E Lhuillier

S Keuleyan

H Liu

P Guyot-Sionnest

Mid-IR colloidal nanocrystals

Keywords: HgTe, quantum-dots, infrared

Colloidal quantum dots presenting optoelectronic properties into the mid-infrared are reviewed with an emphasis on HgTe. Interband transitions with narrow band gap material and intraband transitions with wide band gap semiconductors can both address the infrared range of wavelengths. Semimetals are particularly promising since, by controlling the particle size, a gap can in principle be opened and tuned through the full infrared spectrum. HgTe has been quite successful recently, and colloidal synthesis has allowed the tuning of the absorption edge from 1 and 5 µm at room temperature. The cut-off wavelength and absorption coefficient of these materials have been discussed. Electrochemistry experiments have demonstrated that both n and p nanoparticles are stable. Films of the HgTe colloidal quantum dots have shown room temperature photoconduction when excited in the mid-IR. With novel processing leading to improved detectivity, these materials are becoming of significant interest as an alternative to the much more expensive epitaxial detector technologies.

I. Introduction.

The infrared (IR) spectrum is typically divided into the near-IR (800 nm to 2.5 μm), mid-IR (2.5 μm to 20 μm), and the far-IR (above 20 μm, including THz). Near-IR imaging relies on scattering of an external light source while, above 3µm, the intensity of the black body emission allows thermal imaging. The mid-IR range is also the range of operation of typical infrared spectrometers used for molecular vibrational spectroscopy. In the mid-IR range, there are two transparency windows of the atmosphere, 3-5 µm and 8-12 µm, where ground access to satellite and star light is possible. These ranges are typically called mid-wave IR (MWIR) and long-wave IR (LWIR) respectively, and infrared thermal imaging is targeting these two separate ranges. 1 For single detector applications in scientific spectrometers as well as in infrared motion sensors, pyroelectric materials such as Deuterated Triglycine Sulfate (DTGS) are widely used. However the low sensitivity and low speed of pyroelectric materials does not favor imaging.

Thermal imaging is best provided by photovoltaic or photoconducting detectors relying on inorganic semiconductors. In particular, HgCdTe (MCT for mercury cadmium telluride) can be composition tuned to any cut-off wavelength in the mid-IR and therefore optimized for either MWIR or LWIR. MCT has dominated the field for the past 50 years but it has a number of drawbacks including high material cost, cooling requirements, difficulties in obtaining homogeneous chips, and the delicate lithography due to the relative softness of the material. Imaging chips based on MCT provide high quality imaging devices with megapixel cameras, used for military and astronomy applications.

However, their more widespread use is limited by high costs, typically in excess of $50,000 for a full imaging system. Alternatives to the MCT are therefore actively sought. The quantum engineering of wider gap materials has led to infrared imaging based on intersubband and intraband transitions of Quantum Well Infrared Photodetectors (QWIP) and Quantum Dot (QDIP) of III-V AlGaAs/GaAs epitaxial materials. [START_REF] Schneider | Quantum well infrared photodetectors: physics and applications[END_REF][START_REF] Levine | [END_REF] QWIP provide a viable 4 although still expensive technology. Type II superlattices of InAs/GaSb also allow LWIR photodetection. 5 6 This epitaxial technology allows higher operating temperatures and is compatible with well-established lithography processes and it is emerging as a very promising alternative. 7 . However, it is still costly and in the development stage. For lower price and lower performance, microfabrication now allows for r microbolometer cameras which operate at room temperature and have good performance in the LWIR but poor performance in the MWIR. At the time of writing this review, the cost of microbolometer imaging systems is still high, on the order of $10,000 for cameras with less than 0.1 megapixels, but they have started to penetrate the consumer market, with a widening range of applications in home insulation diagnostics, hot-spot trouble shooting in machinery, etc. Novel applications are likely to emerge as the cost of the imaging devices goes down.

Colloidal quantum dots (CQDs) were first studied as rather monodisperse semiconductor nanocrystal precipitates in transparent glassy matrix. 8,9 The liquid phase colloidal synthesis 10 and the ease and precision with which the electronic structure and thereby the optical properties of the material could be tuned with size 11 have since generated much scientific interest. 12,[START_REF] Klimov | Nanocrystal Quantum Dots[END_REF] In particular, CdSe colloidal quantum dots in the visible spectrum have been widely studied. CQDs can have bright and robust visible luminescence when capped by an epitaxial shell, such as with CdSe/ZnS 14 and other combinations. 15 The fluorescence properties have led to the exploration of biolabelling 16 and light emitting diode 17 applications.

Despite great progress, the visible emitting CQDs have not yet displaced technologies based on organic chromophores. The interest in CQDs has extended to the near-infrared with other materials, such as the widely studied PbS/Se CQDs, first in a glassy matrix, 18 and then in liquid solution. , 19,20,21,22,23 Much of the current research on the near-IR CQDs is motivated by the hope of CQD thin film solar cells 24,25,26 that might compete with the existing silicon or CdTe technologies and emerging organic photovoltaics. Further into the near-IR, above about 1.5 microns or twice the energy of a CH stretching vibration, the electronic excitation of organic chromophores is very quickly degraded to vibrations. This leads to very low photoluminescence (PL) quantum yields for organics, at best 2 to 3 orders of magnitude lower than typical PbSe CQDs in the same spectral range. 20 In principle, the inorganic cores of the CQDs can allow sustained electronic transitions much further into the mid-IR until close to the optical phonon frequencies in the far-IR. Since CQDs promise low-cost fabrication, with simple processes such as dip-coating, spin-coating, spray-coating, printing etc, they provide an attractive alternative to the expensive procedures required for epitaxially grown materials.

In the very high price mid-infrared market, combining these low-cost processing opportunities with the expectation of good optoelectronic properties is a strong motivation for investigating mid-IR CQDs.

A size-tunable mid-infrared response of the CQDs can be obtained by using either intraband or interband transitions. The attractiveness of the former is that any wide band gap semiconductor CQD can, in principle, be used for mid-IR applications including non-toxic materials such as ZnO as well as many other CQDs with well developed colloidal synthesis such as CdTe/Se/S or PbTe/Se/S. However, observing these transitions requires electrical doping. At present, the intraband transitions have been observed transiently after optically excitation 27 as shown in Fig. 1a or by doping by electron transfer 28 29 30 31 as shown in Fig. 1b.

Although the intraband transitions have been used to learn about the CQDs electronic properties, their further application will require successful impurity-doping 32 34 These resonances are plasmon resonances and may be too short lived for some infrared applications but they deserve further study. 36 Other binary narrow-band semiconductors including InAs, 37 Cd2As3, 38 and silver chalcogenides, 39,40 nanoparticles of PbSnTe were shown to tune to longer wavelengths than particles of either PbTe or SnTe.

In contrast to semiconductors, semimetals have no intrinsic limitation towards long wavelength. For example, when graphene will be synthesized as strips or dots with well defined sizes, it could provide an interesting IR material. 42 Among the binary materials, the mercury chalcogenides have a special place since they are semimetals with a particularly simple band structure. In particular, bulk HgTe has been extensively studied 43,44,45,46,47,48 HgTe material parameters where the conduction band is the light hole and the valence band is the heavy hole. The band structure is shown in Fig. 2. This inverted structure crosses back to the normal ordering at the free surface or at the interface with CdTe which leads to a Dirac cone for the dispersion at the surface/interface and this opened the use of HgTe as a topological insulator. 53 The discovery of the quantum spin Hall effect 54 was done with a quantum well structures of HgTe/HgCdTe with a 6.3nm or thicker HgTe layer. In addition to transport studies conducted on pure HgTe, the HgCdTe alloy has long been investigated for infrared detection purposes. As mentioned earlier, this is the dominant mid-IR material for photodetection where, with a precise cadmium concentration, the optical gap tunes from zero to the gap of pure CdTe.

-0.04 -0.02 0.00 0.02 0.04

II. Synthesis of HgTe CQDs.

The indication that HgTe was promising as CQDs was first provided by Rogach et al. 55 In an aqueous colloidal synthesis, HgTe nanoparticles of 3 to 6 nm diameter were produced with luminescence near 1.3 mm, covering the first telecom band. Weller and coworkers continued to investigate the synthesis of HgTe CQDs in the early 2000's 56 , including work on HgTe/CdS core/shell structures 57 . Their method was based on the reaction of Hg(ClO4)2 dissolved in a basic aqueous solution with H2Te. This allowed the bandgap to be tuned across the near-IR as far as 1.8 m. Kovalenko et al modified this method in 2006. By adding an

Ostwald ripening step at elevated temperature, followed by transfer to a nonpolar phase and size-selective precipitation, particles of ~10 nm with PL close to 3µm were obtained. 58 In addition to size, the bandgap can be controlled through composition, as is done in bulk MCT.

Alloying was reported starting directly with a mixture of mercury and cadmium precursors with however very modest tuning into the near-IR up to 850 nm 59 HgxCd1-xTe alloy particles have also been prepared via Hg 2+ cation exchange on CdTe particles 60,61 as well as one-pot reactions 62,63 showing high PL quantum yields, but these studies were also limited to the near-IR. In parallel with the development of the aqueous synthesis, Green et al 64 reported a synthesis of HgTe directly in organic solvent. The synthesis was based on the reaction of mercury bromide with tellurium complexed with trioctylphosphine (TOP), noted as TOP:Te, in octadecylamine as a coordinating solvent. The resulting nanocrystal colloidal material was HgTe and crystalline but polydispersed with a broad absorption spectrum that did not allow assigning a band edge, and there was no detectable PL. In 2006, Piepenbrock et al 65 developed a related synthesis using mercury acetate in chilled alcohol in presence of hexadecylamine and TOP:Te. Using this approach, a clear absorption edge was observed in the near-IR and PL up to 1.6 m was reported. With pure HgTe, the k.p band structure in Fig. In a typical synthesis, 0.1 mmol HgCl2 and 3.5 g ODA are heated under vacuum to 120°C for 1 hour to degas. The solution is then cooled under Ar to the growth temperature determined by the desired size. 0.1 mL 1M Te in TOP is then quickly injected and the solution darkens at a rate dependent on the temperature, typically appearing black within 30 s or less. TEM images, X-ray Diffraction, and absorption spectra of the CQDs are shown in Fig. 3. The HgTe materials are in the stable bulk zinc-blend phase. After the reaction, the HgTe particles can be capped with dodecanethiol, which stops the growth and allows stable dispersion in tetrachloroethylene (TCE) for storage. Since the excess ligand and solvent are opaque in the infrared, the dots need to be extracted for optical characterization. The dodecanethiol capped HgTe dots are easily precipitated using methanol. The precipitate can then be rinsed, dried and redispersed in neat TCE, yielding a solution that is stable in air for weeks to months, and which can be used for absorption and PL measurements. 68 Films of the HgTe CQDs can be obtained by directly drying the TCE solution. Better uniformity can be achieved by precipitating and redispersing in a 9:1 (vol.) hexane:octane mixture. Thicker films can be prepared by depositing multiple layers, fixing each before depositing the next using a dilute solution of shorter ligands such as ethanedithiol. Improved adhesion to the substrate can be achieved by treating the substrates with mercaptopropyltrimethoxysilane for glass substrates or a dithiol for metal electrodes.

Safety:

The acutely lethal organomercury compounds are avoided, but mercury(II) salts are toxic and appropriate precautions must be taken to avoid exposure or release into the environment. HgTe is considered a toxic compound as well as a hazardous product for the environment. The handling of the material requires the appropriate safety equipment including eye protection, gloves and labcoat. Associated risk codes are R26, R27 R28, R33, R50 and R53, while the safety phrases are S13, S28, S45, S60 and S61.

III. Optical properties of HgTe CQDs.

The size control afforded by the methods described above allows the room temperature optical bandgap of HgTe CQDs to be tuned between 1.2 and 5 μm. A two-band k.p model provides a fair prediction of the bandgap as a function of particle size as shown in Fig. 3e.

68

Within the two-band k.p model, an analytical expression of the particle bandgap is given by

0 2 2 2 2 3 2 4 2 m k E E E E P G G QD BE  + + = (1)
where G E is the bulk band gap of HgTe, P E is HgTe Kane parameter and

QD R k  =
is the wavevector associated with the ground state transition assuming a complete confinement of the carrier within the CQD. The parameters used are given in Table 1. Using Eq.1 and a best fit to the data, the wavelength of the first interband transition (in µm) as a function of nanoparticle radius ( QD R in nm) is given by the empirical expression

( ) 1 / 43 1 5 . 30 2 - + = QD QD BE R  (2)
In addition to varying with particle size, the optical-gap also changes strongly with temperature. Like bulk HgTe, HgTe CQDs show a redshift of the band edge transition upon cooling. As shown in Fig. 4a, the absorption band-edge wavelength of the largest size samples shifts from 5m at room temperature to 7µm below 80K. dT dE QD BE is weaker for the smaller particles, as shown in Fig. 4b. This is a similar behavior to that seen for PbS CQDs. 69 In contrast, CdSe 70 shows a size-independent negative dT dE QD BE . Most semiconductors exhibit a negative dE/dT which is attributed to the thermal expansion and to the second order perturbation of the intraband electron-LO phonon coupling. However, several narrow gap semiconductors, including the II-VI Hg-chalcogenides and the IV-VI Pb-chalcogenides have a positive dE/dT. An interpretation is that, in narrow band gap materials, the interband electron-LO phonon coupling second-order contribution which is positive, can be larger than the intraband part. As the size is reduced and the gap increases, the positive interband contribution decreases, leading to a smaller temperature tuning of the band edge. Such effect is seen in CQDs of PbS, 69 PbSe 71 and HgTe. 68 A key parameter for the use of the HgTe CQD in optoelectronic devices is the material absorption strength. In contrast to self-assembled epitaxial quantum dots, which have been investigated in QDIP, the CQDs allow close-packing. In addition, there is a transfer of the oscillator strength to the QD band edge. Therefore the optical absorption edge of HgTe CQDs films could be even stronger than the bulk MCT. This absorption can be quantified in solution as an absorption cross-section per particle, or for a film as an absorption coefficient.

Using acid-digestion and Hg titration with dithizone, the molar concentration of Hg was determined for CQDs of various sizes and compared to the optical density. 68 To get an absorption cross-section per particle, the particle size was estimated from the XRD peakwidth. Far above the band edge, at 415 nm, the cross section per mercury atom is almost constant and equal to 2.6×10 -17 cm 2 and this can then be used for analytical purposes. The lowest-energy transition cross-section per particle was found to be ~1.5×10 -15 cm 2 with a weak size dependence. The integrated band edge cross-section, which better accounts for the sample to sample polydispersivity variation, was ~10 12 cm, as shown in Fig. 4c and it has been satisfactorily compared to the k.p model. For films, the absorption coefficient has been measured as well. Experimental and k.p modeling of this parameter are displayed on Fig. 4d.

The absorption coefficients of the films of HgTe CQDs are indeed as high or higher as those for bulk HgTe [START_REF] Nimtz | Narrow gap semiconductors[END_REF] or bulk HgCdTe alloys 73,74,75 with a similar cut-off wavelengths. Fig. 5b shows PL spectra of HgTe CQDs. It has been noted that organic ligands not merely reduce transmission of the materials in the mid-IR, but more importantly provide an extremely efficient near-field energy transfer from the inorganic core electronic excitation to surface molecular vibrations. 78 Such an effect has been reported to affect the PL quantum yield of near-IR CQDs of InAs 79 and PbSe, 76 77 as well as to cause fast intraband relaxation. 78 80 Therefore inorganic matrices and core/shell structures will be beneficial. Besides the wide tunability of the CQDs, one advantage of HgTe compared to the lead chalcogenides is the greater simplicity of the band structure, including the zero gap and the non-degeneracy of the band edge. This results in a smaller density of states at the absorption edge, which should facilitate population inversion and reduce Auger recombination rates. On the other hand, surface states arising from the inverted band structure that is specific to HgTe might enter the gap at large enough sizes, creating the possibility of additional recombination channels. The HgTe mid-IR PL properties have not yet been thoroughly characterized, but they might be of interest in LEDs or lasers, either by direct electrical pumping or by using near-IR optical excitation. The characterization of the PL of the material will require the determination of the PL lifetime, quantum yield, carrier lifetimes, Auger processes, nonradiative recombination processes, and stimulated emission properties. The HgTe CQDs therefore warrant much further investigation of the PL properties.

IV. Electronic properties of HgTe CQD

Transport in HgTe CQDs has been first characterized using thin solid film field-effect transistors (FET). 81,82,83 By annealing films of thioglycerol capped nanocrystals, high mobility FET, with µ≈1 cm 2 V -1 s -1 and good on/off ratio ≈10 3 , were obtained. However, the material showed only p-type transport and no optical data was reported. With unannealed samples made using an inkjet printer, Heiss and coworkers demonstrated photoconduction up to 3

μmfor their largest particles but there was no study of the conduction mechanism or of the carrier mobility. 84 In general, knowing the absolute energy position of the valence and conduction bands will give some insight into the relative stability of electrons and holes in the quantum dots. Electrochemistry is therefore an advantageous characterization tool for CQDs.

Electrochemistry of HgTe CQDs has allowed determining that both n and p-type charging was possible in anhydrous conditions with moderate reduction or oxidation potentials, as shown in Fig. 6a and6b. In addition, the bleach of the optical spectrum and the rise of intraband absorptions were monitored, confirming that both n and p-charging corresponded to carrier injection in the quantum dot states instead of surface states. Based on the interband bleach and intraband absorption as a function of charge injection, an energy level diagram of the CQD states for two different sizes was reconstructed as shown in Fig. 6c. It was noted that, although the band edge energy was well accounted for by the k.p model of a spherical box, the higher lying transitions of the bleach spectra could not be matched well to the expected states of the spherical quantum dot, an effect which was attributed to the nonspherical shape of the nanocrystals. A tight-binding calculation has been developed for HgTe nanocrystals of different shapes. 85 Although fair agreement was found with the band edge absorption energy, further comparison between theory and experiment will require the synthesis of HgTe nanocrystals with better defined shape.

In contrast to the previous FET results, electrochemistry showed than both n and p-type transport is effective, with similar magnitude of n and p-mobility. It was also noted that the photocurrent was 100x larger for p-type, while photocurrent decays following transient illumination were 100x slower for p-type than for n-type, as shown in Fig. 6d. This was assigned to an electron trap cross-section 100x smaller than that for holes but the recombination mechanism has not been investigated. A difference between transport in the narrow gap CQDs films and earlier studies on the wider gap nanocrystals of CdSe, or near-IR gap PbSe is the significant thermal carrier density. Even in an undoped or uncharged film of CQDs, i.e. intrinsic, there is a thermal carrier density which gives rise to a dark current. For the intrinsic case, the carrier density is given by:

) 2 / exp( 0 T k E N N N p n B G v c -  = = ( 3
)
Where No is the density of dots per volume, EG is the dot band gap, and Nc, Nv are the effective conduction and valence density of states per dot. For an HgTe dot with strong confinement (EG>>kBT), this was estimated as Nc=Nv=2. 86 Eq. 3 then leads to one carrier per thousand nanoparticles for a 5µm cut-off CQD film at temperature. For comparison there would be, at room temperature, only ~10 -17 thermally activated carrier per dot in a CdSe QD with a visible band edge. Transport in films of the aggregated HgTe CQDs that first allowed mid-IR photodetection was observed to be quasi-ohmic with I-V curves that are linear at room temperature and become less linear at low temperature. While such materials could have been naturally doped, it was instead found that between about 150 K and room temperature, the temperature dependence was Arrhenius with an activation energy close to half the optical band gap, as shown in Fig. 7a. 86 Following Eq. 3, these materials were therefore assigned to being intrinsic with a carrier concentration determined by the thermal excitation over the gap. From the current and the estimated carrier concentration, the carrier mobility was estimated in the 10 -2 -10 -1 cm 2 V -1 s -1 range. Using the Einstein relation, the inter nanoparticle hopping time was evaluated as

ps T k eR B QD hop 13 3 2 2    
with QD R the CQD radius and µ is the mobility. It was then proposed that the good photodetection properties likely arose from a hopping time faster than the nonradiative recombination times.

As the synthesis improved allow for monodisperse HgTe CQDs with better spectral properties, it was found that simply dried films were very resistive and showed no photoconduction. This was a side-effect of the better passivation due to the insulating alkane ligands. An organic ligand exchange was performed directly on such films, by dipping in a solution of short ligands dissolved in a solvent such as ethanol, which will not redissolve the film. Exchange with several thiol ligands (ethanedithiol = EDT, hexanedithiol, butanethiol, butanedithiol), and amine ligands (propylamine, butylamine, heptanediamine) was tested, giving qualitatively similar results with measureable mobility and detectivity. 87 However, the mobility of these films was in the 10 -4 cm 2 V -1 s -1 range, two to three decades lower than those obtained with the aggregated material. It was also found that these films were not intrinsic,

showing instead current vs temperature, I(T), curves with a prominent maximum, around 200K, as shown in Fig. 7b. Such rise of the conductance with decreasing temperature was attributed to temperature dependent doping and not to improved mobility at lower temperature. The conductivity was determined to be p-type by FET measurements 87 as for prior work, 81 82 83 while the increase of the carrier concentration with decreasing temperature was attributed to the temperature dependent ionization of a surface-related hole donor. The qualitative argument proposed was that the energy shift of a surface state with temperature is different from that of the HgTe band and that the conductance peak occurs when the levels cross. However, the detailed mechanism of the nonmonotonic I(T) curve has not been worked out. It was then reported that this behavior arises from air-exposure of the films during the ligand exchange, and this was attributed to the likely oxidation of the HgTe surface. Once processed under nitrogen during the ligand exchange and kept air-free, the I(T) curve of the HgTe CQDs films indicated intrinsic behavior and FET measurements showed both n and p-type mobilities.

For infrared applications of CQDs, the vibrational absorption of typical organic ligands is an impediment to high efficiencies. Ultimately, the CQDs should therefore be embedded in an IR transparent matrix, with ideally very low frequency phonons or vibrations. Different strategies have been reported for the preparation of inorganic matrices for CQD films. In one method, CQDs films with the usual ligands coverage are subjected to a ligand exchange/removal and exposed to a few cycles of Atomic Layer Deposition (ALD). 88,89 The ALD inorganic matrix leads to improved mobility and protection of the CQDs film.

However, the removal of the ligands in the films is typically not complete. As pioneered by

Talapin and coworkers, it is much more effective to perform an exchange with charged inorganic ligands directly solution. 90 A following annealing step may then be used to convert the inorganic ligands, such as hydrazinium metal chalcogenide complexes or chalcogenide ions 91 to an inorganic semiconducting matrix. This method allows for the complete removal of the organic ligands and a great flexibility in the choice of the final matrix material. In addition, vast improvements in the electronic mobility have been reported. 92 Recently, Kovalenko et al 93 demonstrated the use of As2S3 sol-gel for making near-IR luminescent films containing PbSe/CdS CQDs. In a related method 87 a dilute As2S3 ligand exchange solution was used to replace the organic ligands of HgTe CQD films, and to fill the interparticle spaces. As2S3 was first dissolved in propylamine and diluted in ethanol.

Then films of HgTe CQD were dipped for one minute into the As2S3 solution and rinsed with ethanol. This was a simpler procedure than the liquid phase transfer of the colloidal dots but it only allowed for about a 90% removal of the organic ligands as judged by the remaining infrared CH stretching vibrations. As with the thiol ligand exchange, the ligand exchange with As2S3 must be performed in an inert atmosphere to prevent oxidation and p-type doping of the HgTe/As2S3 CQD films. With these precautions, the HgTe/As2S3 system, showed Arrhenius I(T) curves with an activation energy consistent with intrinsic behavior. As shown in Fig. 7c, n-and p-type FET were both possible with on/off ratio as high as 10 6 at low temperature. As shown in Fig. 7d, the mobility was also improved by two orders of magnitude at room temperature compared to the ligand exchange with ethanedithiol. The improved mobility may arise from several factors, including reduced barrier height, shorter dot separation, as well as lowered charging energy from the increased dielectric constant.

For infrared detection, several other properties of the material need to be determined. 1 These are the response time, the noise, the responsivity (conversion of light to electrical current), and the detectivity (normalized signal to noise ratio).

The time response is an important property that determines the potential for imaging applications. While near-IR CQDs PbS detectors have been reported to be very sensitive, their response times were also very slow, 94 a result of some very high internal gain mechanism, associated with long lived traps. Similar results were reported with near-IR PbSe and HgTe photodetectors. 95 In contrast HgTe CQDs photodetectors have shown fast response, of the order of the transit time, and consistent with unity gain. Fig. 8a shows the flat frequency dependence of the photocurrent in the 10Hz to 100kHz range. This is therefore promising for fast imaging applications.

Noise is a key parameter of a photodetector. Most reports on CQD-based photodetectors measure the noise only at a given frequency 84 94 but 1/f noise is important in the HgTe CQDs photodetectors, as shown in Fig. 8b. The magnitude of the 1/f noise depends on temperature and its origin is not yet understood. Using the empirical Hooge equation, the 1/f noise magnitude is characterized by the

H  parameter with f N I i H f   = 2 2 / 1



, where f is the signal frequency, I the current, N the total number of carriers involved in transport and f i / 1 is the noise current density. Values for bulk crystalline materials are 10 -6 -10 -3 . However, a value of H  ~ 0.4 was reported for a HgTe CQD film 86 based on estimates of carrier density and this is in large excess compared to single crystal materials. Future studies of film processing should shed light on the origin of the large excess noise in the CQD films and possibly lead to lowering the 1/f noise. This will then allow for improved detectivity.

The responsivity of a photoconductor is the ratio of photocurrent to incident optical power, in A.W -1 (or V -1 ). It can be expressed as

g E G R  =
where G is the gain,  is the external quantum efficiency, and Eg is the gap energy in Volts. With a unity gain and 100% quantum yield, the responsivity of a 2 micron detector would be 1.5 A.W -1 . HgTe CQDs processed with As2S3 have very fast response time, of the order of the expected transit time suggesting that there is no significant gain. Such a detector at 2 microns showed a responsivity of 0.65 A.W -1 , as shown in Fig. 8c, which is consistent with an external quantum efficiency of 50%. For longer wavelength detection with HgTe CQDs in the mid-IR, at 3.5 microns, the responsivity is lower, around 0.22 A.W -1 indicating that an order of magnitude improvement is still possible in the quantum efficiency. A priori, improvement of the responsivity will arise from an understanding of the nonradiative recombination processes and progress in the surface control of the CQDs. The detectivity of a photoconductor is the inverse of the power equivalent to the noise, normalized per area and bandwidth. It is obtained as

n i A R D = *
where R is the responsivity, A is the detector area, and in is the current noise spectral density (A/Hz

V. Conclusion and Perspective

As soluble materials with a priori well defined infrared electronic properties, CQDs Improved responsivities are likely to arise from understanding and eliminating the nonradiative processes, possibly with the introduction of core/shell systems such as the lattice matched HgTe/CdTe. This would also allow for new applications of the CQDs as mid-IR emitters. They might then be good materials for down converting existing visible LEDs or near-IR laser diodes to make mid-IR light sources and lasers. In parallel the development of inorganic matrix materials with infrared transparency, the complete elimination of organics, and somewhat higher mobility will lead to improved optoelectronic properties, and it is likely that HgTe/As2S3 is not yet the optimum material. There is also a need for understanding the source of the excess electrical noise in these granular nanomaterials, since the understanding may lead to less noisy material and improved detectivity at low frequencies, an important problem for the pixel calibration of infrared cameras. The a priori versatile processing of the colloidal materials will allow an easy integration with plasmonic antenna structures which should further decrease the noise by reducing the volume of the active material, or allow higher operating temperatures. The rapid progress over the past couple years suggests that such improvements may be within reach. At present, the HgTe CQDs appear as a very attractive alternative technology for infrared detection and imaging, justifying further basic studies of their optical properties, further material development, and integration into devices.

Figure 1 :

 1 Figure 1 : (a) Transient intraband absorption of photoexcited solutions of CdSe CQDs of the indicated diameters. 35 Reprinted with permission from J. Phys.Chem. 2000, 104, 1494 Copyright. (b) Absorbance spectra of a CdSe CQD film as a negative electrochemical potential leads to electron transfer to the CQDs, with the concurrent bleach of the band edge and rise of the intraband transition. 31 Reprinted with permission from Appl. Phys. Lett. 2002,

Figure 2 :

 2 photonic band edge

2 shows that 3

 3 to 5 μm band-gaps would be realized with particles of sizes between 8 and 12nm. Expanding on the method introduced by Piepenbrock et al, the reaction between mercury(II) acetate and TOP:Te in butanol in the presence of pyridine led to larger HgTe particles allowing, for the first time, to cover the MWIR.66 Briefly, 0.1 mmol (13 mg) Te powder was dissolved in 2 mL butanol using 0.1 mL TOP at 90°C. Meanwhile, 0.1 mmol (32 mg) mercury(II) acetate was dissolved in 2mL of butanol with 2mL pyridine. The mercury acetate solution was then rapidly injected into the flask containing the TOP:Te, with the solution immediately turning black, indicating the formation of HgTe. Absorption spectra of HgTe CQDs prepared by this method are shown in Fig.3c. The absorption edge is not very sharp, showing an excitonic shoulder rather than a peak. This is due to rather broad size distribution. Furthermore, dynamic light scattering and Transmission Electron Microscopy (TEM) indicated partial aggregation of the dots in the solution. Subsequently, a preparation more similar to that of Green et al. yielded more monodisperse nanocrystals with an exciton peak at the band edge.67 In this preparation, HgCl2 is dissolved in oleylamine and reacted with TOP:Te, giving controlled growth and well defined excitonic absorption peaks from 1.4 to 5 μm. Technical grade (70%) oleylamine presents batch to batch variations which lead to undesireable variability in the synthesis. Octadecylamine (ODA) can be used in place of oleylamine to give more consistent results, with the inconvenience that it solidifies below 52°C.

Figure 3 :

 3 Figure 3: (a) TEM image of HgTe CQDs. 67 (b) X ray diffraction of HgTe CQDs. (c) Absorption spectra of solutions of HgTe CQDs of two sizes. 66 (d) Absorption spectra of more monodispersed HgTe CQDs. 67 (e) Exciton wavelength of HgTe CQD as a function of the CQD size. The dashed line results from a k.p simulation. 68

Figure 4 :

 4 Figure 4: (a) Photocurrent spectra of a film of ethanedithiol cross-linked HgTe CQDs at different temperatures. (b) Thermal dependence of the band edge energy as a function of the room temperature band edge energy for HgTe CQD of different size. (c) Integrated cross section per quantum dot as a function of the exciton wavelength. The solid line results from a k.p simulation. (d) Absorption coefficient of HgTe CQD film as a function of the exciton wavelength. The solid line shows results from a k.p simulation. 68

Figure 5 :

 5 Figure 5: Room temperature PL spectra of (a) PbSe CQD of different sizes (dashed curve measured at 77K), adapted with permission from J. Am. Chem. Soc. 2004, 126, 11752, Copyright (2004) American Chemical Society(b) HgTe particles prepared in ref.67 adapted with permission from J. Am. Chem. Soc. 2011, 133, 16422-16424

Figure 6 :

 6 Figure 6 : (a) Bleach spectrum of a HgTe CQD film at different positive bias of the hole injection. (b) Bleach spectrum of a HgTe CQD film at different negative bias of the electron injection. (c) Reconstructed energy level diagram of HgTe CQDs with an excitonic feature around 3 µm. (d) photo (black) and dark (red) conductance of a 3µm HgTe CQD film under illumination and under dark condition as a function of the electrochemistry potential. also shown is the photocurrent decay time (minority carrier lifetime) following a short pulse from a 800nm diode laser.

Figure 7 :

 7 Figure 7 : (a) Current as a function of the inverse of the temperature for a film of aggregated HgTe quantum dots 86 under 2V. Inset current as a function of the applied bias at 210K. (b) Current as a function of the inverse of the temperature for a film of dispersed HgTe quantum dots ( λ≈3µm), processed with ethanedithiol in air and in a glovebox. (c) Current as a function of the gate bias for a film of dispersed HgTe quantum dots ( λ≈3µm), processed air free with As2S3. (d) Mobility as a function of the temperature for films of HgTe CQDs processed airfree with ethanedithiol or As2S3.

Figure 8 :

 8 Figure 8: (a) Photocurrent as a function of the frequency of the light modulation for an illumination by a 800nm laser (b) Noise current spectral density at room temperature and under different bias. (c) and (d) are respectively the map of the responsivity and detectivity as a function of the bias and temperature for a film of HgTe CQD film with a cut-off wavelength at 2µm

  Photodetectors made with HgTe CQDs films have been demonstrated across the mid-IR, and up to 7 μm due to the red shift of the band edge upon cooling. The close-packed HgTe CQDs films present a large absorption (α>10 4 cm -1 ) compatible with thin film devices. Anhydrous electrochemistry has shown that the HgTe CQDs are stable under both n-and p-type charging. Ambipolar mobility of the HgTe CQDs films has also been observed with solid state FETs. The transport properties of films of HgTe CQDs have been improved by exchanging the organic ligands for an inorganic As2S3 matrix. At present, there is a little more than one order of magnitude to gain with the HgTe CQDs to bring the single detector

Table 1 :

 1 parameters related to the HgTe material

	Parameter		Value	Unit	Reference
	Molar mass		328.2		g.mol -1
	Lattice parameter	0.646		Nm	49
	Lattice structure	Zinc Blende
	E bulk G	( ) T		-	303	+	T 63 + . 11 0	meV	50
	Bulk band gap
	EP				18		eV	50
	Kane Energy
	ΔSO			1			eV	50
	spin orbit coupling
		0				20		51
	Static	dielectric
	constant
						14		49
	Dielectric constant at
	optical frequency
	 	LO			17±3		meV	52
	LO phonon energy
	ρthe mass density	8.1×10 3	kgm -3	49
	Several of its properties are listed in Table 1. Unlike CdTe, the Hg s-like orbital is lower in
	energy than the Te p-like orbital. This results in an inverted band structure with a zero-gap

  1/2 ) at the chosen frequency. The unit of D* is Jones=cmHz 1/2 W -1 . If the noise depends on frequency, D* must then specify the frequency at which it is measured. For imaging, the minimum current noise arises from the background thermal radiation noise, typically at 300K. In addition, there is the Johnson noise, shot noise, recombination-generation noise due to the thermal excitation of the carriers, and 1/f noise. The standard DTGS detectors used in commercial FTIR have a detectivity of ~ 2x 10 8 at a 1 kHz frequency, but they operate without cooling and cover the whole IR range. The liquid Nitrogen cooled MCT detectors in commercial FTIR with response up to 12 microns have a detectivity of ~ 6 x 10 10 Jones. With near-IR PbSe CQDs, a detectivity of 10[START_REF] Klimov | Nanocrystal Quantum Dots[END_REF] Jones has been reported, although with a slow response of tens of Hz. In general, the maximum D* decreases rapidly with longer wavelength due to the increased thermal background. For a 3.5 µm cut-off HgTe/As2S3 at 200 K, with faster than 100kHz response, a detectivity of 3.5×1010 Jones at 1kHz was obtained.8787 This was not an optimized value as the films remained too thin to fully absorb the IR radiation and further improvements are expected. To put this result into context, a photoconductor at 3.5 microns limited by the 2 300K thermal radiation would have an ideal detectivity of 8x1011 Jones and this is nearly reached with commercial cooled InAs detectors. Therefore there is a factor of about 20 to gain with further material development of the HgTe CQDs.

Acknowledgements: The work on HgTe CQDs is funded by the DARPA COMPASS program with a grant from ARO. The infrared photophysics research program is funded by the National Science Foundation DMR-1104755, and studies of electrical transport are funded by the US Department of Energy (DOE) under Grant No. DE-FG02-06ER46326.


Emmanuel Lhuillier

Isabelle Ribet-Mohamed

Emmanuel Rosencher

Gilles Patriarche

Amandine Buffaz

Vincent Berger

Mathieu Carras

Interface roughness transport in THz quantum cascade detectors

Keywords: 72.10Fk, 85.60Gz, 68.37Lp

Infrared Detectors based on a Quantum Cascade have been proposed to suppress the dark current which is a limiting factor in Quantum Well Infrared Photodetectors. Those detectors have been mainly designed for the mid-infrared wavelength. Operating in the Terahertz range involves a complete change of regime of transport since the photon energy is lower than the optical phonon energy. Thanks to a two dimensional model of transport, we have identified interface roughness as the key interaction in such a structure. Interface parameters, evaluated by scanning transmission electron microscopy, are used to study their influence on the resistance of the device.

In order to overcome the limiting factor of Quantum Well Infrared Photodetectors (QWIPs), a high dark current, photovoltaic detectors inspired from Quantum Cascade Laser (QCL) and named Quantum Cascade Detectors (QCD) have been designed 1,2, [START_REF] Hofstetter | [END_REF]4,5,6 . They consist in several periods composed of different quantum wells. As in QWIPs, this device is unipolar and the infrared absorption occurs between two subbands. The separation in energy between the subbands corresponds to the detection wavelength.

Unlike the QWIP, where the excited electron is extracted from the top subband by an external electric field, in QCDs, it escapes by resonant tunneling to the adjacent quantum well. Then, the electron cascades from subband to subband down to the next period. Such detectors were initially designed in 3-5µm and 8-12µm wavelength ranges. In this case, the involved energy differences between subbands and the operating temperatures are high enough for efficient longitudinal optical (LO) phonon emission and absorption. A previous study has determined LO phonons as the dominant scattering process in the transport without illumination. 6 However, this argument fails in Terahertz (THz) range as lower energies are involved. In this range, the dominant scattering process has still to be identified.

In this letter, first, we highlight the dominant interaction in a THz QCD, interface roughness (IR) scattering. Then, we use scanning transmission electron microscopy (STEM) measurements to extract the key parameters of IR:  the magnitude of the roughness and ξ the mean distance between defects. Only few studies have been carried out to design a THz QCD, since only one other device has been presented. 7 The QCD under study is a GaAs-Al0.27Ga0.73As heterostructure. It consists in forty periods of 5 coupled quantum wells. The layer sequence in Ǻ, starting from the first barrier, is as follows: 75/52/30/48/50/44/60/50/68/54. Barriers are in bold. The first two wells of a period are doped with Si donors (310 17 cm -3 ) in the central third of the well. The photonic transitions E5-E1 and E4-E1 are expected to lead to a maximum absorption at 70µm. This device, designed and grown by the University Paris Diderot and the III-V Lab, will be presented and characterized in a future publication.

In the THz range, the dominant interaction is a priori difficult to identify. For this purpose, we used a hopping transport code 8 between two dimensional states based on the Fermi Golden Rule. Wave functions were evaluated using a two band kp model. All the following scattering processes are taken into account in our simulation: interaction between electron and LO phonon (LO), acoustical phonon (AC), alloy disorder (AL), interface roughness (IR), ionized impurities (II) and electrons (EE). We have evaluated the different relaxation rates in the cascade for each of the six processes. Calculations have been made for an electron hopping from the bottom of the initial subband (Ki = 0), at a temperature of 10K. The interaction parameters are given in ref 8. Table I presents the intracascade scattering rates (1/ij), and between two different cascades (1/i'j). The numbering system of the levels is the same as in ref 6. Interface roughness is identified as the main interaction, the corresponding ratse 1/τij are at least one order of magnitude above the others. LO phonon interaction in 3-12µm QCD, IR can be considered as the interaction which has to be taken into account when evaluating the transport properties in THz QCDs. Since these scattering rates have been appraised, the net scattering rates Gij between two levels can be deduced. The Gij are taking into account both population of each level and scattering rates. Gij are given by:

i f i ij f FD i FD ij dE E E E f E f G ) ( 1 . )) ( 1 )( ( 0 - - =   
with fFD the Fermi Dirac distribution. We assume an electronic population at the equilibrium, since the intrasubband relaxation is faster than the intersubband one 9,10 . tab. I: Scattering rates (Hz) for different processes at Ki=0, under an arbitrary low electric field and at T=10K. The numbering of the level allocates the number five to the highest level in energy and the number one to the lowest level of a cascade. The prime sign is used for the levels in the neighbouring cascade. 2.610 -8 4.310 -8 5.610 -8 2.810 -8 6.110 -10 1.310 -9 9.110 -8 2.110 -7 The Gij are used for the calculation of R0A (where R0 is the resistance per period of the pixel and A the area of the pixel), via the following relation 6

Process 1/τ54 1/τ43 1/τ32 1/τ21 1/τ53 1/τ42 1/τ31 1/τ5'1 1/τ4'
    = C i C j ij b G e T k A R ' 2 0
, where C designates one cascade and C' the following one. Figure 1 represents R0A as a function of 1000/T for the different scattering processes.

Electron-electron scattering calculation has not been performed due to incertainties on position in energy of the different subbands and a high consuming calculation time. As predicted by the scattering rates, the limiting process is the IR. It leads to a low value of R0A:10 -5 Ω•cm 2 per period, whereas around 10 Ω•cm 2 per period has already been reported in THz QCD 7 and 10 2 Ω•cm 2 per period are often reported for 10 μm QCDs. Such a difference results from three combined effects. First the doping level of our structure is thirty times higher than the previously reported doping 7 , which also reduces the R0A of two to five decades (depending on the temperature), but in compensation increases the absorption of the device. Add to this the injection of the electron on the upper levels of the cascade is not so good due to an unoptimized design of the associated wave function concerning the transport. This results from a bad positioning of the fourth level of the cascade which tends to send back the electrons to the lowest level rather than into the cascade. Last point, the interfaces are not as sharp as expected, see FIG. 2, which also reduces the R0A as explained below. A way to improve the R0A value may consist in applying a higher electric field on the QCD, thus the transition i->i' would lose its elastic behaviour and the transport via interface roughness interaction would become less efficient. The increase of the R0A factor at high temperature (low 1000/T) may be explained by possible transitions between levels with higher energies in the cascade. For a moderate temperature, the R0A presents a low dependence on lattice temperature as the initial Fermi factor can be considered as constant. At low temperatures, R0A increases, it may be attributed to the population filling of the final subband. Now, we only focus on the interface roughness interaction, since it prevails on the others at least from one order of magnitude. The evaluation of the net scattering rate allows us to conclude that: (i) The relaxation inside a cascade is done step-by-step. This is a major difference from the 10µm structure where LO phonon scattering allows the electron to jump towards different subbands. Here, since the coupling is driven by an elastic process, the first neighbour jump prevails. (ii) Once an electron is promoted over the excited state 5, it tends to go back to the previous cascade (G1'5=810 16 m -2 s -1 ~G5'4'=3.610 17 m -2 s -1 ). We can thus assume a drop of the Fermi level between two cascades.

In order to evaluate the parameter  (roughness magnitude) and ξ (mean distance between defects) most of the authors used indirect ways such as mobility measurements 11,12 or measurements under magnetic field 13 . We choose an alternative way by using STEM imaging, giving direct access to  and ξ.

A different sample is used for this experiment. It is described in ref. 5 and corresponds to a 9µm QCD. As mentioned above, in such a device, dark transport is not dominated by IR. Despite the absorption wavelength, the 9 µm QCD has an aluminium proportion in the barriers close to the THz one, 34% vs 27%, respectively. Since this parameter rules the IR scattering, in a first approximation, we can assume roughness to be similar in both QCDs. Figure 2 were obtained with a half-angle probe of 30 mrad, the inner and outter half-angles of the annular detector (called upper-HAADF detector in this machine) were, respectively, 100 mrad and 170mrad. A cross section along the growth direction (FIG. 2 (b)) presents the shape of the interfaces, they are not rigorously abrupt. We observed a gradient in the alloy concentration. The gradient of alloy concentration is as large as two or three monolayers (ML). FIG. 2 (c) presents a cross section along a well interface. We observe an irregularity at the interface between GaAs and AlGaAs. The averaged value of the correlation length is equal to ξ=10±3nm period. The magnitude of the irregularity has the same order of magnitude as the interface alloy gradient, i.e. two or thee ML: Δ=0.6 to 0.9nm. Those values are in a good agreement with those found in the literature 8 ξ=6.5nm and Δ=0.3nm for typical GaAs heterostructure. Those measured parameters are corroborating the dominant role played by IR a posteriori.

Finally we studied the influence of ξ and Δ on the R0A factor, for typical values around the experimental ones. As expected, an increase of the magnitude of the IR leads to a lower R0A. An increase of the roughness magnitude leads to a shift of R0A curve with a quadratic dependence in Δ. The dependence of R0A upon the ξ parameter is not obvious since IR acts as a filter in momentum space for the electron. FIG. 3 presents the typical behaviour of R0A as a function of 1000/T for different ξ values.

An increase of ξ leads to a decrease of R0A.

To summarize, IR is identified as the main scattering process in a THz QCD. This interaction prevails over all the others at least from one order of magnitude. We used STEM imaging system to estimate the IR characteristic parameters and we finally studied how those growth-dependant parameters influence the performances of the QCD. Decreasing the number of wells leads to a reduced number of interfaces and as a consequence limits the elastic behaviour of each transition, and thus should improve the structure performances.

The authors want to thank N. Péré-Laperne for his helpful suggestions and careful examination of this paper.

FIG. 1 R0A per period as a function of 1000/T for the different processes. The LO phonon process is not indicated on this graph since the associated R0A is much larger.

  (a) presents the STEM image of the 9 µm QCD. It was acquired at 200 keV with a JEOL JEM 2200FS scanning transmission electron microscope equipped with a CEOS aberration-corrector. HAADF-STEM images (also called high-resolution Z contrast images)

FIG. 2

 2 FIG. 2 (a) High resolution Z-contrast image of the 9 µm QCD structure prepared by <110>. Corrected

FIG. 3

 3 FIG.3R0A per period as a function of 1000/T, for Δ =2ML and different values of ξ, for the THz

12 3.410 11 4.510 11 1.310 11 4.110 11 7.210 9 1.410 10 9.910 11 2.410 12

	LO	310 -7
									1
	AL	9.510 10 8.310 9	110 10	3.310 9 1.210 10 2.110 8	3.510 8	310 10	6.310 10
	IR 4.210 II 1.510 11 6.310 9 7.510 10 4.610 10 1.710 9	2.510 8	3.110 9	3.110 7	2.310 8
	EE	1.4 10 9 2.4 10 8 5.3 10 8 5.1 10 8 2.4 10 8 1.3 10 6 4.6 10 6 1.3 10 8 8.3 10 8
	AC	5.710 9	4.710 8	7.110 8	610 7	5.510 8 1.2310 7 2.310 7	2.110 9	4.810 9

M.A. Reed and G. Frazier, Optically pumped quantum coupled device, US patent 4878104 (1989).


E Lhuillier

I Ribet-Mohamed

M Tauvy

V Berger

E Rosencher

Ultimate performance of Quantum Well Infrared Photodetectors in the tunneling regime

Keywords: 73.63.Hs, 85.60.Bt, 85.60.Gz Infrared photodetector, Quantum wells, Electronic transport. 1

Thanks to their wavelength diversity and to their excellent uniformity, Quantum Well Infrared Photodetectors (QWIP) emerge as potential candidates for astronomical or defense applications in the very long wavelength infrared (VLWIR) spectral domain. However, these applications deal with very low backgrounds and are very stringent on dark current requirements. In this paper, we present the full electro-optical characterization of a 15µm QWIP, with emphasis on the dark current measurements. Data exhibit striking features, such as a plateau regime in the I(V) curves at low temperature (4 to 25 K). We show that present theories fail to describe this phenomenon and establish the need for a fully microscopic approach.

Introduction

Quantum well infrared photodetectors (QWIP) provide nowadays a well established technology to realize infrared focal plane arrays in the 8-12µm spectral range. Indeed, these detectors rely on the use of GaAs materials, and thus benefit from the maturity of the III-V molecular beam epitaxy (MBE) growth and processing techniques. Focal plane arrays as large as 1024x1024 pixels are now available [1,2], with an excellent uniformity and a very low defect density. In addition, the excellent GaAs material quality results in the absence of 1/f noise [3], making dynamic two point correction unnecessary. Moreover, the peak wavelength of QWIP can easily be tuned thanks to band gap engineering, that is by changing the wells width and the aluminium concentration in the barriers. This flexibility allows QWIP to address the very long wavelength infrared (VLWIR) domain, i. e. wavelengths higher than 12µm [3,4].

Contrary to the well known infrared atmospheric windows (3-5µm and 8-12µm) which are generally used for ground-based applications, the VLWIR domain can be very useful for space-based applications, either in the astronomy [3] or in the defense domain. These applications call for very specific needs in terms of infrared focal plane arrays. For example, as far as space surveillance and space situational awareness are concerned [3,[START_REF] Singh | Proc. SPIE[END_REF], most of the information on target relies on radiometry (and not on imagery). As a consequence, excellent uniformity and linearity are required; the needs in terms of pixel operability are also important, since degraded or dead pixels could miss a target. With their excellent uniformity and high pixels operability [START_REF] Costard | Proc. SPIE[END_REF], QWIP could emerge as potential candidates for these applications. But space surveillance and space situational awareness often require the observation and the tracking of very faint objects against very dark backgrounds. As a consequence, the internal detector noise must be extremely low, in order to remain lower than the background equivalent noise. This places very stringent requirements on the dark current.

During the past decade, much work has been devoted to reducing the dark current of 8-9µm QWIP, and thus to increase their operating temperature up to 75K [START_REF] Costard | [END_REF]. But the issue of VLWIR detectors is totally different: indeed, in the applications considered here, a very low operating temperature is generally not an operational problem, so that the detectors are no longer operated in the thermionic regime, but in the tunnel regime. Paradoxically, apart from the early work of Levine et al in strongly tunnelcoupled QWIPs [8], this regime, which determines the ultimate performance of QWIP, has never been studied in detail, mainly because of both theoretical and experimental difficulties.

In this paper, we report the full electrooptical characterisation of a 15µm QWIP realised by Alcatel-Thales III-V Lab. The components are described in section 2, and the experimental setup in section 3. The dark current measurements are presented in section 4, while section 5 gives complementary measurements, namely spectral response, responsivity and noise. Section 6 compares the dark current measurements with present theories. Section 7 studies the possibility to attribute the current plateau to electric field domain formation.

Samples details

The QWIP detectors studied here were designed to have a peak wavelength at 14.5µm. They rely on a forty periods structure, with 7.3nm (Lw) GaAs wells and 35nm (Lb) Al15.2Ga84.8As barriers. The wells are n type Si doped in their central third with a concentration of 3.10 11 cm -2 . The structure is sandwiched between two n type contacts with a Si concentration of 10 18 cm -3 . QWIP detectors were processed into mesa. The sample is a matrix of 384*288 pixels of 25µm pitch (pixel size 23.5µm) in which some pixels are addressed individually. This specific design guarantees that the results obtained on these pixels will be representative of the performance of a focal plane array. The electromagnetic coupling is realized by a grating etched on top of the pixel, with a period of 4.4µm, i.e. very close to the peak wavelength in the material. The potential wells are thus 127meV deep, the ground state (energy: E1) is located 40meV above the bottom of the GaAs conduction band. The Fermi level (energy: Efw) is 10.6meV above the ground state. The excited state (energy: E2) is quasi resonant with the top of the barrier (energy Vb) since it is only 2meV below the continuum.

Experimental Setup

All the measurements were done with the QWIP detectors on the cold finger of a Janis continuous flow helium cryostat. A Lakeshore 330 controller was used to reach the desired temperature. To perform measurements in the tunnel regime, it is necessary to cool down the detectors at any desired temperature between 4K and 60K, with i) an excellent stability in time, ii) a very good reproducibility, iii) an accurate measurement of the detectors temperature and iv) a very low noise level. For this purpose, a dedicated cold shield was designed, and three DT 470 temperature probes were implemented: on the cold finger, on the sample holder and on the sample itself, respectively. To measure very low currents, we used a 6430 Keithley sub-femto amperemeter, which allowed us to measure currents as low as a few tens of fA with this experimental setup. Particular care was also devoted to limit parasitic noise by using shielded wires and by getting rid of the surrounding 50Hz noise.

This experimental setup allowed us to perform detailed dark current measurements which will be presented in section 4. It was also used to perform complementary measurements, namely: spectral response, responsivity and noise measurements.

The spectral response measurements were carried out with a Fourier Transform Infrared Spectrometer or FTIR (Bruker, Equinox IFS55). The transmission of the cryostat window was taken into account. The spectral response measurements are presented in section 5.

Responsivity measurements were realized by means of a CI SR 80 blackbody, with a 33° field of view.

To complete these experiments, noise measurements were performed. Because of the wide frequency range (1 to 10 3 Hz), a particular care is required to prevent electromechanical parasitic noise. Thus we used a mechanical decoupling system between the ground and our cryostat. Specific doubly shielded wires were also required. The noise signal was sent to a Femto-DLPCA 200 transimpedance amplifier with a tunable gain from 10 2 to 10 9 . The data was acquired using a spectrum analyser (Ono Sokki -CF5220Z).

Dark current measurements

Fig. 1 presents the dark current measurements as a function of the operating temperature and for bias voltage between 0.5V and 2.5V. The curve exhibits two regimes: for T>35K, the dark current is strongly dependent on the temperature, which is consistent with the expected thermionic regime; for T<25K, a plateau appears: decreasing the temperature further has no effect on the dark current. The absence of dependence of the dark current on the temperature is the signature of the tunnel regime. A plot of the current as a function of 1000/T allows one to extract the activation energy (Ea)) in the range of T>35K. For our QWIP, the value of 81meV was extrapolated for the activation energy under null bias. This experimental value is consistently The same dark current measurements were also plotted as a function of the applied bias for a temperature in the 4K-60K range (see Fig. 2). These I(V) curves exhibit clearly two different regimes: at high temperature (T>25K), a strong dependence on the temperature is observed; whereas under 25K all I(V) curves are superimposed, which is the evidence that the tunnel regime has been reached. It corresponds to the plateaus in the current-temperature curves of Fig. 1. The low temperature I(V) curves can be split into three regimes: first an ohmic regime (V<0.5V) with a linear increase of the current with the bias. For higher bias (0.5V<V<1.5V) the I(V) curves present a plateau behavior, where the dependence with the bias voltage is surprisingly low. Such a low dependence on bias voltage has already been observed on photocurrent I(V) curves [9] (they were attributed to electric field domain formation under illumination). It also has been observed on dark current, see ref [START_REF] Singh | Proc. SPIE[END_REF], [10] and [11], but to the best of our knowledge, such a plateau behaviour has never been deeply investigated. We underline that the coupling in this QWIP (miniband width of a few ten nanoeV) is particularly weak and we do not expect to be on the same regime of transport as much coupled superlattice [10]. It is also interesting to underline that neither hysteresis, nor saw tooth pattern have been observed in those I(V) curves. The plateau current value stays several orders of magnitude above the setup current resolution, which allows us to study this fine effect. For bias voltage higher than 1.5V, a rapid increase with the bias is observed. This rise, which necessarily involves a strong electric field on the first barrier in the vicinity of the contact [12], can be further attributed in the center of the structure to current from ground state to continuum. Indeed for a bias voltage of 1.5V, the potential drop over one period becomes equivalent to the ground-toexcited states transition energy (~80meV). Thus, the resonant transport from the ground state of a well to the excited state of the next well may also occur. This resonance may be broadened by the resonance with other states in the continuum.

included between meV E V b 86 1 = - (
The I(V) curves in Fig. 2 also exhibit an asymmetry with the polarity of the bias. The ratio I+/I-between the currents at +1V and -1V bias voltages is typically of 1.5. This asymmetry was attributed to the doping segregation [13]. The possibility of an asymmetric aluminium profile, due for example to an aluminium overshoot while opening the Al cell, could also be considered. Indeed, the phenomenon is not so simple, since for low temperature the ratio I+/Iis higher than unity at low bias, while it becomes lower than unity at high bias.

Spectral response, responsivity and noise measurements

Spectral response measurements are reported in Fig. 3. As expected, the peak wavelength is typically 14.5µm. The high wavelength cut-off presents a dependence on the bias voltage, which is attributed to photon and electric field-assisted tunneling, see inset of Fig.

3 [14]. This point will be further discussed in Section 7.

Responsivity measurements allowed us to determine the external quantum efficiency (product of the internal quantum efficiency by the photoconductive gain): 5% under 1V (responsivity of 0.58A.W -1 ) and 18% under 2V of bias (responsivity of 2.1A.W -1 ) for a temperature of 20K and a wavelength of 14.5µm. Such responsivities values are quite good for QWIP [START_REF] Schneider | Quantum well infrared photodetectors -Physics and applications[END_REF]. QWIP optimisation always requires a trade-off between dark current and responsivity. Since 15µm QWIP are operated in very specific conditions (low operating temperature, low incident power) this trade-off necessarily leads to a specific optimisation concerning the doping. For this purpose, a full understanding of electronic transport in these specific structures is necessary, as will be shown later.

Finally, we measured the noise spectral density, under dark condition. We were able to measure a current noise density as low as Hz fA/ 3 , for T under 20K, in a range of frequency (400-800Hz) where the noise spectral density presents a plateau. To evaluate the detectivity, we only considered the noise due to the dark current since we are considering low flux and thus low photocurrents. We thus obtain a detectivity, under dark condition, as high as 5.2 10 12 Jones at 20K (3.10 10 Jones at 50K) and under 2V bias voltage.

6.

Comparison of dark current measurements with theory

In this section we compare our dark current measurements with present theories. We start by summarizing the main expressions that can be found in the literature, together with their physical background, and then compare their theoretical predictions with our experimental results.

A. High temperature regime modelling

Different expressions of the thermionic current (Jth) can be found in the literature. The expressions (1), ( 2) and ( 3) are respectively due to Levine [8], Schneider and Liu [START_REF] Schneider | Quantum well infrared photodetectors -Physics and applications[END_REF] and Kane [START_REF] Kane | Intersubband transitions in quantum wells[END_REF]. (2)

T k E E eFL V b d Kane th b ex fw w b e T k m ev J + - - -       = 2 2 / 3 2 2 * 2   (3)
where e is the elementary charge, m* the effective mass of GaAs,  the reduced Planck constant, the scattering time from ground state to continuum, Ld the period width and Eex the exchange energy [START_REF] Bauer | [END_REF]. The value of Eex~-13.6mev is obtained following the approximate expression from ref [18]:

) 32 . 0 1 ( 4 2   w F F ex L k k e E - -  (4)
with F k the Fermi wave vector and  the GaAs permitivity.

These current expressions all rely on an Arrhenius law can either be seen as a population factor near the continuum energy or as a probability for an electron on the ground state to be scattered to the continuum. However, these expressions are quite different since both the expression of the activation energy and the factors before the exponential are different. Considering the activation energy, a bias dependence is introduced via a factor w eFL . Such a dependence reflects the effective barrier lowering due to the electric field, assuming that no Stark shift will affect the ground state energy. The 2 w eFL dependence, preferred in equations ( 2) and ( 3) is equivalent to considering that the electrons are globally in the center of the well. The drift velocity also includes a bias dependence:

2 1         + = sat d v F F v  
with  the mobility and sat v the saturation velocity. Both µ and vsat are adjusting parameters. Typical values can be found in the literature, for example in ref. [19]: µ=0.1 m 2 .s -1 .V -1 and vsat=5.10 4 m.s -1 .

Fig. 4 shows the comparison of the different current expressions with the experimental current. Liu and Kane expressions give very similar results whereas their prefactor expressions are quite different. For our QWIP the best agreement with the experimental current is obtained by Levine expression. In particular the extra dependence with the electric field in the prefactor leads to a better agreement. Moreover the high number of adjusting parameters in ( 2) and (3) makes these equations less trivial to use for any structure. Therefore, we will only consider the first expression from now on. The following physical picture can be used to interpret Eq.( 1): the dark current is the product of the elementary charge by a three dimensional (3D) carrier density and by a velocity. The 3D carrier density is built from a two dimensional (2D) density of states divided by the well width. Finally, this 3D density of states is multiplied by the typical energy of the problem b eFL to give a 3D carrier concentration.

B. Low temperature regime modelling

The low temperature current is mostly attributed to sequential tunneling. In this regime, electrons tunnel through the barrier. The modelling of this tunneling requires the use of a transmission coefficient often evaluated using the Wentzel-Kramers-Brillouin (WKB) approximation. Akin to the high temperature regime, several expressions of the current can be found in the literature. Equation ( 5) was derived by Levine [8], (6) by Gomez [20] et al, and ( 7) by Schneider and Liu [START_REF] Schneider | Quantum well infrared photodetectors -Physics and applications[END_REF].

            + +  + = - - - T k E eFL E T k E E w w b Levine ST b b fw b fw e e eFL E P L T ek J 1 1 1 1 ln ) ( . 1 2  (5)             + +  = - - - T k E eFL E T k E E w b Gomez ST b b fw b fw e e E P L T ek J 1 1 1 1 ln ) ( . 2 1 2  (6)    - -  - - = b eFL E FD E FD iu SchneiderL ST dE E f E m e dE E f E m e J 1 1 2 1 1 2 ) ( ) ( * ) ( ) ( *       (7)
Here P is the WKB probability for an electron to cross the trapezoidal part of the barrier and is given by

( )  ( ) ) . 3 * 2 4 exp ) ( 2 / 3 2 / 3 E eFL Vb E Vb eF m E P b b - - -     - - =  (8)
Expressions ( 5) and ( 6) in fact result from simplifications of equation [START_REF] Costard | [END_REF]. In this last expression the current is the product of the elementary charge by the 2D density of states, and by a time constant associated to the electron tunnelling through the barrier. This quantity is integrated over all initial states and weighted by the Fermi Dirac (fFD) population factor of each state. The current expression is the sum of two terms: the first one represents the current which goes down the structure. The second term is the current which flows back from the lower energy quantum well. The tunneling time constants can be written with a "ping pong" approach [21,22,[START_REF] Eyvind | Berkeley physics courses: Quantum physics[END_REF], where the electron goes from one side of the well to the other in a time equal to two well widths (one well width in Levine et al. [8] expression) divided by the electron velocity.

Each time the electron reaches a side of the well, it gets a probability P to tunnel through it. Thus the tunneling time constant is given by

1 1 2 - = P v L w 
, with a factor two if we consider that only one side of the well contributes to the current. 1 v is then linked to the energy by

2 1 * . 2 / 1 v m E = .
A key approximation to obtain the expressions ( 5) and ( 6) is to consider that the tunneling time constant is independent of the energy, in order to extract the tunnelling rate of the integral. Such a hypothesis is a better approximation for low doped QWIP for which the electronic population has a limited extent in energy. Expressions ( 5) and ( 6) mostly differ by the value at which the probability to cross the barrier is evaluated. Levine also increases the field dependence by introducing a factor 6) and ( 7) give very similar shape, and they only differ by a factor two (due to the fact that [START_REF] Costard | [END_REF] includes the energy dependence of the tunnelling probability). As expected (5) exhibits a higher bias dependence due to the factor w eFL . It is difficult to determine which expression gives the better agreement with the experiment, since none of them reproduces correctly the plateau behavior. In the following we have chosen to use the expression [START_REF] Costard | [END_REF] rather than the simplified ones, particularly because our doping is not so low.

Finally the global theoretical current can then be written as the sum of the thermionic (Jth expression (1) ) and the sequential tunneling (Jstexpression (7) ) contributions. A comparison of the experimental and theoretical I(V) curves is shown in Fig. 6(a) (high temperature) and Fig. 6(b) (low temperature).

The high temperature modelling presents quite a good agreement with the experimental dark current, particularly at low bias voltage (V<2V).

The disagreement at high bias voltage (V>2V) is due to the change of transport mechanism, not taken into account in our current expression. At low temperature, because of the superposition of experimental I(V) curves, we only have plotted the case for T=10K, see Fig. 6(b). The agreement between experimental and theoretical current is low. Indeed, neither the order of magnitude nor the shape are correctly reproduced.

7.

Is the plateau regime a consequence of electric field domain formation?

The plateau regime in the I(V) curves clearly corresponds to a situation where a change in the bias voltage has no effect on the current flowing through the structure. A possible explanation of this phenomenon could be the following: since the current is controlled by the electric field reigning at the contact [START_REF] Rosencher | [END_REF], this surface electric field could stay unchanged when the bias voltage increases, while the potential drop is localized in one or a few periods of the structure. This would result in high electric field domain formation either in the vicinity of the contact [25] or in the bulk of the structure [9]. However, when electric field domains are formed, the I(V) curve presents saw tooth patterns or equivalently peaks in the conductance curve [10,26]. This is not the case in our sample. Moreover the spectral response measurements reveal that the high wavelength cut-off increases with the bias voltage, which gives strong indication that the potential drop is not concentrated only on a few periods, but distributed over the whole structure [21]. We underline the fact that this distribution is not necessarily homogeneous.

Conclusion

The tunneling regime undoubtedly gives access to the ultimate performances of QWIP detectors in terms of dark current. Because the dark current is a crucial issue for VLWIR low background applications, we have studied its behaviour with great attention in 15 µm-peaking devices. We have evidenced a striking feature in I(V) curves at low temperature, i.e. a plateau regime. Let us note that this plateau behaviour is not sample dependent and is observed in samples presenting different technological parameters.

Simple theoretical expressions allowed us to reproduce I(V) curves at high temperature with a good agreement. However, we have shown that these simple expressions are not able to describe the I(V) curves at low temperature. In particular, the plateau structure in the I(V) curves cannot be explained with the WKB approximation. Spectral response measurements allowed us to show that this plateau could not be attributed to high electric field domain formation either.

In other words, the available theoretical tools don't allow us to understand the physical phenomena involved in dark current in the tunnel regime. We believe that this theoretical problem could be solved thanks to a microscopic model using a scattering approach. Thus, all the relevant interactions could be taken into account. Such a microscopic modelling of the dark current is absolutely necessary to identify the major contribution to dark current, and to finally improve the design of QWIP structure.

Figure captions

  Fig.1presents the dark current measurements as a function of the operating temperature and for bias voltage between 0.5V and 2.5V. The curve exhibits two regimes: for T>35K, the dark current is strongly dependent on the temperature, which is consistent with the expected thermionic regime; for T<25K, a plateau appears: decreasing the temperature further has no effect on the dark current. The absence of dependence of the dark current on the temperature is the signature of the tunnel regime. A plot of the current as a function of 1000/T allows one to extract the activation energy (Ea)) in the range of T>35K. For our QWIP, the value of 81meV was extrapolated for the activation energy under null bias. This experimental value is consistently included between meV E V b 86 1 = -(corresponding to the transition from the lower populated state to the continuum) and meV E V fw b 76 = -(corresponding to the transition from the upper populated state to the continuum).

  w eFL . Such an extra dependence is not in favour of the plateau prediction. Then the integration of the Fermi-Dirac factor and the use of log property leads

Fig. 5 .

 5 Fig.5. presents the current associated with the different tunnelling current expressions. Expressions (6) and (7) give very similar shape, and they only differ by a factor two (due to the fact that[START_REF] Costard | [END_REF] includes the energy dependence of the tunnelling probability). As expected (5) exhibits a higher bias dependence due to the factor

Fig. 1 .

 1 Fig.1. Dark current density as a function of the temperature for bias voltage values ranging from 0.5V to 2.5V by steps of 0.5V. Inset: activation energy as a function of the bias (dots) and linear fit (solid curve).

Fig. 2 .

 2 Fig. 2. Dark current as a function of the applied bias for different temperatures.

Fig. 3 .

 3 Fig. 3. Normalized spectral response as a function of the wavelength for different values of the bias. Inset : explanation of the tunnel effect assisted by photon and electric field.

Fig. 4 .

 4 Fig. 4. Comparison of the different current expressions as a function of the bias at 50K. Following [15] the parameter

Fig. 5 .

 5 Fig. 5. Comparison of the different expressions for the tunneling current as a function of the bias, at T=10K.

Fig. 6 .

 6 Fig. 6. (a) Theoretical dark current density (solid line) and experimental (scatter line) as a function of the applied bias for temperatures from 30K to 60K by step of 10K. (b) Theoretical (solid line) and experimental (scatter line) dark current density as a function of the applied bias at 10K.