplastics. We also showed that the photocatalytic process can be effective for polymers with various molecular structures, such as polystyrene and PMMA and also with different average sizes of nanobeads. It is now necessary to optimize the process, but also to better understand the degradation mechanisms. However, one of the most important challenges to take up in this work will be to find an analytical procedure that is easy to implement. In the rest of this work, we have planned to couple several photoreactors in a row for processing in a single pass, which will be better for an application in the case of real effluents.

Fig. 1 .

 1 Fig. 1. Photocatalytic tubular reactor made with (1) a 4 UV-A lightning system + a quartz tube + 3

Fig. 2 .

 2 Fig. 2. a) Mineralization kinetic of NBs-PMMA on TiO2-P25/β-SiC without and under UV-A irradiation

Fig. 3 .

 3 Fig.3. a) pH influence and b) UV-A light intensity influence on the photocatalytic mineralization of

Fig 4 :

 4 Fig 4: Comparison of photocatalytic mineralization of NBs-PMMA and NBs-PS1/PS2 (Flow rate: 10

Acknowledgements

The authors of this article thank Campus France for the grant awarded to Paul Henri ALLE.


M Hakl

Q Y Lin

S Lepillet

M Billet

J-F Lampin

S Pirotta

R Colombelli

W J Wan

J C Cao

H Li

E Peytavit

S Barbieri
email: stefano.barbieri@univ-lille.fr

Ultrafast quantum-well photodetectors operating at 10µm with flat frequency response up to 70GHz at room temperature

    Ultrafast quantum-well photodetectors operating at 10µm with flat frequency response up to 70GHz at room temperature M Hakl, Q y Lin, Sylvie Lepilliet, M Billet,

Introduction

Thanks to their intrinsically short electron relaxation time, on the ps timescale, midinfrared (MIR-3-12 µm) quantum-well infrared photodetectors (QWIP) based on III-V semiconductor materials were identified as ideal candidates for ultra-high-speed operation at the end of the 80s. Since then, several experiments have been carried out to determine their RF bandwidth using both pulsed mid-infrared excitation or heterodyne detection [1][2][3][4][5][START_REF] Grant | An ultrafast quantum well infrared photodetector[END_REF][START_REF] Grant | Room-temperature heterodyne detection up to 110GHz with a quantum well infrared photodetector[END_REF].

The exploitation of QWIPs as heterodyne receivers with IF bandwidth of tens of GHz is particularly attractive for a number of applications, including free-space communications, gas sensing and spectroscopy, atmospheric and space science, or military countermeasures [START_REF] Pang | Gigabit free-space multi-level signal transmission with a mid-infrared quantum cascade laser operating at room temperature[END_REF][START_REF] Liu | Mid and Long-Wave Infrared Free-Space Optical Communication[END_REF][START_REF] Macleod | Broadband standoff detection of large molecules by mid-infrared active coherent laser spectrometry[END_REF][START_REF] Weidmann | Ground-based prototype quantum cascade laser heterodyne radiometer for atmospheric studies[END_REF]. Besides enabling the implementation of coherent detection schemes, another advantage brought by heterodyne detection is the possibility to operate QWIPs in the shot-noise regime, overcoming the noise contribution of the thermally activated dark current, which severely impacts the NEP of MIR QWIPs at high temperatures [START_REF] Brown | CO2-Laser Heterodyne Detection with GaAs/AlGaAs MQW Structures[END_REF]. So far, the largest heterodyne detection bandwidths at room temperature are those obtained by Grant et al. using a 100-quantum wells (QWs) QWIP operating at a 10µm [START_REF] Grant | An ultrafast quantum well infrared photodetector[END_REF][START_REF] Grant | Room-temperature heterodyne detection up to 110GHz with a quantum well infrared photodetector[END_REF]. The device was processed in a 16µm-side square mesa, illuminated from a 45° polished substrate. At room temperature, a ~25GHz 3dB cutoff and ~10dB attenuation at 110GHz were demonstrated, clearly illustrating the high frequency potential of QWIPs.

In the context of high-speed QWIPs the possibility of coupling the detector element to an antenna opens interesting perspectives. Indeed, the antenna allows a reduction of the detector active volume without sacrificing the radiation collection area, thus avoiding a reduction of the quantum efficiency.

Nano-antennas were first applied to MIR bolometers as a way to increase both their sensitivity and speed [START_REF] Grossman | Lithographic spiral antennas at short wavelengths[END_REF][START_REF] Chong | Antenna-coupled polycrystalline silicon air-bridge thermal detector for mid-infrared radiation[END_REF][START_REF] González | Antenna-Coupled Infrared Detectors for Imaging Applications[END_REF]. QWIPs based on arrayed patch antennas resonators (PARs) were first proposed in 2001 [START_REF] Beck | Microstrip antenna coupling for quantum-well infrared photodetectors[END_REF]. PARs are ideally suited for QWIPs as they allow illumination at normal incidence, which is clearly advantageous compared to facet illumination [START_REF] Grant | An ultrafast quantum well infrared photodetector[END_REF][START_REF] Grant | Room-temperature heterodyne detection up to 110GHz with a quantum well infrared photodetector[END_REF], while confining the electro-magnetic field inside a sub-wavelength volume [START_REF] Beck | Microstrip antenna coupling for quantum-well infrared photodetectors[END_REF][START_REF] Palaferri | Ultra-subwavelength resonators for high temperature high performance quantum detectors[END_REF][START_REF] Le Perchec | Plasmon-based photosensors comprising a very thin semiconducting region[END_REF]. Compared to QWIPs based on standard mesa geometry and of comparable collection area, this enables the realization of "thin" detectors (including a small number of quantum wells) with a higher detectivity while keeping a small capacitance, which is clearly relevant for high speed operation. QWIPs based on 2D arrays of PARs were recently demonstrated showing over one order of magnitude improvement in detectivity compared to a mesa reference QWIP [START_REF] Chen | Antenna-coupled microcavities for enhanced infrared photo-detection[END_REF][START_REF] Palaferri | Room-temperature 9-μm wavelength photo-detectors and GHz-frequency heterodyne receivers[END_REF]. However, in these works the potential in terms of high-speed operation was largely underexploited, with reported maximum heterodyne detection frequencies (no reported 3dB bandwidth) up to only 4 GHz, limited by a device design (parasitics) leading to a large capacitance [START_REF] Palaferri | Room-temperature 9-μm wavelength photo-detectors and GHz-frequency heterodyne receivers[END_REF].

In this work we have fully addressed this issue by demonstrating PARs-based QWIP detectors specifically designed for ultra-broadband operation. Thanks to this design we demonstrate experimentally that the detector capacitance can be reduced down to the (unavoidable) static capacitance of the antenna resonators. With these devices we demonstrate room temperature heterodyne detection at 10.3 µm with a nearly flat frequency response up to 70GHz (limited by the detection electronics), and state of the art responsivities of ~0.15A/W. To the best of our knowledge this represents the broadest experimental RF bandwidth reported to date for a QWIP detector [START_REF] Grant | An ultrafast quantum well infrared photodetector[END_REF][START_REF] Grant | Room-temperature heterodyne detection up to 110GHz with a quantum well infrared photodetector[END_REF][START_REF] Rodriguez | Room temperature, wide-band Quantum Well Infrared Photodetector for microwave optical links at 4.9 μm wavelength[END_REF], and extends by over 65 GHz the results presented in Ref. [START_REF] Palaferri | Room-temperature 9-μm wavelength photo-detectors and GHz-frequency heterodyne receivers[END_REF]. Moreover, we develop a smallsignal equivalent circuit model that can quantitively reproduce the observed device frequency response, which we find to be strongly dependent on bias and temperature. From this model we extract an RC limited 3dB cutoff of ~150GHz at 300K, and an intrinsic roll-off time constant ≲1ps, providing the first experimental evidence that QWIP detectors can indeed reach RF-bandwidths limited by electron capture on the ps timescale at room temperature. These results pave the way to the development of ultrafast MIR optoelectronics.

Device design and fabrication

The structure is grown by molecular beam epitaxy (MBE) on a semi-insulating GaAs substrate: 100nm-thick lattice-matched Ga0.51In0.49P etch-stop layer followed by an Al0.2Ga0.8As/GaAs heterostructure. The PAR active region consists of seven, 6.5nm GaAs quantum wells (QWs) with a central, 5.3nm-thick region, n-doped at a level of 6.7x10 17 cm -3 .

The wells width is chosen to obtain a bound-to-bound transition energy of ~120meV. The QWs are separated by 40nm barriers, and the active region is sandwiched between 50nm and 100nm-thick top and bottom n-doped contact layers with concentrations of 3x10 18 cm -3 and 4x10 18 cm -3 respectively. In Fig. 1(a)(b), we present, the SEM images of the fabricated detector. It consists of a 5x5 periodic array of square PARs of side s=1.85µm and period p=3.9µm, sitting on top of a Ti/Au (100/400nm) ground plane. As detailed in the next Section, the values of p and s are chosen to obtain a maximum PAR array absorption as close as possible to the intersubband transition energy. At the same time, to minimize the array capacitance, the number of patches is kept to the minimum needed to allow collecting 100% of the incident radiation (~20µm-diameter laser spotsize, see next Section).

Particular care was taken in the detector microwave design, aimed at reducing the effect of parasitic capacitances brought by electrical connections and contact pads, which limited RF operation up to a few GHz in Ref. [START_REF] Palaferri | Room-temperature 9-μm wavelength photo-detectors and GHz-frequency heterodyne receivers[END_REF]. As shown in Fig. 1 The fabrication of the PARs begins with the realization of a buried metal layer serving as electromagnetic ground plane and bottom Schottky contact metallization (we have deliberately chosen Schottky rather than ohmic contacts to avoid the risk of metal diffusion in the QWIP active region due to high temperature annealing, potentially leading to high MIR losses). This is obtained by transferring the epi-layers onto a 2"-diameter high-resistivity (>5 kW.cm) silicon wafer using a Au-Au thermo-compression bonding technique detailed in [START_REF] Peytavit | Wide-band continuous-wave terahertz source with a vertically integrated photomixer[END_REF], followed by the wet etching of the GaAs substrate and the etch-stop layer. Next, the Ti/Au (8nm/300nm) top Schottky contact metallization is realized through e-beam lithography, followed by e-beam evaporation and lift-off. The epi-layers are subsequently ICP etched using the top metal layer as etch-mask. The ground metal layer is finally dry-etched by an Ar + ionbeam around the PARs array down to the silicon substrate. A 100-nm-thick Si3N4 layer is then deposited on the silicon by plasma enhanced chemical vapor deposition (Fig. 1

(a)(b))

To electrically connect the patches together, suspended ~150-nm-width Ti/Au (20nm/600nm) wire-bridges are finally fabricated by a two-step e-beam lithography process.

To this end a first resist layer is used as support after deposition, e-beam lithography and reflow, followed by a second one to define the wires by standard lift-off process. The same process is used to realize the air-bridge connecting the 2D array to the 50W coplanar line. The latter is deposited on the Si3N4: this avoids any leakage currents between the electrodes of the coplanar waveguide coming from the silicon.

Spectral and dc electrical characterisation

In Fig. 2 we report the results of the infrared spectral characterization of the PARs array. Fig. 2(a) shows the absorption spectra at 300K, corresponding to the fraction of the incident power absorbed by the QWIP detector with two polarizations of the incident light: orthogonal (black) and parallel (red) to the wire bridges [START_REF] Chen | Antenna-coupled microcavities for enhanced infrared photo-detection[END_REF][START_REF] Palaferri | Room-temperature 9-μm wavelength photo-detectors and GHz-frequency heterodyne receivers[END_REF]. The absorption is defined as 1-R(w), where R(w) is the reflectivity spectrum obtained through FTIR micro-reflectivity measurements. At the cavity resonance for the orthogonal polarization (116meV -10.7µm) we find that, 1-R(w) = 0.9, i.e. 90% of the incident photons are absorbed. Indeed, the period p=3.9µm, was selected to operate the QWIP as close as possible to the critical coupling regime, compatibly with the targeted intersubband transition energy [START_REF] Palaferri | Ultra-subwavelength resonators for high temperature high performance quantum detectors[END_REF]. In this condition, the single PAR collection area at the resonant frequency is given by 0.9xp 2 , yielding a total collection area of ~340µm 2 (=(18.5µm) 2 ) for the PAR array. As shown in the insets of Fig. 2(a), for the parallel polarization, the spatial distribution of the cavity mode is modified by the presence of the wire bridges, This yields a blue shift of the cavity resonance, as well as a reduced integrated absorption.

Energy (meV) Norm alised photo

In Fig. 2(d) we report the measured photocurrent spectrum at 77K, obtained by FTIR spectroscopy with the QWIP structure processed in a mesa geometry, therefore showing the effect of the bare intersubband transition. We find a peak at 107meV, in good agreement with the expected bound-to-bound transition energy. By multiplying this spectrum by the cavity absorptions in Fig. 2(a) we obtain the dashed spectra shown in panel (b), in good agreement with the QWIP detector photocurrent spectra measured a 77K, represented by the solid lines. This indicates that the PARs array absorption is dominated by absorption in the metal and contact layers (see Supporting Information for more details). From the black solid line we find that the QWIP operates in the ~10-12µm range, with a maximum response at ~10.8µm (115meV).

In Fig. 3(a),(b) we report the dark current and dc photocurrent vs bias characteristics at 77K and 300K, obtained by illuminating the QWIP with a 10.3µm (120meV) DFB quantum cascade laser (QCL), polarized at 45° with respect to the wire bridges (the full polarization dependence at 77K is reported in Fig. 2(c)). For these measurements, the collimated beam from the QCL was focused on the detector using an AR coated aspheric lens (NA = 0.56; 5mm focal length).

At 10.3µm we measured a waist diameter of 20µm using a knife-edge technique, i.e. approximately equal to the side of the 5x5 PAR array collection area (18.5µm ~ %340µ) * ).

Therefore, for the rest of this work, we assume that all the QCL power, measured after the lens, is incident on the QWIP. This corresponds to the power values reported in Fig. 3

(a)(b).

As expected, at 300K the dark current dominates the photocurrent for all power levels. At 77K the situation is reversed, showing that at this temperature the QWIP can be potentially operated in the photon-noise regime with only a few mW of incident power [2]. At 77K and 3.5-4V (Fig. 3(a)) we observe a pronounced saturation of the photocurrent, that we attribute to negative differential drift velocity, resulting from intervalley scattering [START_REF] Schneider | Optically induced electric-field domains by bound-tocontinuum transitions in n-type multiple quantum wells[END_REF]. Saturation fields in the 10-20kV/cm range have been found in previous works. Here, at 3.9V (Fig. 3(a)) the average electric field is ~100kV/cm, indicating that a large fraction of the applied bias drops on the Schottky contacts.

The photocurrent and responsivity as a function of incident power at 77K and 300K are reported in Fig. 3(c), respectively at 3.4V and 2.5V. Responsivities are corrected by the polarization factor (Fig. 2(c)), and their value corresponds to the situation where the incident field is polarized orthogonally to the wires, which is the ideal condition to operate the QWIP.

At low power we obtain responsivities R = 1.5A/W and 0.15A/W at 77K and 300K. From the reflectivity spectrum of Fig. 2 Rs, (see Table 1) are respectively the dc internal photoresistance of the PARs array, and the leakage resistance of the Schottky contact biased in reverse breakdown, both under illumination (the forward biased Schottky junction is considered as a short circuit). ZL ~50W is the load impedance seen by the QWIP. centered at 107meV with a FWHM of 10% (see the spectrum of Fig. 1(d)), we find that the responsivity measured at 77K at high bias and low optical power, is compatible with a photoconductive gain , = . / /. 01 ≅ 2.5, where . / and . 01 are respectively the electron's capture and transit time (see Supporting Information) [START_REF] Schneider | Quantum Well Infrared Photodetectors: Physics and Applications[END_REF]. The decrease of responsivity at 300K is attributed to a decrease of the drift velocity and capture time (see Table 1). Finally, by increasing the power we observe a clear decrease of responsivity at 77K. This is attributed to

(a) (black line) and assuming an intersubband transition energy

I s Z L R 0 C PAR C s R s bias ( ) I L
the presence of a series resistance provided by the Schottky contacts (RS in the circuit of Fig. 3(d), see next Section and Table 1). As a consequence, for a given applied bias, the decrease of the detector photoresistance (R0 in the circuit of Fig. 3(d)) with increasing incident power produces a progressive lowering of the electric field across the QWIP active region [START_REF] Ershov | Photoconductivity nonlinearity at high excitation power in quantum well infrared photodetectors[END_REF] At room temperature Rs is instead negligible (see Table 1), therefore the saturation effect is much less pronounced.

Heterodyne mixing and frequency response

In Fig. 4 we report the heterodyne frequency response (FR) of the QWIP in the 10MHz-67GHz range, measured at 77K and 300K, at low and high applied biases. To record these spectra, we used a 67GHz-bandwidth cryogenic probe, positioned at the edge of the coplanar waveguide shown in Fig. 1 The traces are corrected by (i) the propagation losses from the QWIP to the SA measured with a vector network analyzer (VNA), and (ii) the power changes (2dB max) of one QCL due to temperature/current tuning (see Supporting Information).

The top two traces in Fig. 4 show the detector FR at high bias, i.e. 3.4V(77K) and 2.5V(300K).

From Fig. 2(c), the corresponding responsivities are 0.75A/W and 0.13A/W. At 77K we find a monotonic decrease with frequency, with a 3dB-cutoff frequency of ~30GHz, while at 300K the response is much flatter, with a ~2dB increase from 0 to ~40GHz, followed by a 3dB drop at ~67GHz.

At low biases the shape of the FR is rather different. As shown by the two bottom traces, recorded at 1.1V(77K) and 0.9V(300K), the FR is virtually flat up to 67GHz, except at low frequencies where we observe a pronounced drop below ~5GHz(77K) and ~10GHz(300K).

To gain insight in the behaviour of the QWIP, we used a VNA analyzer to derive the device impedance vs frequency in the operating conditions corresponding to the spectra of Fig. 4. We find that at low bias (1.1V, and 0.9V spectra in Fig. 4) the detector's RF impedance can be well reproduced using the simple small-signal circuit displayed in Fig. 3(d) (see Supporting

Information for the for the complete derivation of the circuit parameters) [START_REF] Coleman | Mixing and Detection of Coherent Light[END_REF]. Here R0, Rs, and where m is a modulation index given by ) = *%6 7 ×6 9 6 7 :6 9 = 0.77, ; <= is the dc (i.e. average) measured photocurrent, > ? is the PARs array dark resistance, and ; @ = ; <= (> ? + > C )/> ? . The term under the square root at the denominator takes into account the frequency roll-off of the intrinsic transport mechanism, with t approximating the carriers capture or transit time [START_REF] Schneider | Quantum Well Infrared Photodetectors: Physics and Applications[END_REF][START_REF] Coleman | Mixing and Detection of Coherent Light[END_REF]. Concerning Rd, based on the fact that at 77K (300K) the QWIP current under illumination is dominated by the photocurrent (dark current) component (see Fig. 3(a)(b)), we have made the following approximations: > ? ≈ > @ at 300K, and > ? ≫ > C at 77K. These approximations allow de facto to eliminate the QWIP dark resistance as independent variable from the source term in Eq.(1) (see Supporting Information for more details)

The solid curves corresponding to the two bottom FRs in Fig. 4

; C = W %J:(X Y Z) 9 ; <= [ \ :[ ] [ \ = W %J:(X Y Z) 9 ; @ , (1)
thus eliminating the power loss in the contact resistance. In this case, from the small-signal circuit model, we have that:

; I = ; C J J:[ ^/[ _ :`X Y [ ^abcd (2)
yielding a parasitic roll-off time constant > I U 6e[ /(1+> I /> @ ) ≲1ps (see Table 1). We also find (see Supporting Information) that, for the chosen PAR array size, the QWIP impedance at low bias is close to 50W for frequencies ≳20GHz(300K) and 30GHz(77K), which is ideal for RF impedance matching.

Table 1. Measured photocurrents (; <= ), ; @ , and small-signal circuit resistances under illumination (> @ , > C ), used to compute the solid lines in Fig. 4 for different operating conditions (bias and temperature). The value of the roll-off time constant (τ in Eq.( 1))

is the one yielding the best fit of the experimental data. At high biases the effect of CS is much less pronounced and the power drop at low frequencies disappears (Fig. 4, top two spectra). From the small-signal circuit this can be explained by a reduction of RS due to the Schottky barrier becoming more transparent, therefore effectively shunting U C at low frequencies. As a result, the QWIP impedance does not display the strong increase at low frequency found at low biases (see Supporting Information). From the small-signal circuit we find a good agreement with the measured FRs using (i) R0 =40W, Rs=20W, for the spectrum at 3.4V (77K) with, t ~ 8ps and Iph =15.2mA; and

(ii) R0 =40W, Rs=0W, for the spectrum at 2.5V, (300K) with t ~2-3ps and Iph =2.2mA (see Table 1, 3 d and 4 th column). As a result, at 300K the QWIP is almost impedance matched to 50Ω at all frequencies. We note that our small signal circuit model does not explain the ~2dB increase in the FR from 0 to ~40GHz observed at 2.5V. As shown above, thanks to the very small device capacitance, by fitting the measured FRs using the small signal circuit model we can extract the intrinsic detector response times, which, as shown by the solid curves in Fig. 4, dependent on the operating conditions. From the values of . and from the photoconductive gain derived from the responsivities, we can then obtain the values of . / and . 01 shown in Table 1 (see Supporting Information): except at 77K under high bias, the QWIP intrinsic response time appears to be dominated by electron capture. We also find the expected decrease/increase of . 01 with increasing bias/temperature [START_REF] Schneider | Quantum Well Infrared Photodetectors: Physics and Applications[END_REF]. Interpreting the dependence of . / on bias and temperature is beyond the scope of this work and will require more systematic measurements that are presently under way. At the same time, on this subject there appears to be a lack of experimental data in the literature [4].

In Fig. 5 we report two examples of heterodyne beatnote spectra recorded in single shot at 77K, under an applied bias of 3V and without any amplification (Fig. 5(a)), and at 300K, with an applied bias of 1.1V and with a narrow band amplifier of 50dB gain (Fig. 5(b)). In the first case the noise floor is limited by the spectrum analyzer, while in the second spectrum the noise floor is determined by the amplifiers noise. We find instantaneous linewidths of ~100kHz, limited by the QCL thermal and current fluctuations. At 77K the RBW is set to 100kHz, yielding a SNR of ~77dB, while at 300K we find a SNR of 72dB with a RBW of 50kHz. Reducing further the RBW produces a decrease of the beatnote intensity because the RBW goes below the instantaneous heterodyne beatnote linewidth.

The dependence of the SNR of the heterodyne beatnote frequency, obtained with the spectrum analyzer without amplification (P1 = 27.5mW and P2 = 6mW), can be directly extracted from the spectra recorded max-hold trace mode (see Supporting Information). At 30GHz and 60GHz, with a RBW of 3.5MHz, we obtain SNRs of 50dB and 35dB, and of 35dB and 25dB, respectively at 77K (3.4V) and 300K (2.5V).

Conclusions

Antenna-coupled QWIP detectors operating in the 10µm-12µm range are demonstrated, exhibiting a flat frequency response up to 67GHz at 77K and 300K. At 300K, from our experimental results and with the help of an equivalent circuit model, we find an RC-limited 3dB cutoff frequency >150GHz and ~ps intrinsic response times. These results are achieved thanks to a detector specifically designed for ultrafast operation, and provide the first experimental evidence that QWIPs can indeed reach RF-bandwidths limited by electron capture over timescales of ~1ps at room temperature.

We believe that the detectors demonstrated here, in combination with QCLs, will open up new perspectives in MIR photonics, namely by extending to the MIR range the possibilities offered by ultrafast near-infrared optoelectronics, so far the only frequency range benefitting from the availability of ultrafast photodetectors. Envisaged applications are free space communications with data rates >10Gb/s, coherent multi-species gas sensing, high precision spectroscopy and metrology, astronomy, as well as to study real time dynamics on the 10ps time scale [START_REF] Pang | Gigabit free-space multi-level signal transmission with a mid-infrared quantum cascade laser operating at room temperature[END_REF][START_REF] Liu | Mid and Long-Wave Infrared Free-Space Optical Communication[END_REF][START_REF] Macleod | Broadband standoff detection of large molecules by mid-infrared active coherent laser spectrometry[END_REF][START_REF] Weidmann | Ground-based prototype quantum cascade laser heterodyne radiometer for atmospheric studies[END_REF][START_REF] Hale | The Berkeley infrared spatial interferometer: a heterodyne stellar interferometer for the mid-infrared[END_REF]. More specifically, on this last topic, we expect that broadband devices, such as those demonstrated in this work can shed new light on the ultrafast electron's dynamics in QWIPs.

A final intriguing perspective is the use of these structures as QCL-pumped photomixers for the generation of sub-THz radiation [START_REF] Lepilliet | Milliwatt-level output power in the subterahertz range generated by photomixing in a GaAs photoconductor[END_REF][START_REF] Peytavit | CW Source Based on Photomixing With Output Power Reaching 1.8 mW at 250 GHz[END_REF]. To this end we note that the actual responsivity of ~0.75A/W at 77K, 3.4V (high power -see Fig. 3(c)) could be theoretically brought up to ~4A/W if the ISB transition peak and the PAR resonance were perfectly matched. In this case, from the top trace in Fig. 4, we would expect microwave power levels in the mW range.

Replacing the Schottky contacts with non-diffusive ohmic contacts (not to increase MIR photon absorption) should also reduce the saturation of the responsivity at high incident powers (Fig. 2 dominated by ohmic losses (i.e. absorption in the metal and contact layers). Indeed, results from FDTD simulations performed on a single resonator yield an upper limit of ~15 for the PAR array Q-factor (without intersubband absorption). This is smaller than 1/H IJK () UT ) ≈ 20 obtained from Eq.(S2) showing that even without spectral shift, cavity absorption would be dominated by ohmic losses). The responsivity " #$%& ()), obtained from Eq.( S1) and (S2) with A = 2.5, is shown in Fig. S1: for ℏω ≈ 120meV, corresponding to the QCL photon energy (λ = 10.3bm), we obtain " #$%& in accordance with the measured experimental responsivity at 3.4V, 77K and low incident power (Fig. 3(c) in the main text).

Heterodyne mixing experimental setup and frequency response spectra

The schematic of the experimental setup up is shown in Fig. S2. To minimize the linewidth of the heterodyne beatnote the QCLs were driven with low noise current drivers (Koheron, DRV110) with a current noise of 300pA/Hz 1/2 .

Fig. S2

Schematic of the experimental setup used for the measurement of the QWIP FR. VA -variable attenuator; M -mirror; BS -beam splitter; l/4 -quarter waveplate; P -polarizer. Fig. S3 shows the procedure followed to extract the fzrequency response (FR) spectra displayed in Fig. 4 of the main text. As an example we consider the FR at 300K and 2.5V. First, by sweeping the frequency of one QCL, we have recorded the heterodyne beat signal using the spectrum analyzer (SA) set in max-hold trace mode as described in the main text. The corresponding SA trace is displayed in Fig. S3 (a) Example of extraction of the FR. (a) Raw heterodyne spectrum, collected with the SA set in max-hold trace mode, with a RBW of 3.5MHz (red solid line). Measured attenuation due to due to the insertion loss in the probes, cables and bias-tee (red dots, in dB). Measured attenuation corrected by the QCL power change (blue dots, in dB). (b) Heterodyne beat spectrum obtained by subtracting the blue dotted trace in panel (a) from the raw heterodyne spectrum. The black circle corresponds to the data reported in Fig. 4 of the main text (300K, 2,5V). Fig. S3(a) (solid red line). Here, each vertical line corresponds to a heterodyne beat between the two QCLs. Next, with a VNA analyzer, we have measured the power attenuation from the QWIP to the SA, due to the insertion loss in the probes, cables and bias-tee (red dots in Fig. S3(a)). This curve has been finally corrected to include the power variation of the QCL due to the frequency tuning. The resulting blue dotted curve in Fig. S3(a) has been subtracted by the raw heterodyne spectrum, yielding the spectrum shown in Fig. S3(b). The black circles, corresponding to the line peaks recorded every 500MHz, are those displayed in Fig. 4 of the main text. We note that the heterodyne spectra were recorded with a RBW of 3.5MHz. This is larger than the actual heterodyne beat linewidth, which was of the order of ~100kHz (see the main text), therefore guaranteeing that the intensity of the heterodyne beats is not reduced by filtering. In Fig. S4 we report the other heterodyne spectra (corrected by losses) used to extract the FRs shown in Fig. 4 of the main text.

Optical beam

Bias-T

Spectrum Analyzer

Small-signal Circuit model

In an optical heterodyne experiment as described in this work, a dc biased photoconductor is illuminated by two laser beams of power P1 and P2 with a difference frequency wb. The incident optical power on the photoconductor can be expressed as:

c(d) = c U + c T + 2fc U c T sin () K d). (S3)
The photocarrier density in the photoconductor follows the time variation of the incident power. By assuming that the photoconductor exhibits a linear I-V characteristic, it can be modeled as a timedependent conductance, which can be written as:

j(d) = j k + j U sin () K d + l), (S4)
where j k and j U are respectively a dc and a dynamic conductance term. They are given by [3]:

j k = j m + j Sn , (S5) j U = o fU7(6 5 p) O j Sn , (S6) with j Sn = % Nq r st . ( S7
)
In Eq.( S6) and (S7), the term j m (1/Rd) is the dark conductance, while j Sn (1/Rph) is the internal photoconductance given by the ratio between the dc (i.e. average) conduction photocurrent, u Sn , generated by the two laser sources, and the dc bias voltage, v mC applied to the photoconductor. In the expression of j U the denominator reflects the frequency roll-off of the intrinsic recombination or transport mechanism, with t approximating the carriers capture or transit time. The term m is a modulation index given by:

w = Tf& < ×& O & yzy , ( S8
)
where c E{E = c U + c T . For the powers used to record the spectra of Fig. 4 in the main text (c U = 27.5mW, c T = 6mW) we obtain w = 0.77.

An accurate model of the QWIP detector should include the electrical capacitance of the PARs array, &2\ , in parallel with j(d). The Schottky contact biased in reverse breakdown should also be added (the forward biased Schottky junction is considered as a short circuit). The resulting electrical circuit model is shown in Fig. S5. This circuit also includes the inductor and capacitor (Ä KIÅJRÇ , KIÅJRÇ ) of the bias-T used in the experiment, allowing the decoupling between dc and ac currents. Finally, Ñ Ö is the load impedance seen by the QWIP (see Section 4). Voltages and currents in the circuit are time periodic (period T=2p/wb) and can be expressed in a Fourier series. By neglecting high orders harmonics [3], the voltage across the photoconductor takes the simple form: Load lines under illumination (pink) and in the dark (blue), obtained from the circuit of Fig. S6(b). Operating point under illumination (black circle) and in the dark at 300K (orange circle) and 77K (purple circle). For simplicity, but without loss of generality, we have neglected the temperature dependence of the Schottky IV characteristic.

v(d) = v mC + v ÅC cos() K d + l). (S9)
The current is given by:

u(d) = u mC + u ÅC cos() K d + à). (S10)
The quantities v mC , v ÅC , u mC , u ÅC , l, à can be derived from the circuit of Fig. S5 by applying Kirchhoff's laws at ) = 0 and ) = ) K (i.e. exploiting the decoupling between dc and ac currents thanks to the bias-T inductance and capacitance) and by using the constitutive relation:

u(d) = j(d)v(d) (S11).
From the equations above, an ac small-signal circuit (w = wb) and a dc circuit (w = 0) can be derived, as shown in Fig. S6(a),(b) [3]. Here, " â is the Schottky junction differential resistance under illumination

I(t)

V bias V L C bias-T L bias-T G(t) V(t) Z L C PAR R d (300K) R d (77K) V bias V t V s I s V Schottky ˚(V) Current˚˚(A) ˚Schottky˚diode˚IV ˚˚˚˚˚˚˚˚˚(V Schottky ˚-˚V t )/R s ˚(V bias ˚-˚V Schottky )/R 0 ˚˚˚˚˚˚˚˚˚(V bias ˚-˚V Schottky )/R dark 0 I photo (300K) I photo (77K) (a) (b) V Schottky Current dc (77K) (300K) C s R s I L PAR reverse biased Schottky contact (a) ω = ω b I s Z L R ph C PAR R d R ph R s R d V V bias DC I dc R ph R d V dc R h C PAR C s R s R d V V bias I dc (b) ω = 0 R ph C PAR R d V dc V 0
at the operating point (u mC , v J ): 1/" â = äu/äv| % st ,r 4 . In the ac circuit the small-signal behavior of the Schottky junction is modelled by adding a junction capacitance J in parallel to " â . Instead, in the dc circuit we must model the full Schottky junction IV curve. As shown in Fig. 5(b) (green lines) this is done by replacing the latter (red line) by a linear characteristic u = (v -v E )/" J passing through (u mC , v J ) for v > v E , while for v < v E we consider the Schottky as an open circuit (u = 0). The threshold voltage v E can then be incorporated in the voltage source of the dc circuit by writing that v k = (v KIÅJ -v E ) for v KIÅJ > v E , and v k = 0 for v KIÅJ < v E (v KIÅJ is the bias effectively applied to the device, see Fig. S5(a)). Clearly, both Vt and Rs will depend on the value of the QWIP point of operation under illumination.

In the dc circuit the PAR array is modeled by its dc photoresistance under illumination,

" k = \ s \ Nq \ s 7\ Nq , ( S12
)
where Rd is the PAR array dark resistance. The corresponding load line (purple) is shown in Fig. 5(b), with a slope given by 1/R0. The blue lines are instead the load lines of the PARs array in the dark at 300K and 77K, with slopes given by 1/Rd. At 300K the QWIP current under illumination is dominated by the dark current component, as can be seen from Fig. 3 At 77K, the QWIP current under illumination is instead dominated by the photocurrent component (Fig. 3(a) in the main text and Fig. S7(a),(c)), i.e. Rd >> R0 , Rs, resulting into a large change of slope of the load line (Fig. S5(b)). In this case, by linearizing the Schottky using its resistance under illumination, we completely neglect the fact that in the dark the slope of the Schottky IV is much larger. On the other hand, as shown schematically in Fig. S5(b), the larger is Rd the smaller will be the slope of the load line, thus reducing the difference between the effective dark current and the dark current obtained through the dc circuit model (purple and green circles).

In the ac circuit (Fig. S6(a)) the PAR array is modeled by an equivalent ac current source, u J , with its internal impedance " k in parallel with the intrinsic capacitance of the array &2\ . The current source can be computed as [3]:

u J = v mC × j U = o fU7(6 5 p) O v mC × U \ Nq , ( S13
)
where u J is in general a phasor (from now on we assume that all currents and voltages are represented by phasors). The dc equivalent circuit (Fig. S6(b)), can be used to derive v mC :

v mC = r é \ é 7\ 4 " k = u mC " k . (S14)
From this equation we note that, due to the Schottky contact resistance, v mC < v k (note that the effective dc voltage applied to the PAR array is equal to v mC + v E ). From Eqs.(S12), (S13) and (S14) we obtain:

u J = o fU7(6 5 p) O u mC \ é \ Nq = o fU7(6 5 p) O u mC \ s R\ é \ s , (S15)
where u mC is the dc current under illumination that can be measured experimentally. It is also useful to express the current source Is as a function of the dc photocurrent of the QWIP, which is obtained by subtracting the dark current from u mC . From the dc equivalent circuit of Fig. S6(b) the photocurrent is given by:

u Sn = u mC -u mÅFè = v k 1 U \ é 7\ 4 - U \ s 7\ 4 = = u mC \ s R\ é \ s 7\ ê (S16)
By comparing Eq.(S15) and (S16) we finally obtain:

u J = o fU7(6 5 p) O u Sn \ s 7\ 4 \ s . ( S17
)
From Eq.(S16) we find that the dc photocurrent is equal to the measured dc current only if " k and " â are negligible compared to " m . As can be deduced from Fig. 3(a)(b) in the main text, this is the case at 77K for sufficiently high power levels, but not at 300K. In this case the correction factor (" m + " J )/" m (Eq.S17) cannot be neglected.

QWIP impedance measurements

In Fig. S8 we report the real and imaginary parts of the QWIP impedances vs frequency (black and red lines) obtained from the ë UU parameters measured with a VNA analyzer, after de-embedding the 50 integrated coplanar line. At T=77K, the ë UU parameters were measured under the same operating conditions (bias, temperature and illumination) used to record the FRs, while at 300K they were measured in the dark. This last choice stems from the fact that, contrary to 77K, at 300K the dark current is much larger than the photocurrent even under illumination at high power (see Fig. the main text), i.e. the QWIP impedance under illumination is very well approximated by the dark impedance (" m ≪ " Sn , see Fig. S6(a)). The impedances at low biases (Fig. S8(a),(b)) are well reproduced by the equivalent circuit of Fig. S6(a), where the QWIP impedance (blue lines) is given by the sum of the PAR array and Schottky contact impedances:

Ñ #$%& () K ) = \ é U7I6 5 \ é ì 89: + \ 4 U7I6 5 \ 4 ì 4 . (S18)
In Fig. S8(a),(b), Ñ #$%& ()) is computed using the values of " k , " J -reported in the first and second column of Table 1 in the main text, with &2\ = 30fF and J =0.7pF (see next Section). In particular, when L → 0, we see clearly the effect of J , producing a fast increase of the real part of Ñ #$%& () K ), until, at L K = 0, "G[Ñ #$%& () K = 0)] = " k + " J . At higher frequencies (L K ≫ (2ò" J J ) RU ), " â is shunted by J , and Ñ #$%& () K ) coincides with the impedance of the PAR array, with a roll-off corresponding to a time constant equal to " k &2\ (first term in Eq.S18). As shown in Fig. S8(c),(d), at high biases the QWIP impedances change completely. Firstly, the fast increase as L K → 0, disappears, which we interpret as the evidence that the Schottky junction becomes more transparent, i.e. " â shunts J at all frequencies (see the next Section). At higher frequencies both the real and imaginary parts of Ñ #$%& () K ) show a maximum, followed by a slow decay. As shown by the blue lines this behavior cannot be fully reproduced by our simple circuit model using the parameters reported in the third and fourth column of Table 1 of the main text. In particular the imaginary part becomes inductive around 15-30 GHz. This phenomenon is probably linked to the fact that the QWIP is operated close to the onset of intervalley scattering. A more detailed analysis is needed, which is beyond the scope of this work. In Fig. S9 we report the measured load impedance, Ñ Ö , i.e. the impedance seen by the QWIP in the plane of the coplanar probes. This was extracted from ë UU parameter measurements. As can be seen, Ñ Ö can be approximated by its real part "G[Ñ Ö ] = " Ö ≅ 50Ω.

Determination of QWIP circuit parameters

The various elements, " k , " J , &2\ and J in the circuit of Fig. S6 depend in principle on the QWIP operating temperature, bias, and illumination conditions. To determine their values we rely on the experimental FR spectra displayed in Fig. 4 of the main text and on the corresponding QWIP impedances shown in Fig. S8.

The first equation used to determine " k , " J , is given by (see Fig. S6):

" k + " J = "G[Ñ #$%& () K = 0)]. (S19)
The second equation is obtained instead by noting that the values of the experimental FRs in Fig. 4 of the main text, correspond to the power dissipated in Ñ Ö :

c Ö = U T ℛG[Ñ Ö ] • |u Ö | T = U T " Ö • |u Ö | T . (S20),
Indeed, we recall that the FRs are corrected by the attenuation from the QWIP to the SA, which, in turn, has an impedance of 50Ω, i.e. perfectly adapted to Ñ Ö (Fig. S8). Since c Ö in Eq.(S20) depends on " k and " J , by comparing it with the power levels in the FRs of Fig. 4, gives the second equation, which, together with Eq.(S19), allows the determination of the QWIP and Schottky resistances separately. Concerning the capacitances, we begin by fixing &2\ = 28fF, which corresponds to the computed static capacitance of the PAR array using a parallel plate model. J is instead determined by fitting the decay of the experimental FRs at low frequency.

The details of the calculations are given below, respectively at low bias (Vbias = 1.1V, 77K, and Vbias = 0.9V, 300K) and high bias (Vbias = 3.4V, 77K, and Vbias = 2.5V, 300K). The values of the measured dc photocurrent of the QWIP, u Sn , and the values of " k and " J can be found in Table 1 of the main text.

Low bias

We start by assuming to be at a sufficiently high frequency such that " â is shunted by the parallel capacitance J (L K ≫ (2ò" J J ) RU ) and can therefore be neglected (see Eq.(S18)). In this case we obtain:

u Ö = u J U U7\ ù /\ é 7I6 5 \ ù ì 89: , (S21)
where we have approximated Ñ Ö with its real part " Ö ≅ 50Ω (see Fig. S9). Now, provided that the frequency is not too high, e.g. L K ≈ 10GHz, the last term at the denominator can also be neglected thanks to the extremely low value of &2\ (note: the validity of these last two assumptions can be verified a posteriori from the values of " k and " J ). The power dissipated into the load is then given by:

c Ö = U T u J T " Ö 1 \ é \ é 7\ ù = T . (S22)
At T=77K we have that " m ≫ " J (see Section 3), hence, from Eq.(S17), we have that u J ≈ w × u Sn = 0.38mA (at L K ≈ 10GHz ) K B~0). At this point Eq.(S22) can be used to determine the value of " k by comparing c Ö with the measured value of the FR at 10GHz (1.1V, 77K curve in Fig4 of the main text ). The value of " J can finally be obtained from Eq.(S19) with "G[Ñ #$%& () K = 0)] = 550Ω (see Fig. S8(a)). We find " k = 200Ω and " ¢ = 350Ω (first column of Table.1).

The last step consists in determining the value of J . This is obtained by fitting the decay of the experimental FR at low frequency (see Fig. 4 in the main text), yielding J ≈ 0.7£ §. We note that this value is in agreement with the theoretical capacitance expected for a Au/GaAs Schottky junction with a doping density of 4x10 18 cm -3 (~15nm depletion region width) [4]. The computed QWIP impedance is represented by the blue curves in Fig. S6(a), showing a good agreement with the impedance derived from the S11 parameter. Also, the computed FR using Eq.(S19) reproduces very well the experimental one as shown in Fig. 4 for B~1ps.

Concerning the measurement at T=300K and 0.9V, we have that " • ≈ " k . In this case, from Eq.(S17) and Eq.(S22) we obtain: where, again, we used the fact that at L K ≈ 10GHz ) K B~0. In this last equation the term " k + " J is known from Eq.(S19) and Fig. S8(b) (" k + " J = "G[Ñ #$%& () K = 0)] = 200Ω). Then, again, " k is determined by comparing c Ö in Eq.(S23) with the measured value of the FR at 10GHz (0.9V, 300K curve in Fig4 of the main text). From this procedure we obtain " k = 105Ω and " ¢ = 95Ω, which, however, do not allow to reproduce the QWIP impedance in a satisfactory way, as shown by the purple traces in Fig. S8(b) (here we used J = 0.7pF). We find that the values " k = 75Ω and " ¢ = 125Ω, allow to obtain the closest agreement with c Ö , compatibly with a good fit of the QWIP impedance (blue traces in Fig. S8(b)). The resulting computed FR, shown in Fig. 4 of the main text, is ~2dBm above the measured FR. This spectrum was obtained with J = 0.7pF, yielding, as for the 77K,1.1V FR, a decay at low frequency in good agreement with the measurement.

High bias

As already pointed out, at high bias we don't observe anymore the drop in the FR asL K → 0. In other words, J is shunted by " J , which can be taken as the Schottky contact impedance at virtually all frequencies. As we did at low bias, we also assume that the frequency is sufficiently low that 2òL K " k &2\ ≪ 1 (e.g. fb = 1GHz). Under these assumptions we have that with u Ö = u J × " k /(" k + " â + " Ö ), yielding:

c Ö = U T u J T " Ö 1 \ é \ é 7\ ê 7\ ù = T , (S24)
where, as usual, " k + " J = "G[Ñ #$%& () = 0)]. At T=77K, since " m ≫ " J , we have that u J ≈ w × u Sn = 11.7wH, and " k + " J = 60Ω (see Fig. S8(c)). By using c Ö from Eq.(S24) to fit the value of the measured FR at 1GHz (3.4V, 77K curve in Fig4 of the main text) we obtain " k = 40Ω and " J = 20Ω. As shown in Fig. 4 of the main text, from Eq.(S20) we obtain an excellent agreement with the measured FR using B ~8ps.

At T=300K we still have " m ≪ " Sn , i.e. " • ≈ " k . Hence, from Eq.(S17) and (S24) we have: with " k + " J = "G[Ñ #$%& () K = 0)] = 40Ω (see Fig. S8(d)). Since Eq.(S25) also depends on the sum " k + " J , in this case the values of " k and " ¢ are determined by fitting the measured FR over the full frequency range using Eq.(S20) (with w × u Sn = 1.7mA). The best agreement is obtained with " k = 40Ω and " J = 0 (Fig. 4 in the main text). As was done at low bias, by subtracting " J from the QWIP differential resistance in the dark (Fig. S7(d)), we obtain the value of the PAR array dark resistance at 2.5V (300K): Rd ~ 42W.

As shown in Fig. S8(c),(d), contrary to what happens at low bias, the computed impedances at high bias provide only an approximated value of the actual QWIP impedance (see Section 4).

Evaluation of carriers capture and transit times T = 77K, Vbias = 3.4V.

From the responsivity reported in Fig. 3(c) of the main text at c ™´™ = c U + c T =33.5mW, we obtain A(77K, 3.4V) = B C /B EF ≃ 1.25 (i.e. ~ half the value at low incident power A ≃ 2.5see Section 1). The roll-off time constant B can therefore be approximated by the transit time [5]. From the fit of Fig. 4 in the main text, we then have that τ ≃ B EF ≃ 8ps and B C = 1.25 × B EF ≃ 10ps T = 77K, Vbias = 1.1V.

We have that u Sn ∝ A, therefore (see Table 1 in the main text):

  Fig. 1 (a) SEM image of the 5x5 PAR array with an integrated coplanar waveguide. (b) Closeup on panel (a) showing the full 5x5 PAR array used in the experiment (s=1.85µm; p=3.9µm), and the air bridge connecting the coplanar waveguide to the array. (c) Individual resonators incorporating the multi-QW structure are connected by suspended Au wires of ~150nm diameter (an array with p = 5µm is shown in the panel).

Fig. 2 .

 2 Fig. 2. (a) Absorption spectra of the PARs 2D array measured at 300K in two orthogonal polarizations: perpendicular (black) and parallel (red) to the wire bridges (spectra recorded at 77K, not shown, are virtually identical). The measurements are performed using a MIR microscope connected to the spectrometer. Insets. PAR fundamental modes in the two polarizations: computed 2D spatial profiles of the electric field component perpendicular to the surface (blue -positive; red-negative). Plots were obtained using a commercial FDTD solver. (b). Photocurrent spectra measured at 77K in the two orthogonal polarizations (solid lines). Both spectra are normalized to the peak of the photocurrent spectrum at 90°. Dashed lines: spectra obtained by multi plying the spectrum of panel (d) by the absorption spectra of panel (a). (c) Normalized photocurrent vs polarization angle, measured at 300K, with a quantum cascade laser emitting at 10.3µm (120meV -dashed blue lines in panels (a), (b), and (d)). The red line indicates the polarization angle (45°) used for the measurements displayed in Fig.3 and Fig.4. (d) Photocurrent spectrum measured at 77K (Vbias = 0.25V) with the QWIP processed in a mesa geometry.

Fig. 3 .

 3 Fig. 3. Photocurrent vs applied bias at (a) 77K and (b) 300K for different incident QCL powers. The dark current I/V characteristics are shown in dashed. (c) Photocurrents (black dots) and responsivities (red dots) vs power, measured at 2.5V, 300K (squares) and 3.4V, 77K (circles). (d) Small signal; equivalent circuit of the QWIP detector (see text). CPAR ~30fF and Cs ~1pF are respectively the 2D PARs array and Schottky contact capacitances. R0 and

  Photocurrent (mA)

  (a). The photodetector was connected to a wideband bias-T and simultaneously illuminated by two 10.3µm-wavelength DFB QCLs driven with ultra-low noise (~300pA/Hz 1/2 ) current generators (see Supporting Information for a schematic of the experimental setup). The current of one QCL was kept constant while the current and temperature of the second one were fine-tuned in order to sweep the heterodyne frequency in the range 0-67GHz. The powers incident on the QWIP from the two QCLs are P1 = 27.5mW and P2 = 6mW (33.5mW total). The spectra of Fig.4correspond to the intensities of the heterodyne beat signals recorded with a spectrum analyzer (SA) set in max-hold trace mode.

Fig. 4

 4 Fig.4 QWIP detector FRs at different temperatures and biases (dotted curves). The powers incident on the QWIP from the two QCLs are P1 = 27.5mW and P2 = 6mW (33.5mW total). The spectra are corrected by the attenuation from the QWIP to the SA, measured with a VNA analyzer. The solid lines correspond to fits obtained using the small-signal circuit model of Fig.3(d) for different carrier's lifetimes (see main text).

  ZL~50W represent (i) the dc internal photoresistance of the 5x5 PAR array under illumination, (ii) the resistance of the reverse biased Schottky contact under illumination (the forward biased Schottky junction is considered as a short circuit), and (iii) the measured load impedance seen by the QWIP detector. CPAR ~30fF and Cs ~1pF, are respectively the 2D PARs array and Schottky contact capacitance. The former corresponds to the computed static capacitance of the PARs array The current source in the circuit represents the photocurrent generated in the patch array oscillating at the beat frequency wb. It is given by:

  represent the power dissipated in the load: H I = J * ℛL[N I ] • |; I | * . They are computed from the circuit of Fig.

Fig. 5

 5 Fig.5 Examples of single shot heterodyne beatnote spectra recorded (a) at 77K without amplification, and (b) at 300K with a low noise, narrow band amplifier of 50dB gain.

Fig. S1 ( a )

 a Fig. S1 (a) Computed responsivity spectrum from Eq.(S1) and (S2)

Fig. S4 ( a )

 a Fig. S4 (a) Heterodyne beat spectra corrected by the attenuation and QCL power change, following the same procedure used to obtain the spectrum of Fig.S3(b). The black dots are those displayed in Fig.4 of the main text (300K, 0.9V -77K, 1,1V -77K, 3.4V).

Fig. S5 ( a )

 a Fig. S5 (a) Electrical circuit model of the heterodyne mixing experiment. (b) Schematic Schottky diode IV characteristic in reverse breakdown (red). Modelled Schottky diode IV characteristic (green).Load lines under illumination (pink) and in the dark (blue), obtained from the circuit of Fig.S6(b). Operating point under illumination (black circle) and in the dark at 300K (orange circle) and 77K (purple circle). For simplicity, but without loss of generality, we have neglected the temperature dependence of the Schottky IV characteristic.

Fig

  Fig. S6 (a) Equivalent small-signal ac circuit (w = wb). (b) Equivalent dc circuit (w = 0).

  (b) in the main text, and from Fig.S7(b),(d). In other words Rd ~ R0, and the change of slope of the load line from illumination to dark (Fig.S5(b)) is small, i.e. the linearization of the Schottky IV through Rs is a good approximation.

Fig. S7 (

  Fig. S7 (a),(b) Current under illumination and in the dark at 77K and 300K. (c),(d) Differential resistance under illumination and in the dark at 77K and 300K.

Fig. S8 .

 S8 Fig.S8. Real (black) and imaginary (red) parts of the QWIP impedance, extracted from the S11 parameters measurements, after de-embedding the 50W integrated coplanar line. The measurements at 77K (panels (a), (c)) were recorded under illumination with a power c E{E = c U + c T = 33.5mW. The measurements at 300K (panels (b),(d))were done in the dark. The blue and purple lines (see text) represent the impedance computed from the small-signal equivalent circuit of Fig.S6(a) using the values of the resistances shown in the legends, with &2\ = 30fF and J =0.7pF.

Fig. S9 .

 S9 Fig.S9. Real and imaginary part of the impedance seen by the QWIP in the plane of the coplanar probes (Ñ Ö ).

  The capture time (. / ) and transit time (. 01 ) are obtained from τ and the photoconductive gain (see text). The corresponding drift velocity (i ? ) is obtained from the ratio between the thickness of the QWIP active region (365nm) and . 01 .

	T(K)	77	300	77	300
	j S`kC (V)	1.1	0.9	3.4	2.5
	; <= (mA)	0.49	0.14	15.2	2.2
	; @ (mA)	0.49	0.38	15.2	2.2
	> @ (W)	200	75	40	40
	> C (W)	350	125	20	0
	τ(ps)	1.5	≲1	8	2.5
	τ / (ps)	1	≲1	10	2.5
	τ 01 (ps)	25	90	8	14
	i ? (x10 6 cm/s)	1.5	≳0.4	4.6	2.6

Computed responsivity

The responsivity of the PAR array for an incident electromagnetic wave of frequency ω, polarized perpendicularly to the wire bridges, can be computed from [1]:

2 345 (6)7# 89:

where [1-R())] is the PAR array absorption spectrum shown in Fig2(a) of the main text, A = B C /B EF is the photoconductive gain, G is the electronic charge, and H IJK ()) is the intersubband absorption coefficient of the PAR given by the following expression:

Here L M = 0.088 is the overlap factor between the PAR mode and the quantum wells; ℏω S =27.5 meV is the intersubband plasma energy [2]; ℏω TU ≈ 107meV is the intersubband transition energy extracted from the photocurrent spectrum (see Fig. 2(d) of the main text), and Γ ≈ 10.7meV is the FWHM of the intersubband transition, which we assume to be approximately equal to 10% of ℏω TU . Q &2\ ≈ 8 in Eq.( S1) is the quality factor of the PAR array (i.e. excluding intersubband absorption), that we obtain from the FWHM of [1-R())] (Fig. 2(a) of the main text). This is a good approximation since, due to the spectral shift between H IJK ()) and Following the same procedure described above we obtain A(300K, 0.9V) ≃ 0.011. From the fit of Fig. 4 in the main text, we then have that τ ≃ B C ≲ 1ps, and B EF ≳ B C /0.011 ≃ 90ps.

T = 300K, Vbias = 2.5V.

Following the same procedure described above we obtain A(300K, 2.5V) ≃ 0.18. From the fit of Fig. 4 in the main text, we then have that τ ≃ B C ≃ 2.5ps, and B EF ≃ B C /0.18 ≃ 14ps.