E Lhuillier

I Ribet-Mohamed

A Nedelcu

V Berger

E Rosencher

Quantum transport in weakly coupled superlattices at low temperature

Keywords: number(s): 73.63.Hs, 72.10.-d, 85.60.Gz I

We report on the study of the electrical current flowing through weakly coupled superlattice (SL) structures under an applied electric field and at very low temperature, i.e. in the tunneling regime. This low temperature transport is characterized by an extremely low tunneling probability between adjacent wells. Experimentally, I(V) curves at low temperature display a striking feature, i.e a plateau or null differential conductance. A theoretical model based on the evaluation of the scattering rates is developed in order to understand this behavior, exploring the different scattering mechanisms in AlGaAs alloys. The dominant interaction in our typical operating conditions is found to be the electron-ionized donors scattering. The existence of the plateau in the I(V) characteristics is physically explained by a competition between the electric field localization of the Wannier-Stark electron states in the weakly coupled quantum wells and the electric field assisted tunneling between adjacent wells. The influence of the doping concentration and profile as well as the presence of impurities inside the barrier are discussed.

INTRODUCTION

Electronic transport in superlattices (SL) has been extensively studied since the early work of Esaki and Tsu 1 . However most studies deal with strongly coupled structures in order to observe high field domain formation 2 or coherence effects such as Bloch oscillations 3 . In this paper we focus on the transport in very weakly coupled SL at low temperature. Only little work has been devoted to the microscopic understanding of this type of tunnel transport 4 , despite the observation of new phenomena such as phase transitions 5 and the fact that Quantum Well Infrared Photodetectors (QWIPs) operate in the low coupling regime at low temperature. Most of the existing models are based on the Wentzel-Kramers-Brillouin (WKB) approach. However, as we will show, this model fails to explain the experimental results. We thus developed a microscopic model of transport at low temperature for very weakly coupled SL, based on a scattering approach. This study is of a particularly large scope: indeed, the maturity of GaAs-based materials (low number of defect levels) and the unipolar character of QWIPs (no passivation needed) eliminate unwanted parasitic material effects, and thus only fundamental microscopic interactions are involved in the transport. Our model takes into account six interactions: electron-optical phonon, electronacoustical phonon, alloy disorder, interface roughness, ionized impurities and carrier-carrier interactions. Due to the very narrow ground miniband we expect that coherent transport 6 and second order effects 7 (two successive tunneling processes via a (virtual) state) stay moderate. As a consequence we investigate hopping transport [START_REF] Pollack | Hopping transport in solids[END_REF] between ground subbands of adjacent wells [START_REF] Tsu | [END_REF] . Several papers already addressed this regime 10,11 , but generally the coupling between wells investigated by the authors is far larger than ours and their model fails to explain our experimental data. Our model provides a full quantum description of current transport in weakly coupled SLs, validated by experiments.

In this paper we first present (sec. II) sample measurements (I(V) curves and spectral response). The I(V) curves at low temperature exhibit in particular a striking null differential conductance (i.e. plateau) behavior. In sec. III, the usual WKB approximation is shown to fail in reproducing this striking feature. Our model, based on the calculation of different scattering rates, is developed in sec. IV. Section V presents the results of our model concerning the scattering rates and the resulting current as a function of the electric field. The dominant interaction in our experimental conditions is found to be the electron-ionized impurity scattering. The existence of the plateau in the I(V) characteristics is explained by a competition between the electric field localization of the Wannier-Stark electron states in the weakly coupled quantum wells and the electric field assisted tunneling. Finnaly, the influence of both the doping profile and the presence of defects in the barrier is presented in Section VI.

II. EXPERIMENTS A. Structure

The experiments have been done on a QWIP structure 12 composed of forty periods with a 73Å wide GaAs well and a 350Å wide Al 15.2 Ga 84. 8 As barrier. The central third of the well is silicon doped with a concentration of n 2D =3 10 11 cm -2 . The structure is sandwiched between two n-type, silicon doped contacts ([Si]=10 18 cm -3 ). This QWIP is obtained by MBE growth, and then processed into mesas of 23.5µm lateral size. The barrier is 127meV high and the ground state is located approximately 40meV above the bottom of the GaAs conduction band. The doping value leads to a Fermi level 10.6meV above the ground state.

B. Measurements

The device was placed on the cold finger of a Janis helium cryostat. The temperature regulation was made with a 330 Lakeshore control unit. Currentvoltage measurements were carried out with a 6430 Keithley sub-femtoampere source meter. Special care was dedicated to the fine control of the sample temperature. FIG. 1 (a) presents dark current measurements, which displayed a good repeatability in time and between pixels. For T>25K, the current increases monotonously with the temperature. This regime has been extensively studied and is well understood, see for instance ref 13. Below 25K, however, the current is independent on the temperature, which is the sign of the tunneling regime. The low temperature I(V) curve displays three different parts:

• First, an ohmic regime (0V→0.5V) where the current increases linearly with the bias. • Second, a plateau regime (0.5V→1.5V) where the dependence of the current with the bias is surprisingly low. This plateau is attributed to the transport between the ground states of two adjacent wells. The main goal of this paper is to explain the very low dependence of the current with the bias in this plateau regime. • The high bias regime (V >1.5V) where the current increases very rapidly. This rise is the sign of a change in the transport mechanism.

The origin can be attributed to impact ionization [START_REF] Gendron | [END_REF] in the vicinity of the contact or to transport from the ground state to the continuum in the center of the structure. In the following we will not address this high bias part of the I(V) curve, since this transport mechanism has already been largely investigated in a previous paper 15 . One should notice that the I(V) curves present a slight asymmetry: we will address this effect in section V. Neither hysteresis, nor saw tooth pattern 16 have been observed in our I(V) curves.

The spectral response was measured by a Bruker Equinox 55 Fourier Transfrom InfraRed spectrometer (FTIR) in which the signal is amplified by a Femto -DLPCA 200 amplifier. The measurements are presented in FIG. 1 (b). The QWIP displays a spectral response peaked at 14.5µm, with a full width at half maximum (FWHM) of 2µm. The inset of FIG. 1 (b) shows the variations of the spectral response with the applied bias in the high wavelength part of the spectrum. This point will be further discussed in section III B.

III. WKB MODELLING

Tunnel transport in QWIP is generally described using the WKB approximation 5,16,17 , which relies on two assumptions: (i) The variation of the potential barrier is small compared to the electron wavelength. (ii) The tunneling probability from the final state is negligible. The WKB expression, which gives the tunneling probability of a particule of energy E through a potential barrier U(x) between points a and b, is given by the expression (50.9) from the Landau-Lifchitz book [START_REF] Landau | lifchitz in Mécanique quantique, théorie non relativiste[END_REF] : ( )

( )         = ∫ - b a dx x p D
( ) ( ) x U E m x p - = * 2
is the electron momentum . Such an approximation leads to the following expression for the current density:

∫ ∫ ∞ - - ∞ - - = b eFL E FD WKB E FD WKB WKB dE E f E m e dE E f E m e J 1 1 2 1 1 2 ) ( ) ( * ) ( ) ( * τ π τ π h h (1)
where e is the elementary charge, E 1 is the ground state energy,

1 ) exp( 1 ) ( -       - + = T k E E E
L F e E V E V F e m P - - - - × - = h (2)
is the WKB probability that the electron tunnels through the trapezoidal barrier. Here V b is the barrier height and m b * the effective mass in the barrier. FIG 2 shows a comparison between WKB prediction and the experimental results. Clearly, WKB approximation fails to reproduce the I(V) plateau. This result is consistent with the probability P in Eq.(2) being a strict monotonic function of the applied bias V. This discrepancy is however unexpected since WKB approximation generally yields a good agreement with experience for similar devices 5 . The reason of this discrepancy is the following. In Ref 18, it is clearly stated that WKB approximation is valid if one neglects the reflected wave from the final state (x>b) (see the discussion below expression 50.2 in ref 18). This can be easily explained in simple terms. In the WKB approximation, the potential barrier between points a and b is split into thin slices {x i , x i+1 } of thickness ∆. If one neglects the reflected waves in the potential barrier (i.e. condition ii), the particle wavefunction between x i and x i+1 is:

( )

x i i i p e A x h - = ψ so that the probability D i of the wave to reach x i+1 from x i is ∆ - = h i p 2 i e D
if the variation of p(x) is small over ∆ (condition i). The probability of the particule to tunnel through the potential barrier is thus:

( ) ∫ - ∑ ∆ - ∏ = = = b a 2 i i 2 dx x p i p i e e D D h h
As clearly stated in the reference 18, this is valid if the electron wave function is delocalized for x>b, but not if the wavefunction is localized for x<b, in which case the reflected wave from b to a is of course not negligible. Consequently, the WKB approximation cannot take into account the effect of scattering in the neighbouring wells, which is the main coupling mechanism for transporting the electrons from well to well in this hopping regime.

IV. SCATTERING APPROACH

We thus chose to develop a scattering approach of transport in multi Quantum Wells (MQW). Scattering methods have already been used to model the quantum transport in heterostructures for resonant tunnel diodes [START_REF] Chevoir | [END_REF] , MQW structures 20,21 and more recently in Quantum Cascade Lasers 22,23 (QCLs). But only little work has been devoted to applying this method to weakly coupled SLs [START_REF] Etteh | [END_REF] , mainly because of the difficulty to deal with the low coupling effects.

The high quality of the GaAs material, grown by Molecular Beam Epitaxy (MBE), allows us to evaluate a scattering rate and a current from microscopic Hamiltonians, since no uncontrolled or detrimental material defects (deep levels, hopping on defects,…) prevail. Our model includes the six main interactions observed in GaAs-based materials: optical phonon, acoustical phonon, alloy disorder, interface roughness, ionized impurities and interactions between carriers (see appendix A to F for details on the scattering rate evaluation). No a priori hypothesis is made concerning the magnitude of each process. However, we assume that the GaAs material grown by MBE is of high enough quality to disregard scattering due to dislocations. We also assume that no neutral impurities are involved in the transport mechanism.

It is important to understand that the tunnel transport between ground states is a very inefficient mechanism in the weakly coupled Quantum wells (QW) considered here. Indeed, we can assume that a MQW is a stack of doped planes with a typical doping of 3⋅10 11 cm -2 . Considering that the current density in the plateau regime is 10 -6 A.cm -2 , (see FIG. 1a) we can conclude that the typical scattering rate is given by: This means that an electron is scattered from one well to the next every ten milliseconds. This time should be compared to the intra-well scattering time which is less than one picosecond 20,25 within the conduction band, (ten orders of magnitude smaller). Consequently we are dealing with very unlikely events.

Our model is based on the evaluation of the inter-well scattering times 1 -Γ using the Fermi Golden Rule (FGR). The tunnel transport between ground states is rather simple to model since it only couples two dimensional (2D) levels. The time

1 - Γ is included in the current expression: , ) ( )) ( 1 ).( , ( . * 1 2 i i FD f FD E d f f F E em J ε ε ε π - Γ = ∫ ∞ h (4)
where 2 * h π m is the 2D density of states (DOS). The use of an equilibrium population factor is motivated by the fact that the inter-well scattering rate is several orders of magnitude lower than the intra-well rate, which leads to a thermalized subband for each well 26 . Our model includes the direct current (J+, the electron relaxes from the upper well to the lower one) and the reverse current 4 (J-, the electron flows up the structure), see FIG. 3. In this Wannier-stark approach, the current writes: The expression of the scattering rate given by the FGR is:

i i FD f FD eFL E i i FD f FD E d f f F E m e d f f F E m e J J J d ε ε ε π ε ε ε π ) ( )). ( 1 ).( , ( . * ) ( )). ( 1 ).( , ( . * - Γ - - Γ = = - = ∫ ∫ ∞ - ∞ - + h h (5)
∑ - = Γ Kf f i i i H f K ) ( 2 ) ( 2 ε ε δ π h (6)
In this expression i and f point out the initial and final states,

2 m K E i i i h + =
ε with E i the energy of the ground state. Non-parabolicity for the in-plane dispersion is neglected in the current calculation. Indeed the exchanged energies remain very low compared with the inverse of the coefficient of non-parabolicity of

GaAs ( * 2 ) 1 ( 2 2 m k E E h = + α , 1 61 . 0 - = eV α
) 29 . The wave functions and the ground state energies are evaluated for each value of the electric field, so that Stark effects are taken into account in our model. The electric field is denominated by F and the period of the superlattice is L d . The periodicity allows us to replace the

f i E E -quantity by . d eFL
In order to compare the theoretical J(F) curve with the experimental J(V), we assume that the electric field on the structure is homogeneous. It is well known that the electric field distribution leads, for a given bias, to a higher electric field in the vicinity of the contact than in the center of the structure 30,31 . Typically the difference between the homogeneous electric field and the "real" electric field is about a few ten % 32 . Nevertheless, the higher the number of periods, the lower the associated correction. Our structure contains forty periods and this "mean field" approach should be adequate. There are two main consequences to this homogeneous electric field hypothesis: first we neglect all contact effects 33 and then we assume that no electric field domain [START_REF] Schneider | Quantum well infrared photodetectors -Physics and applications[END_REF]34 exists in the QWIP. To justify the last point, we used the high wavelength part of the FTIR measurement (Fig. 1 (b)), (assuming that the electric field profile is the same with and without photon flux). In the high wavelength part of the spectrum, the photon energy is lower than the bound-to-extended state transition energy, so that the electron does not have enough energy to be excited directly into the continuum. In fact, the electron is rather subject to tunneling assisted by photon and electric field, through the triangular part of the barrier 35 . This tunneling probability depends on the electric field value. Thus, the translation of the photocurrent spectrum with the bias reflects the field reigning on each quantum well, which allows us to conclude that the bias is effectively applied on the QWIP.

Because of the large barrier involved in our structure, quantum wells are very weakly coupled and the miniband width is in the nano-eV range 15 , whereas the potential drop per period is some tens of meV. Electrons are thus highly localized and their wavefunctions are consistently described by their unperturbed quantum well wavefunctions. Consequently our approach is based on a hopping mechanism from one well to the next one. This is an important difference with the paper of Castellano 15 et al in which the I(V) plateau is attributed to a saturation of the electronic velocity in a very narrow miniband (Esaki-Tsu approach).

V. RESULTS

A. Parameters used for modeling A temperature of 10 K is used. The other parameters used for the evaluation of the scattering rates are given in the following table:

)) ( 1 ( ) , ( f FD f F E ε - × Γ
of the scattering rates, for a null initial wave vector (K i =0), by the population factor of the arrival level, as a function of the bias. One of the main results of this graph is the fact that at low field (V<2V) the dominant interaction is the one between the electrons and the ionized donors. In the plateau regime (0.5V<V<1.5V), this interaction is at least one order of magnitude higher than the others. At higher bias (V>2V), other interactions such as LO phonon and alloy disorder also become important. Concerning LO phonon, we have to underline that this effect will not happen in a higher wavelength QWIP because of smaller exchanged energy.

One should also notice that for very low bias (V<0.5V) the product of the scattering rate by the population factor is increasing, which is the consequence of the

)) ( 1 ( f FD f ε - factor.
Electronelectron interactions do not show this behavior because this factor was not included, due to the difficulty to evaluate the energy of the arrival states [START_REF] Harrison | Quantum wells, wires and dots: Theoretical and Computational Physics of Semiconductor Nanostructures[END_REF] .

The FIG. 4 shows that a plateau is theoretically obtained in the scattering rates vs bias, which will lead to a constant current in this range. One might wonder what the physical origin of this plateau is. In fact, the plateau regime results from the competition between two different effects of the electric field. On the one hand, as described above, an increase of the electric field tends to enhance the wave function in the neighbouring well, see FIG. 5 (c), enhancing the scattering and thus the electrical current. On the other hand, the increase of the electric field tends to localize the wave function in each well (Wannier-Stark effect 2 ), see FIG. 5 (b), leading to a decrease of the electrical current, i.e. a negative differential resistance (see FIG. 5).

C. Level broadening When the electric field is very low in the structure, the electron wavefunctions, which are unperturbed in our first order perturbation theory, tend to be degenerate and delocalized. This leads to an unrealistic infinite conductivity as it is well known in transport theory [START_REF] Seeger | Semiconductor Physics, an introduction[END_REF] . Rott et al. [START_REF] Rott | [END_REF] and Wacker 40 have already demonstrated that for Wannier-Stark hopping a 1/F n law is expected at low field, where the n value depends on the considered Hamiltonian. To correctly describe the ohmic regime it is of course necessary to take into account the decoherence effects on the transport mechanism. The question of the decoherence may be treated using a non equilibrium Green's function method, but this method is highly computationally demanding 41 . Other teams have also tried to include decoherence using the density matrix formalism, see the work of Iotti et al. 42 , Callebaut et al. 43 and more recently Gordon et al. 44 . In order to take this effect into account while keeping a simple first order calculation, the easiest way is to introduce a lifetime broadening [START_REF] Seeger | Semiconductor Physics, an introduction[END_REF] .

Following a Wannier-Stark approach 2 , the delocalized part of the wave functions magnitude is given by

        ∆ d eFL J 2 0 1
, with J 1 the first order Bessel function (see FIG. and has been obtained with the same scattering method. Such a value is consistent with previous theoretical 20 and experimental 25 results. This value is also very close to the broadening energy (50fs) extracted from our spectral measurements. The expression of the current is thus:

2 0 1 2 /             + ∆ ∝ τ h i eFL J J d (7)
Using the fact that eFL d >> 0 ∆ , and 2 ) ( FIG. 6 shows the predicted current as a function of the electric field for the six interactions. The theoretical curve taking into account the 6 mechanisms (full line) can be compared with the experimental curve (squares). Our model is able to reproduce the null differential conductance behavior. We can observe that on the plateau regime the agreement between theory and measurement is quite good, typically a factor three. Because of the quadratic dependence with the doping of the current due to ionized donors (see Appendix E) this factor three may result of an uncertainty of "only" 70% on the doping value. Other effects such as uncertainty on aluminum concentration or segregation of aluminum and silicon may also be involved in this difference.

VI. INFLUENCE OF THE DOPING DENSITY AND PROFILE

One of the main advantages of our microscopic approach is that we can describe the effects linked to the doping density and profile. Such effects are expected to be quite important since we have demonstrated that the electron-ionized donors interaction dominates the plateau regime. We underline that our model gives a quadratic dependence of the current with the doping density:

) (n n e J Γ p (8)
with an explicit sum over the number of electrons (equal to the doping) and one implicit sum included in the scattering rate over all the scattering centers. To experimentally validate this dependence of the current as a function of the doping magnitude we have grown two series of samples which only differ by the magnitude of the doping. The first structure is very close to the previous sample with a well (barrier) width of 7.2 nm (34 nm), the aluminium content in the barrier is 15% and a Si doping in the central part of the well. The sheet densities are repectively 1×10 11 cm -2 (component B 1 ) and 2×10 11 cm -2 (component B 2 ). The structure includes sixty periods. FIG. 7 presents the associated dark current. The ratio of the two plateau magnitudes is 4.6, whereas 4 was expected. Similar results have been obtained in a second structure (L w =8nm, L b =40nm, %Al=13%, forty periods and a doping of 2×10 11 cm -2 and 4×10 11 cm -2 ), where a ratio very close to 4 was effectively measured. There is a clear added value relatively to the Esaki Tsu-like 15 model which is independent of the doping. Even more, the current is not only sensitive to the doping value, but also to its position.

This last part will study effects such as segregation or the influence of impurities inside the barrier, since those two parameters are difficult to control precisely and may have significant effect on the current.

A. Doping position We have first theoretically studied the dependence of the current as a function of the position of these impurities. We have scanned the position of an ideal delta doping (1Ǻ) trough the well and plot the associated current in FIG. 8. The electron sheet density is kept constant through the scan. In our model, we expect the current to decrease while driving the doping layer away from the wavefunction maximum, since the scattering overlap integrals are strongly reduced. This behavior is clearly observed in FIG. 8. Let us note that, because of the applied electric field of 10kV.cm -1 , the maximum of the curve is not at the center of the well, but shifted of nearly 15 Ǻ in the direction of the electric field. Thus it may be possible to reduce the dark current by shifting the doping position from the center of the well to an other position. To confirm this prediction, we have grown two samples (L w =6.8nm, L b =39nm, %Al=15%, 40 periods, sheet density 3×10 11 cm -2 ). The doping is respectively in the central third of the well (component C 1 ) and on the last third (grating side) of the well (C 2 ).

We have plotted on FIG. 9, the magnitude of the dark current as a function of the temperature under a bias of -1.5V. Using X ray diffraction and spectral measurements we have measured that the structure C 2 presents a lower confinement (transition energies respectively of 88.3 meV for C1 and 87.4 meV for C2). Consequently at high temperature (T> 35 K), this sample presents a higher thermoionic dark current due to a more efficient thermal activation of the electron. At low temperature however, in spite of its lower confinement, this C2 structure displays a less important tunnel current, which results from the fact that the interwell hopping scattering rate has been effectively reduced, as expected fro our theory. B. Segregation and I(V) asymmetry Since our model takes into account the doping profile, it can also be used to predict the effect of doping segregation [START_REF] Carras | [END_REF] . The segregation length is a function of the growth temperature 47 and partial pressure of the different deposited elements. As a realistic approximation 31 , we assumed that our doping distribution is an asymmetric trapezoid. This trapezoid is composed of three zones, see the inset of FIG. 10.

The first zone of length L 1 represents the segregation in the direction opposite to the growth. This segregation is quite low and thus the segregation length (L 1 ) is taken equal to 5 Ǻ.

The second zone corresponds to the nominal place of doping.

The last zone corresponds to a segregation in the direction of the growth, and consequently shows a higher segregation length (L 2 ). For our typical growth temperature the segregation length is in the range 25-50Ǻ, as reported by Wasilewski et al 47 .

The volume doping density has been chosen such that, whatever the values of the segregation lengths, the sheet density remains unchanged. We observe that the segregation reduces the dark current (FIG. 10). This is explained by the fact that the segregation tends to move the doping away from the center of the well.

The asymmetry of the I(V) curves is often attributed to the doping segregation. Rather than changing the polarity of the applied field in our simulation (which implies to re-evaluate all wave functions and energy values), we changed the direction of segregation, L 1 becoming L 2 and vice versa.

Using our segregated profile (L 1 =5Ǻ, L 2 =50Ǻ), our model predicts a ratio of I + (current under positive bias) over I -(current under negative bias of 1.3, for a bias of 1V (on the plateau). The experimental value of this ratio is included in the 1.5-1.6 range, leading to a difference between experimental and theoretical value of 20%.

As expected, the segregation introduces an asymmetry with the bias polarity. The quantitative agreement is acceptable if we consider the hypothesis made on the doping profile shape.

C. Importance of the growth method. As we have shown that the ionized impurities play a major role in the value of the dark current, it is important to study the influence of the growth method. Indeed, because of the high reactivity of the aluminum [START_REF] Asplund | Proc. SPIE[END_REF] , non desired impurities can be present in the barrier. Impurities such as carbon, oxygen, silicon, sulfur, tellurium and germanium can be incorporated with a concentration which is dependent on the growth method. With MBE the residual concentration is below SIMS resolution, [START_REF] Asplund | Proc. SPIE[END_REF] typically ~some 10 14 cm -3 . With Metal Organic Chemical Vapor Phase Epitaxy (MOVPE) this concentration is typically one order of magnitude higher. As shown in FIG. 11, the presence of these impurities in the barrier, added to the nominal doping, has no influence for MBE, and is also negligible in the MOVPE case. Such a result is very important for the QWIP designer, since both methods can be used without major impact on the device performances. We have studied the electronic transport under dark condition of weakly coupled QWs at very low temperature, i.e. in the tunneling regime. The I(V) curves exhibit a plateau region, where the current displays a very low dependence with respect to the applied electric field. We have checked that this does not originate from electric field domain effects. We have shown that the usual WKB approximation is unable to reproduce this striking plateau regime. Consequently we developed a full quantum scattering approach of the transport, based on the Fermi golden rule and taking into account all the main interactions met in AlGaAs heterostructures. Our model suggests that the plateau regime is due to a competition between two mechanisms when the electric field is enhanced in the QWs: a decrease of the current due to the electric field localization of the carriers (Wannier-Stark effect) and an increase due to a higher scattering probability due to an increasing tunnel effect. We conclude that the electron-ionized donors interaction is the dominant one and obtain a good agreement between theory and experiment for the plateau value. We have applied our model to predict the influence of the doping density and profile on the dark current. We showed that our model is able to reproduce the I(V) curves asymmetry, at low bias, by the use of a segregated doping profile. We also demonstrated the very low effect of the choice of the growth method on the dark current. This work may promote the development of new doping profiles for QWIP operating in the low photon flux regime.

AKNOLEDGEMENTS

Authors thank Angela Vasanelli and Robson

Ferreira for fruitful discussions.

APPENDIX A: ELECTRON-LONGITUDINAL OPTICAL (LO) PHONON

The typical value of the energy drop per period (10 to 35 meV in the plateau regime), remains lower than the GaAs LO phonon energy (36meV). Consequently there is a very low probability that this mechanism is the main one, at least for F<10kV.cm -1 . Moreover the very low operating temperature is not favorable to this type of scattering. The Hamiltonian describing the interaction between electron and longitudinal phonon can be written as 19,20 hc b e q H q q r q i phonon e

+ = ∑ + - r r ) ( / α
, where q is the momentum of the phonon, ) (q α describes the strength of the interaction and + q b the phonon creation operator. In the particular case of optical phonon ) (q α is linked to the Frölich interaction [START_REF] Rosencher | Optoélectronique[END_REF] , where the electric field due to the dipole of the GaAs doublet interacts with an electron:

2 0 2 2 2 1 2 ) ( q q w e q p LO LO + Ω = ε α h , (A1) with s p ε ε ε 1 1 1 - = ∞ .
Here

) ) ( * 2 ( ) ( 4 ) ( 2 2 2 2 0 2 2 2 LO f i f i z LO z f f LO i LO w K K m E E q q q F dq d dK K K h h h - - + - × + = Γ ∫∫∫ δ θ π α (A2)
By evaluating the integral over K f one finds [START_REF] Harrison | Quantum wells, wires and dots: Theoretical and Computational Physics of Semiconductor Nanostructures[END_REF] :

∫ ∫ + = Γ 2 0 2 3 2 2 ) ( ) 2 ( * ) ( q q q F dq d m K z LO z LO i LO θ π α h (A3) where 2 2 2 z q Q q + = , (A4) and ) cos( 2 2 2 2 θ f i f i K K K K Q - + = (A5) and ) ( * 2 2 2 2 LO f i i f w E E m K K h h - - + = . ( A6
)
In such an expression, the Bose Einstein factor which describes the population of phonons is taken equal to unity. This choice is justified by the fact that temperature is very low, which implies that no optical phonon absorption is possible.

APPENDIX B: ELECTRON-LONGITUDINAL ACOUSTICAL (AC) PHONON

As for optical phonons, the very low temperature is again not favorable to the interaction between acoustical phonons and electrons. However, because of their lower energy, our model needs to include both emission and absorption of acoustical phonons. The considered Hamiltonian is a classical electronbulk acoustical phonon interaction, based on Debye dispersion i.e. hc b e q H q q r q i phonon e

+ = ∑ + - r r ) ( / α 19 with: q c D q s c AC . 2 ) ( 2 2 ρ α Ω = h (B1)
where D c is the acoustic deformation potential, Ω the volume of the sample, ρ the density and c s the sound velocity, q the phonon wave vector and

( )). ( 1 ( 4 2 2 0 0 2 2 2 z AC BE z f f s AC AC q F w n q q d dK K c h h + = Γ ∫ ∫ ∞ π θ π α (B4) with s f i f i c K K m E E q h h ) ( 2 2 2 * 2 - + - = , (B5) and ) cos( 2 2 2 2 θ f i f i K K K K Q - + = (B6) and 2 2 Q q q z - = (B7)
Here n BE is the Bose Einstein distribution. In the case of absorption, this 1+n BE factor is replaced by n BE .

APPENDIX C: ALLOY DISORDER (AL)

In Al x Ga 1-x As alloy the presence of aluminium in substitution of the gallium induces scattering, because of the different atomic potential of the two atoms. It is quite hard to evaluate a priori the magnitude of this interaction. Such a scattering is usually treated by a potential proportional to the deformation.

) ( . r x V V δ ∆ = 21 where V ∆
is the band offset between GaAs and AlAs. Generally, the calculation consists in defining a statistical correlation function between the aluminium atoms positions , following the Nordheim rule 51 :

) ' ( )] 1 ( [ ) ' ( ) ( 0 r r x x r x r x - - Ω = δ δ δ (C1)
where 0 Ω is the size of the primitive cell. We can now evaluate the mean value of the matrix element

2 2 f x V i M AL δ ∆ = (C2) and so 2 3 * 2 2 ) ( ) ( ) ( r d r r x r V M i f AL Ψ Ψ ∆ = δ (C3)
We define the associated form factor

∫ = alloy i f AL dz z z F 2 2 ) ( ) ( ξ ξ . (C4)
To conclude, the expression of the scattering rate is given by:

∫ ∫ - - Ω ∆ = Γ ) ( 2 )] 1 ( [ 2 * 0 2 f i f AL AL d d m F x x V ε ε θδ ε π h h , (C5)
which is equal to

AL AL F x x V m )] 1 ( [ 0 2 3 * - Ω ∆ = Γ h . ( C6
)

APPENDIX D: INTERFACE ROUGHNESS (IR)

Because of the wide barriers and the small number of interfaces we do not expect this process to be dominant. This is a main difference between QWIP and Quantum Cascade Detector 52 (QCD) or QCL. In the latter, wells are highly coupled and barriers are quite thin, making interface roughness a non negligible interaction at low temperature 22 . The treatment for interface roughness 21,53 is very close from the one made for alloy disorder: Unuma 21 makes the remark that interface roughness is the sheet equivalent of the alloy disorder. As for alloy disorder we start by defining a linear potential with the perturbation:

) ( ) ( ) ( r F z z V r V i b IR - ∆ = δ , ( D1
)
where V b is the band offset between GaAs and Al x Ga 1-

x As, ∆ is the magnitude of the interface defects and F(r) is the spatial distribution of defects. The delta function underlines the local character of this interaction. Most often F(r) is chosen to follow a Gaussian correlation function:

) ' exp( ) ' ( ). ( 2 2 ξ r r r F r F - - = (D2)
with ξ the correlation length. The matrix element is now given by 2

2 ) ( ) ( . . i r F z z Vb f M i IR - ∆ = δ (D3) or 2 * 3 2 2 2 ) ( ) ( ) ( ) ( ∫ - Ψ Ψ ∆ = r F z z r r r d V M i i f b IR δ (D4)
and the very simple form factor can be written:

2 2 ) ( ) ( i f i i IR z z F ξ ξ = .
(D5)

In the case of multiple interface the form factor is summed over all interface positions (z i ). Considering the elastic character of the interaction

) ( * 2 2 2 2 f i i j E E m K K - + = h , ( D6
) the exchanged wave vector

f i K K Q - = becomes: ). cos( ) ( * 2 2 ) ( * 2 2 2 2 2 2 2 θ f i i i f i i E E m K K E E m K Q - + - - + = h h (D7)
To finish, the scattering rate is:

θ ξ π ξ d e F V m Q IR b IR ∫ - ∆ = Γ 2 0 4 3 2 2 2 2 2 2 * h . ( D8
)

APPENDIX E: IONIZED IMPURITIES (II)

Scattering by ionized impurities is involved in two different ways in QWIP: first, through the doping which is generally localized in the well. Then the high reactivity of the aluminum in the barrier leads to the inclusion of undesirable impurities (mostly carbon) in the barrier. Concentration of residual impurities is highly dependent of the growth method [START_REF] Asplund | Proc. SPIE[END_REF] . As shown in Sec. VI-C, our model can conclude on the importance of residual impurities. It is also able to take into account the segregation of the doping, which leads to asymmetric I(V) curves with bias polarity.

In the following we assume that impurities are completely ionized, even at low temperature. The Coulombian Hamiltonian is:

r e r V r 1 4 ) ( 0 2 ε πε = . (E1) V is Fourier transformed 39 Q e e e Q V i i r r iQ z z Q Q r ") " ( 0 2 2 ) ( - - - ∑ = ε ε . (E2)
Thus the matrix element is given by 2

" ) " " ( " * 2 2 0 2 2 ) ( ) ( " ) 2 ( ∫ ∫ - - - - × = iKir i ri r iQ zi z Q iKfr f r II e z Q e e e z Q d r dzd e M ξ ξ ε ε (E3)
We can also define the associated form factor

2 * ) ( ) ( ) ( z e z dz Q F i zi z Q f II ξ ξ - - ∫ = , (E4) so we get M II by 2 2 2 0 4 2 ) ( 4 f i f i II r II K K K K F e M - - = ε ε . (E5)
The scattering rate is obtained by summing over all positions of the doping (z ii ) :

θ f i f i f i K K K K K K - + = - (E7) and ) ( * 2 2 2 2 f i i f E E m K K - + = h . ( E8
)
In order to take into account the screening of the interaction, we used a Thomas-Fermi approach with a constant screening length [START_REF] Rosencher | Optoélectronique[END_REF] T k n e q b r ε 2 2 0 = , n the volumic doping and r ε the permittivity of the materials. It leads to an effective wave vector [START_REF] Rott | [END_REF][START_REF] Price | [END_REF] 2 0 2 q q q eff + = . Note that unlike the previous process (LO, AC, AL, IR), Coulombian interactions will lead to a quadratic dependence with the doping.

APPENDIX F: ELECTRON ELECTRON (EE)

Electron-electron interaction is certainly the most difficult interaction to understand. To deal with it, some authors use Green function formalism 56,57 . Only little work has been devoted to the treatment of this interaction using the envelop formalism, we could quote works of Smet 58 , Harrison [START_REF] Harrison | Quantum wells, wires and dots: Theoretical and Computational Physics of Semiconductor Nanostructures[END_REF] and Kinsler 59 . This lack is the consequence of the very time consuming numerical treatment. All theoretical difficulties in the treatment of this interaction are due to the two bodies type of this interaction. Initial states will be noted as i and j , and the final states as f and g . The interaction potential is, as for ionized impurities, the Coulombian potential, so the matrix element can be written This expression depends on the kinetic energy of the two initial states i and j . In order to use the scattering rate of the process in the same way as the previous ones, we sum this expression over all j initial states. . This approximation allows us to obtain an upper limit of the scattering rate, which is not an issue if this interaction is not the main one. Kinsler et al 59

  FIG.1(a) Dark current density as a function of the applied bias for T=4K, 10K, 20K, 30K, 40K, 50K and 60K. (b) Spectral response at T=10K for V=1.5V, inset spectral response for different bias voltages from V=1V to V=2V by step of 0.2V, in the 15.5µm-16µm range.

  the effective mass of the electron in GaAs,

  population factor, k b the Boltzmann constant, T the temperature, fw E the Fermi level in a well, F the electric field, L b the barrier width, 1 -WKB τ is the inverse of the time for which an electron succeeds in crossing the barrier. Following Gomez 5 1 -

FIG. 2 .

 2 FIG. 2. Experimental and theoretical (WKB) currentdensity as a function of the applied bias.

FIG. 3 .

 3 FIG.3. Band profile of the QWIP under an electric field of 3 kV.cm -1 .

ε

  the associated energy and H ˆ the perturbation Hamiltonian. The wave functions are evaluated in the envelope function formalism[START_REF] Bastard | Wave Mechanics Applied to Semiconductor Heterostructures[END_REF]

nm 6. 5 ∆FIG. 4 .

 54 FIG.4. Product of the scattering rates, for K i =0, by the population factor of the arrival level, as a function of the bias for the six considered processes.

.

  5 (b)). The current density is proportional to the part of carrier wave function delocalized in the next well. So the associated In the Wannier-Stark approach the role of the electric field is to localize the wave function when the field is higher than 0 ∆ . When 0 F → , this latter expression diverges as explained above. To take into account the level broadening due to intrawell scattering, we introduce a imaginary part τ h i to the transition energy[START_REF] Cohen-Tannoudji | Quantum mechanics[END_REF] which leads to an effective field τ shown that it is also equivalent to introducing a coherence length for the electron wave function. The value of the dephasing time τ has been taken equal to the intrawell scattering time (scattering between two states of the same subband and the same well)

FIG. 5 FIG. 6 .

 56 FIG. 5 (a)The current variation for an increasing electric field is the result of a competition between the enhanced probability of the electron to be in the neighbouring well (enhancing scattering), and an enhanced Wannier-Stark localisation of the electrons in their well. For very low fields, the dephasing time

FIG. 7

 7 FIG.7Dark current as a function of the applied bias for component B 1 (doping level of 1×10 11 cm -2 ) and B 2 (doping level of 2×10 11 cm -2 ).

FIG. 8 .

 8 FIG.8. Dark current density (full line) as a function of the doping position in the well, under an electric field of 10 kVcm -l . The dotted line is the current value for the same electric field but without ionized impurities. Thus it may be possible to reduce the dark current by shifting the doping position from the center of the well to an other position. To confirm this prediction, we have grown two samples (L w =6.8nm, L b =39nm, %Al=15%, 40 periods, sheet density 3×10 11 cm -2 ). The doping is respectively in the central third of the well (component C 1 ) and on the last third (grating side) of the well (C 2 ). We have plotted on FIG.9, the magnitude of the dark current as a function of the temperature under a bias of -1.5V. Using X ray diffraction and spectral measurements we have measured that the structure C 2 presents a lower confinement (transition energies respectively of 88.3 meV for C1 and 87.4 meV for C2). Consequently at high temperature (T> 35 K), this sample presents a higher thermoionic dark current due to a more efficient thermal activation of the electron. At low temperature however, in spite of its lower confinement, this C2 structure displays a less important tunnel current, which results from the fact that the interwell hopping scattering rate has been effectively reduced, as expected fro our theory.

FIG. 9

 9 FIG. 9 Dark current as a function of the temperature, under a voltage bias of -1.5V, for the component C 1 and C 2 . The grey pattern indicates the doping position into the well.

3 )FIG. 10 .

 310 FIG.10. Theoretical dark current density, as a function of the electric field, for different values of segregation lengths (L 1 and L 2 ). Inset: doping profile in a quantum well.

FIG. 11 .

 11 FIG. 11. Dark current density as a function of the concentration of undesired ionized impurities in the barrier under an electric field of 10kVcm -1 . The two rectangles highlight the typical range of concentration for MBE (leaned pattern) and for MOVPE (horizontal pattern) VII. CONCLUSION

  of the optical phonon is GaAs, Ω the volume of the sample, ∞ ε and s ε are respectively the dielectric constant at infinite and null frequency and q 0 the inverse of the screening length. Such an Hamiltonian considers only bulk phonons. Readers interested in other types of phonons (surface phonon for example) could read ref 49 and 50. The matrix element is equal to:expression can be included in the FGR giving:

)

  be injected in the FGR to obtain the expression of the scattering rate:

  factor of the different states which appear in the expression. Because it is very difficult to obtain separately f

  have however proposed a solution to avoid this approximation. To finish the expression of the scattering rate is given by α and θ two angles. The expression of q xy is detailed calculation one should read Harrison's book[START_REF] Harrison | Quantum wells, wires and dots: Theoretical and Computational Physics of Semiconductor Nanostructures[END_REF] .

TABLE 1

 1

					. Interaction parameters used for
	simulation (see Appendices A to F)
	Parameter	unit	value
	m* 20	kg	0.067.m 0
	s ε ≈ 13, 48 r ε	A 2 s 4 m -3 kg -1	12.9 0 ε
	∞ ε 48	A 2 s 4 m -3 kg -1	10.9 0 ε
	h	w ρ 19 LO	48	meV Kg.m -3	36.6 5320
		c s	19		ms -1	5220
		D c	19		eV	12
	∆ 21,53,36,37 ξ 21,53,36,37	nm	0.3

1

2 Stark -Wannier

L. Esaki and R. Tsu, IBM J. Res. Develop. 14, 61 (1970).

A.Wacker, in Theory of transport properties of semiconductor nanostructures (Chapman & Hall, London, 1998), chap. 10.


Emmanuel Lhuillier

Sean Keuleyan

Philippe Guyot-Sionnest

Colloidal Quantum Dots for Mid-IR Applications

Keywords: quantum dot, HgTe, mid-infrared

The use of colloidal material offers an interesting alternative to top down approaches for the realization of low cost infrared detectors. We demonstrate photoconduction in thin films of a colloidal material in the mid infrared (up to 7µm), using HgTe colloidal quantum dots. Thin films of the colloidal quantum dots have a large absorption coefficient (>10 4 cm -1 ), and the photoconductive response is dramatically improved by encapsulating the nanoparticle into an inorganic matrix of As2S3. Such devices show fast response and large detectivity (>10 10 jones) at temperatures above 200 K.

INTEREST IN COLLOIDAL MATERIAL IN INFRARED DETECTION

Infrared detection is currently driven by mature technologies, such as quantum well infrared photodetectors (QWIP) 12 , type II superlattices 3 , HgCdTe alloys or InSb 4 . In spite of up to fifty years of development, these materials remain expensive, limiting the use of the infrared cameras to primarily defense and astronomy applications. In addition, the push for higher operating temperature makes improvements even more challenging. In QWIP devices, the active layer is already operating close to its quantum limits 5,6,7,8 . New paradigms are thus necessary to achieve both higher operating temperature and reduced cost.

Quantum dot infrared detectors (QDIP) have offered an interesting alternative to quantum well device since the phonon bottle neck was expected to reduce the coupling between carriers and phonons, thus allowing higher operating temperature through a reduction of the dark current. In practice though, the epitaxial growth of the QD in its matrix keeps the coupling to the phonons high, and no significant dark current reduction has been achieved so far.

Colloidal quantum dots (CQD) show some evidence for a phonon bottle neck 9 , which may provide an advantage in devices. The absence of epitaxy between the dot and the surrounding matrix offers a better opportunity to reduce the coupling to phonons and indeed, slow relaxation has been observed in CQD, indicating that the discrete spectrum of the CQD is poorly coupled to the phonon bath. Moreover the synthesis and control of CQD material properties has seen tremendous development over the past two decades [START_REF] Klimov | Nanocrystal Quantum Dots, Second Edition[END_REF] . Control over properties in these materials is almost as advanced as that for molecular beam or metal organic chemical vapor deposition grown heterostructures. Nanoparticles can be zerodimentional (quantum dots [START_REF] Murray | [END_REF] ), one dimentional (nanorods 12 ) or two dimentional (nanoplatelets 1314 ). Core-shell heterostructures are also synthetized with type I 15 and type II 16 band alignments. The use of semiconductor nanocrystals for optoelectronic applications 17 has been demonstrated in solar cells [START_REF] Nozik | Quantum dot solar cells[END_REF] , LEDs and detectors 19 . Materials and such devices though have been developed primarily for visible applications with less work devoted to the infrared (IR).

A possible path to access the mid-IR range of wavelengths is based on intraband transitions in wide band gap materials such as CdSe 20 . To realize these transitions, the conduction band can be populated with electrons (or the valence band with holes) by gating or optical pumping. This requirement, though, increases complexity and may be incompatible with many applications. Near-IR energiesare accessed with lead chalcogenide CQD (PbS, PbSe, PbTe). Their use in converting the near-IR part of the solar spectrum and in infrared detection has been reported by Konstantatos et al 1921 . The long wavelength limitwith these lead based CQD remains restricted by their bulk band gap. 22 Semimetals, such as HgTe are consequently the best candidates for CQD devices for the mid wavelength infrared (MWIR). Many works have reported the synthesis of HgTe for near-IR applications 232425 , but none of them have reached cut-off wavelengths up to the mid-IR, mostly due to difficulties in preparing stable, large HgTe CQD. In addition, the use of CQD in MWIR remains very challenging due to the capping organic ligands, which have vibrational modes in this range of energy.

In this paper we discuss the synthesis and photoconduction properties of CQD material in the MWIR. In Section 2 we discuss the chemical synthesis of the HgTe CQD as well as the device preparation and experimental setup. In the third section we focus our interest on the optical properties (relation size-absorption edge wavelength and absorption coefficient). The final section deals with the photoconduction properties of thin films of HgTe nanoparticles. First, we underline the possible difficulties of dealing with colloidal material in the mid-IR, then we show the limits of a photoconductive response with an organic ligand matrix. Finally we demonstrate that embedding the HgTe CQD into a matrix of As2S3 drastically improves carrier mobility and consequently the film responsivity and detectivity. The material synthesis is performed following the recipe developed in references 26,27. A first generation of material 26 was developed and showed tunable optical properties in the MWIR. The particles were found to be aggregated and the optical spectra were quite broad. In a later development, a new synthetic method 27 was developed to improve the CQD monodispersity and as a consequence the sharpness of the optical spectra. Briefly, HgCl2 is dissolved in octadecylamine, the mixture is heated to 100°C for one hour under vacuum. Then a solution of Te in trioctylphosphine (TOP) (TOPTe) is quickly added to the flask. The solution darkens, indicating the formation of HgTe. The particle size is controlled by the duration and temperature of the reaction, where increasing the duration or the temperature leads to an increase of the size. The obtained solution is quenched into a room temperature solution of TOP, dodecanethiol and tetrachloroethylene (TCE). The absorption spectra shown in Figure 1 (a) shows the possibility to tune the exciton feature wavelength at room temperature between 1.5 and 5 µm.

MATERIAL AND DEVICES

Typical particles are shown in a transmission electron microscopy (TEM) image in Figure 1 (b). CQD are typically composed of an inorganic core (i.e. the semiconductor HgTe in this case) surrounded by an organic shell of ligands, which control the growth and ensure colloidal stability of the solution. As will be discussed later, the presence of this organic shell makes the use of this material in the IR challenging.

Films of CQD can be obtained by dropcasting, spincoating or dipcoating. In order to obtain homogeneous film, the CQD initially dissolved into TCE are precipitated by addition of an alcohol and centrifuged. The obtained solid is redissolved into a 9:1 in volume mixture of hexane:octane. This film can be dropcast over interdigitated electrodes (50 pairs of 5mm long Pt electrodes with a 10µm spacing), see Figure 1 (c). For a single deposition the film thickness is typically 100nm. Multilayer deposition is possible by fixing each layer.This can be achieved by a ligand exchange process where the relatively long ligands (eg. dodecanethiol) are replaced by shorter ones such as ethanedithiol. For electrical characterization the sample is mounted on the coldfinger of a closed cycle He cryostat.

For characterization the I-V curves are acquired using a Keithley 6487 picoammeter. Photocurrent spectra are obtained by biasing the sample using a femto DLCPA 200 amplifier, and the amplified output is processed via a Nicolet Magna IR 550 FTIR. The time response is measured by illuminating the device with a 10 ps pulse from a 1.06µm Nd:YAG laser, with the sample biased by a battery, and the response recorded through a Tektronik TDS 1012B oscilloscope. The responsivity of the system is measured while the system is illuminated by a blackbody at 955°C and the incoming light is mechanically chopped at 25Hz. The optical aperture of the system is f/3. The biasing, magnification of the signal and the acquisition are made with the amplifier and oscilloscope previously mentioned. Noise is measured in the dark for the evaluation of the detectivity. The noise measurements are obtained by biasing and amplifying the signal using the same amplifier as above and the data is acquired with a Stanford SR 760 spectrum analyzer.

OPTICAL PROPERTIES

As for quantum well or superlattice based devices the optical properties are tuned through a change of the size of the system. The confinement takes the same role as the cadmium composition in the HgCdTe alloy to tune the band gap energy of the material. Tuning the size of the particle from 5 to 15 nm results in the cut-off wavelength tuning from 1.5 to 5 μm at room temperature, as shown in Figure 2 (a). This is in good agreement with a prediction coming from a two band k•p model 28 . Pushing the absorption out to the long wavelength infrared (LWIR) will require the growth of CQD as large as 18nm.

An alternative way to the tune the band edge energy is to change the temperature of the sample. As for bulk HgTe, The HgTe CQD have a positive value of dT dE QD BE , meaning that a cooling of the material leads to a redshift of the transition. As an example, sample with a cutoff wavelength of 5µm at room temperature absorbs up to 7 μm at liquid nitrogen temperature. This shift of the band gap energy with temperature exhibits a size dependence which is attributed to the tuning of the relative weight of the inter-and intraband coupling of the carriers to the optical phonons.

For application in photodetectors, a key parameter is the absorption coefficient. In our case the absorption coefficients are measured by depositing a film of CQD onto a Si wafer and measuring the film thickness by ellipsometry (Gartner L116S), while the absorption is obtained through a FTIR spectrometer (Nicolet Magna IR 550). The absorption coefficients measured are typically in the 10 4 cm -1 range (see Figure 2 (b)), which is as high as for the bulk HgTe 29 and HgCdTe alloy 303132 . Compared to QDIP obtained through Stransky-Krastanov growth, these values are considerably higher, due to the larger QD packing density in the colloidal thin films.

PHOTOCONDUCTION AND DEVICES PERFORMANCE 4.1. Challenges for CQD in mid-infrared

Absorption in the mid-IR has been reported recently with other colloidal material 33 , nevertheless obtaining photoconduction remain even more challenging. Here, we first discuss some of the difficulties in realizing IR photoconduction in films of CQD.

• CQD are composed of an inorganic semiconductor core surrounded by organic ligands, which control the growth and the colloidal stability of the nanoparticle. These ligands are typically long alkane chains which are connected to the core through an organic functional group such as an acid, amine, or thiol. The ligands have vibrational transitions in the mid-IR, (eg. the C-H bond has a resonance near 3000cm -1 ). Energy can therefore be lost from the electronic excitation in the core, to the organic ligand vibrations. Such a process is, for example, known to quench the photoluminescence of CQD in this range of wavelength 34 . Removing/replacing these organic molecules is therefore desired. • Typical organic ligands are also very poorly conductive, since they behave as tunnel barrier for the inter CQD transport. Carrier mobilities in drop-cast films of CQD are typically around 10 - 6 cm 2 V -1 s -1 , limiting the photoconductive responsivity. • Depending on the strength of the binding to the semiconductor core, the ligands can be removed which result in incomplete passivation of the particle surface, resulting in the formation of trap states which can further alter mobilities and recombination rates. • Finally a CQD film can be seen as a bulk material full of cracks, which can lead to additive electrical 1/f noise 35 and limited detectivity.

To achieve detectors with large detectivity in the mid-IR all these points have to be addressed.

Limitations of the organic capping

The first strategy to address the previously mentioned issues is the now commonly applied ligand exchange, which aims to reduce the length of the ligands.. Typically, the initial ligands with twelve or more carbons are chemically replaced by shorter ligands (with four or less carbons). This process is experimentally very easy to perform -simply dipping the CQD film into a solution of short ligands in ethanol efficiently achieves the exchange. The number of absorbing C-H bond is reduced and the mobility drastically increased due to the reduced interparticle distance. Ligand exchange with ethanedithiol 36 has been extensively investigated and typically increases the conductance by 3 orders of magnitude.

The ligand exchange process, performed on our films of HgTe CQD gave an increase in mobility, accompanied though, by a non-monotonic temperature dependence of the dark current (see Erreur ! Source du renvoi introuvable. (a)). In addition, responsivities remained low (≈1mA.W -1 , see Erreur ! Source du renvoi introuvable. (b)), a consequence of the mobility increasing to only about 10 -4 cm 2 V -1 s -1 .

Encapsulation into an inorganic matrix

To obtain further improvement of the responsivity and detectivity, we wish to completely remove the organic component of the material, which is poorly conductive and leads to poor passivation. We identify As 3+ as an ion which largely increases the conductance of the HgTe CQD film and As2S3 as a possible IR transparent matrix 37 including this ion with good optical properties in the mid-IR. Once the material is processed air-free (to avoid the activation of acceptor states) and encapsulated into by As2S3 38 , we recover an intrinsic behavior for the temperature dependence of the current, see Erreur ! Source du renvoi introuvable. (a). Moreover, a responsivity one hundred times large is obtained, see Erreur ! Source du renvoi introuvable. (b), and can be attributed to the increase of the carrier mobility. In addition, we note that the frequency cut-off remains high (>100kHz, see Figure 4 (a)) and that the measured detectvity can be as high as 10 10 Jones, (Figure 4

CONCLUSION

We report the synthesis, optical, and electro-optical characterization of mid-infrared active HgTe CQD with a band gap tunable through the near-and mid-infrared. This material can be prepared easily via standard solution synthesis. Thin films have a large absorption coefficient (>10 4 cm -1 ), and once processed air free and encapsulated in the right inorganic matrix this material presents a large detectivity (>10 10 jones) and fast time response. This progress paves the way for the use of this new material in low cost infrared detectors.

Figure 1 (

 1 Figure 1(a) Absorption spectrum for HgTe CQD solution in the IR range for several particle sizes between 5 and 15nm. The sharp feature near 2900 cm-1 is from the capping ligands. (b) TEM image of HgTe nanoparticles. (c) Photo of a film of HgTe deposited over interdigitated electrodes.

Figure 2 (

 2 Figure 2 (a) k•p modeling of cut-off wavelength of HgTe CQD as a function of their size. Experimental data appears as cross. (b) Absorption length as a function of the cut-off wavelength. The continuous line is obtained from k•p modeling.

Figure 3

 3 Figure 3 (a) Current as a function of the inverse of the temperature for a film of HgTe CQD processed in air with EDT and processed air free with As2S3 under 1V bias. (b) The responsivity for the same two films at room temperature.

  (b)).

Figure 4 (

 4 Figure 4 (a) Photocurrent as a function of the modulation frequency of a 800nm laser for a film of HgTe CQD processed air free with As2S3. (b) Measured detectivity, as a function of temperature and bias, of a film of HgTe CQD (λcut-off=3µm) processed air-free and with As2S3.

  Photocurrent(a.u.)


Emmanuel Lhuillier

Sean Keuleyan

Paul Rekemeyer

Philippe Guyot-Sionnest

Thermal properties of mid-infrared colloidal quantum dot detectors

Keywords: Quantum dot, infrared, HgTe PACS: 85.35.Be, 85.60.Gz

HgTe colloidal quantum dot films are studied for photodetection over the 3-5 µm atmospheric transparency window. The temperature dependence of the conductivity indicates that the material behaves approximately as an intrinsic semiconductor. In photoconduction, the responsivity can be as high as several hundred mA W -1 at room temperature. The dark current presents 1/f noise which is larger than for homogeneous conductors, and this noise decreases with temperature. A specific detectivity of 2×10 9 Jones is obtained for a sample with a 6 micron cut-off wavelength at 130K. These values are obtained for the thickest films studied (~ 400 nm) and whose thicknesses are still much less than the optical absorption length. The time response can be faster than 100 nanoseconds.

I. INTRODUCTION

Colloidal quantum dots (CQD) have been studied over more than two decades, and they have potential for various optoelectronic applications 1 . While sharing the advantages of liquid processing with organic molecular compounds, they have a very unique advantage in the infrared due to weaker vibrational absorption and well defined infrared electronic absorption. They have already been extensively investigated in the near-IR for photodetection, 2,3,[START_REF] Tsidilkovski | Electron spectrum of gapless semiconductors[END_REF][START_REF] Nimtz | Narrow-gap semiconductors[END_REF]6 . PbS and PbSe quantum dots are potential materials in the near-IR 7,8,9,10 . For longer wavelengths, HgTe should be a better choice since it is a gapless semiconductor [START_REF] Nimtz | Narrow-gap semiconductors[END_REF][START_REF] Tsidilkovski | Electron spectrum of gapless semiconductors[END_REF] . As CQD, this material had already been shown to function well for near infrared photodetection 12,13 and we recently extended the material's response into the mid-IR. [START_REF] Keuleyan | Mid-infrared HgTe colloidal quantum dot photodetectors[END_REF] CQD may address current demand for a decrease of the cost of the infrared focal plane array and an increase in operating temperature. Already established technologies in the mid infrared, such as InSb or MCT or even promising detectors such as type II superlattices, achieve excellent performances but fail to provide a low cost solution. In part, the high cost is due to the requirement of epitaxial growth. HgTe CQD material can thus bring ease of processing and much lower costs. [START_REF] Kim | [END_REF]16,17 Moreover the development of HgTe CQD benefits from the extensive knowledge of HgCdTe, making this material easily transferable to MCT foundry.

In this report we present an initial study of the temperature dependent conductivity and photoconductivity characteristics of the HgTe CQD with mid-IR gaps. We report measurement of the dark current, spectral response, responsivity and noise as a function of bias and temperature.

The efficiency of the CQD based detectors is evaluated and their optimal operating temperature identified. We identify possible pathways for optimizing the material as a photodetector.

II. EXPERIMENTAL

A. MATERIAL AND FILMS PREPARATION

The synthesis of the QD particles uses the previously published method. 1410 Briefly, 13mg of tellurium are dissolved in 2mL of butanol and 0.1mL of trioctylphosphine. The mixture is heated to 90°C for 20 min under Ar to dissolve the tellurium. The temperature is then adjusted based on the desired particle size, with higher temperatures leading to larger particles. The smallest particles with band gaps reaching the visible region were obtained at temperatures as low as 0°C while the largest particles were prepared at 90°C. Meanwhile mercury acetate is mixed with 2mL of butanol and 2mL of pyridine. This solution is then quickly injected into the flask, immediately forming a dark solution. Finally the reaction is stopped after several minutes by extracting to a room temperature solution of 10% dodecanethiol in tetracholoethylene (TCE).

The diameters of the dots in this study are in the 6 to 12nm range. Particles with a ~8nm diameter (respectively 10.5 nm) present a band edge energy corresponding to 3µm (5µm). Noise measurements are performed with a spectrum analyzer (HP 3561a). The sample is biased with a battery in order to limit the input noise. The mean signal is removed using a resistor bridge and the noise current is then input to a low noise operational amplifier (LF356N, National Semiconductor). The current amplification is typically in the 10 4 to 10 7 range before input to the spectrum analyzer. All the noise measurements were conducted in the cryostat with the optical shutter closed.

III. RESULTS AND DISCUSSIONS

A. AN INTRINSIC SEMICONDUCTOR

Dark I(V) curve for sample C are presented in Figure 2Figure 2. The dark I(V) curves present almost no hysteresis. They are generally linear at high temperature and become less linear as the temperature decrease. The dark current as a function of temperature is also presented in Figure 2Figure 2. The transport decreases about two orders of magnitude from the room temperature value, with an Arrhenius law

      -  T k E T I b a exp ) (
, with Ea an activation energy, kB the Boltzmann constant, and T the temperature until settling to a lower slope. Activation energies are reported in Table 1Table 1 for sample A, B and C. The activation energy is in each case close to half of the energy gap value (at room temperature), the latter being determined as the edge of the photoresponse described below measured at room temperature. Such a relation between the energy of thermal activation of carriers and the gap is expected for an intrinsic semiconductor. At room temperature, we therefore conclude that the carrier density is dominated by the intrinsic properties. There may of course be some carrier doping in our samples, but given the thermal behavior of the dark current, it must be much less that the density of intrinsic carriers at room temperature. In the intrinsic case, the carrier density can be determined as

Mis en fo

) 2 / exp( 0 T k E N N N p n B G v c -  = =
where 0 N is the number of nanocrystals per volume, and Nc and Nv are the numbers of electron and hole states respectively in the nanocrystals that are thermally accessible. Given that much of the confinement energy is in the electron states, we take Nc=2. For the more massive holes, we will also take Nv=2 even though it may be somewhat more. The estimated carrier densities at room temperature are then shown in Table 1.

The measured resistivities of the samples allow an estimate of the mobility using

  ne = -1
, which typically leads to value in the 0.5 cm 2 V -1 s -1 range for sample C. For the discussion later, we also provide an estimate of the hopping time between nanocrystals. Using Einstein's relation,

B. EFFECT OF THE TEMPERATURE ON THE OPTICAL PROPERTIES

Figure 3 shows the spectral response for the three samples. By changing the size and temperature of the particles, the cut-off wavelength tunes from 2.8 to almost 7 µm. Shorter cut-off wavelengths are accessible by growing smaller particles but are not in the scope of this work. In samples B and C, some features in the spectral response are due to vibrational absorption of the C-H bonds from the ligands (3.5µm), H2O (3µm) and CO2 (4µm) coming from water vapor and carbon dioxide in the interferometer.

For all the samples, a decrease in temperature results in a red-shift of the spectral response, as shown in Figure 3Figure 3and Table 1Table 1. This is consistent with bulk HgTe, where thermal dilation of the HgTe lattice leads to a reduced gap with decreasing temperature. 18,19 The thermal dependence of the energy gap is captured quite well with an empirical parabolic fit.

Alternatively, Varshni's 20 expression, given by equation (1) (1), is often used.

T T K T E T E G G + + = =   2 ) 0 ( ) ( (1)
Here, the  parameter is linked to the thermal dilation coefficient.  was originally associated with the Debye temperature although it can deviate significantly 2016 . The parameter values for bulk HgTe are poorly known and reported values cover a wide range, with =

1 4 10 7 . 2 - -   K eV to 1 4 10 5 . 8 - -   K eV
. 1814,1915 Our experimental points are compared with a Varshni's law fit for the thermal dependence. As shown in Figure 2Figure 2, the fit is imperfect and is in fact not very sensitive to the choice of  set at 500K, but the coefficient  is reasonable. Table 1 lists the parameters. With more monodispersed samples it will be important to determine these parameters with more accuracy than in the bulk and to extract their size dependence 21 since the red shift with temperature is relevant to the mid-IR detection.

Increasing the bias increases the magnitude of the response but it does not change its spectral shape. The relative responsivity as a function of temperature is shown in Figure 4b and is obtained by the integral of the photocurrent spectrum. At sufficiently high temperature, the responsivity reaches a regime where it is independent of the detector temperature. However, at a low enough temperature, the responsivity decreases exponentially.

The responsivity is proportional to the exciton ionization efficiency of a single dot multiplied by the gain over the structure which is the ratio of the carrier lifetime over the carrier transit time, [START_REF] Rosencher | Optoelectronics[END_REF] as given by equation (2 (2).

transit LT R     (2)
Future measurements of each of these parameters will be needed. At present we speculate that the ratio of carrier lifetime and transit times is a constant independent of temperature and bias. This will be the case if carrier trapping plays a significant role. A possible example of traps could be very large dots with therefore vanishing gaps. In this case, the major contributor to the temperature variation of R is the ionization efficiency . We then consider that the ionization rate is thermally activated

) / exp( 0 T k E k k b I I - =
and competes with a fixed recombination rate r k due to radiative and nonradiative processes within one dot. The ionization efficiency is then

kT E Io r kT E I I e k k e / / QD single - - + =  (3)
Fitting the experimental data with such an expression leads to a strong discrepancy at the lowest temperature since it predicts 0 =  at 0K. Such a deviation may result from the distribution of activation energies. For a collection of dots, we choose a Gaussian energy distribution such that

dE E E E T E T i i         - - =    - 2 2 QD single ensemble 2 ) ( exp 2 1 ) , ( ) , , (       (4)
This provides a better fit with values of the mean activation energies of 160meV for sample A, 100 meV for sample B and 34 meV for sample C. The broadening of the ionization energies are within 20-50%. In this model, the saturation of the ionization efficiency at moderate temperature implies that the time required for charge separation becomes smaller than the recombination time, radiative and non-radiative, and therefore that the quantum yield of charge generation is already close to unity. This is compatible with the fast hopping time mentioned earlier at room temperature.

The origin of these activation energies for charge separation may lie in the disorder associated with the size inhomogeneity and in an energy barrier associated with charge transfer, even in the absence of disorder. To estimate the latter, we use Marcus theory [START_REF] Marcus | [END_REF] considering that the electron (or hole) transfers from an exciton in a sphere with binding energy Eex to a neighboring sphere.

The Exciton binding energy is given by 24 R e E ex

 2 8 . 1 =
, with R the radius of the dot and  their dielectric constant. Assuming a value of =20 in HgTe 25,26 , this is estimated as 32 meV and 24 meV for samples B (R =4 nm) and C (R=5.3 nm). The Marcus expression for the energy barrier is

2 0 0 0 ) 1 ( 4   ex E G + = 
where 0 is the reorganization energy. For electron transfer in a two-sphere

model         -       + - =  0 2 0 1 1 1 1     R R e
,where R is the dot radius, and  is the matrix thickness between the dot surfaces estimated as ~ 1 nm for the dodecanethiol. The dynamic and static dielectric constants,   and 0  , refer to the matrix. This is unknown for the matrix material, mostly bound alkane thiol, but we take

C. RESPONSIVITY

Using the I(V) curves under illumination, as described in reference 1410, we can extract the absolute responsivity of our detectors. To proceed, we first identify a temperature at which the dark current is far below the photocurrent in our range of flux (room temperature for the A sample, 190K for the C sample). We then measure two I(V) curves under illumination at this temperature, for two different amounts of incident flux. This latter is evaluated as in equation (3) (3)

( )  -   = cutoff BB d e hc A T kT hc d BB            min 5 2 2 , 1 1 2 sin cos , , (3)
Where Ad is the active area of the sample (0.1 cm 2 ), α is the half angle of view of the detector and β is the angle between the normal of the detector and the incident light, h is the Planck constant, c the speed of the light, and TBB the equivalent black-body temperature of the light source.

Finally the responsivity is given by equation (

D. NOISE AND DETECTIVITY

The spectrum analyzer provides the voltage noise density (vn). The current noise density (in) which is the useful quantity to evaluate the detectivity, is then obtained by

BW G v i n n  =
, where G is the current amplification and BW is the bandwidth of the spectral acquisition.

Mis en fo

Mis en fo

The noise for sample C is plotted in Figure 6Figure 6 (a) and (b). At all biases and temperatures, a non-white noise prevails at low frequency, and follows a power law with an exponent in the -0.4 to -0.7 range consistent with 1/f noise. As temperature and bias increase, the dark current and the noise both increase.

1/f noise can be modeled using the Hooge's law 27 ,

f N I i H f   = 2 2 / 1  (5)
where N is the number of carriers, I the average current through the sample, f the frequency and αH is a parameter which describes the magnitude of the 1/f noise. Hooge reported that in most metallic materials, this parameter is rather close to 2×10 -3 . While we do not know of reported values of H  for bulk HgTe, H  has a value in the 10 -3 to 10 -5 range for HgCdTe detectors 28,29 (with a cadmium content in in the 0.17 to 0.3 range), also much smaller than observed here. We than they need to be. One possibility is that the sample inhomogeneity induces current crowding 30,31 such that the actual number of carriers should not be related to the total volume of the sample. Whether the noise level is intrinsically related to the nanoparticle assembly or more

Mis en fo

Mis en fo specific of the particular samples that we have studied, and in particular the aggregation discussed in section I, Figure 1, remains unknown and is a target of further study.

Finally, by combining the responsivity and noise measurements we evaluate the specific detectivity of the samples, using equation ( 6) (6) .

n d i A R D = * (6)
Maps of the detectivity as a function of the operating temperature and applied bias are presented in Figure 7Figure 7.

These maps show that the HgTe CQD material is able to reach detectivity above 10 9 Jones. The optimal operating temperature is just below room temperature (270K) for sample A (λcut-off = 2.8µm) and is around 130K for sample C (with a λcut-off ≈ 6µm at this temperature). The applied optimal biases are respectively for sample A and C 19V (corresponding to an electric field of 8kV cm -1 ) and 8V (19kV cm -1 ), which is above the applied bias of the regular read-out circuit (generally a few volts). Nevertheless this optimal bias can easily be reduced by changing the spacing of the electrodes.

E. INFLUENCE OF THE THICKNESS

The thickness of the CQD film is an important parameter determining some of the detector properties. In particular, based on bulk optical properties of HgTe, the absorption depth of films of CQDs is expected to be around 5 microns, which is an order of magnitude larger than the thickest samples studied. Correspondingly, the optical densities of the samples were rather small, Mis en fo of the order of 0.1. Thicker films will directly lead to improved responsivities. While making uniform thick films will require a dedicated effort, it is interesting to explore how the response of thinner films varies with thickness. Therefore, three samples based on sample C of respective thickness 15, 35 and 450nm have been studied. Their properties are summarized in Table 2Table 2. The thickness is measured using an Atomic Force Microscope, scanning a groove made with a sharp razor blade. Even the thinnest film provides a measurable mid-IR photoresponse, while being very transparent and only a couple layers of dots. Generally the thicker the film the smaller the resistance, see Figure 8Figure 8 (a). An increase of the thickness also results in greater absorption which leads to a higher responsivity (Figure 8Figure 8(c)). In parallel, the noise also increases with sample thickness. According to Hooge's expression, we expect the current 1/f noise to increase as the

Mis en fo

square root of thickness while the responsivity increases linearly with the thickness as long as it remains much smaller than the absorption length. Therefore, there should be a net increase of the detectivity with square root of the thickness. A square root fit is then used as a guide to the eye in Figure 8Figure 8 (e).

The measurement of the transient photocurrent has been used to probe the density of states inside the gap of several materials. 32,33 Here, we focus on the transient photocurrent decay, which is related to the cut-off frequency of the detector. Surprisingly, an increase in film thickness gives a faster response of the detector, see Figure 8Figure 8 (b). It is also observed that an increase of the bias leads to a faster response, both for the rise and decay time while the incident beam energy only slightly affects the transient shape. Increasing the operating temperature of the detector, from 250K to 295K, also leads to a faster response. While these effects are not completely understood, they likely arise from capacitance associated with space charge at the contacts 34 .

While this capacitance is constant for a given electrode geometry, the resistance of the devices drops with increasing film thickness, bias and temperature, leading to a decreased RC time constant. Compared to PbS CQD detector 9 operating in the near-IR, the HgTe times response are much faster but the responsivities are also lower, in accordance with equation (2(2).

Since the optimal detectors need to have thicknesses larger than the thickest one studied here in order to saturate the absorption, we expect that the speed of the detector will be much faster than required for usual imagery application for which the typical integration time is in the 1ms range.

IV. CONCLUSION

The conductivity and mid-infrared response photoconductivity of HgTe CQD solid films are investigated as a function of temperature. By changing the size of the particle and the temperature, the cut-off wavelength has been tuned from the near infrared up to 7µm. The measured responsivity already competes with existing devices such as QWIP 35 or type II superlattices 36,37 and further improvements are expected with optimized thicker films.

The dark current is thermally activated with an activation energy close to half of the energy gap.

This suggests that, at room temperature, most carriers are thermally generated. From the estimated carrier density, the mobility is extracted and is relatively large at room temperature, of the order of 0.1-1 cm 2 V -1 s -1 , especially given that the material is a simple drop-cast film of CQDs without any attempt to eliminate or replace the ligands. These mobility values will need to be confirmed by direct measurements. We observe that the temperature dependence of the responsivity is much weaker than that of the dark current, being essentially constant around room temperature and dropping only at lower temperatures. We interpret this behavior as arising from already efficient charge separation at room temperature. The films show 1/f noise at low frequency with values of the Hooge's constant that are two orders of magnitude larger than for clean homogeneous metals, although the noise decreases with temperature leading to an optimum operating temperature. The best detectivity values for a 450 nm thick film are ~ 2×10 9 jones at 130K and the fastest response times are below 100 ns at room temperature. We stress again that these results have been obtained with no processing of the films other than simply drop-cast from solutions.

Since the mobility and carrier generation efficiency are already high it seems that further improvement will rather come from the improved optical absorption edge, longer carrier lifetime, and reduced noise. Improved optical properties come from thicker films, and materials with improved monodispersivity. Higher monodispersivity will also reduce the concentration of large dots which may act as recombination centers. Reducing the noise will involve investigating the effect of film processing to possibly reduce the crowding effect which maybe the source of excess 1/f noise. It is expected that detectivities above 10 10 Jones in the mid-IR will be achievable while maintaining a fast response time and moderate cooling temperatures consistent with imaging operation.

V. ACKNOWLEDGEMENTS

The

  Figure 1Figure 1 gives a transmission electron microscopy (TEM, Tecnai F30, 300kV) image showing aggregated particles. Dynamic light scattering (DLS, Malvern, Zetasizer) measurements suggest the nanocrystals are similarly aggregated in colloidal solution. The aggregate size distribution follows approximately a log normal distribution typical, with a mean size which is typically 20 times larger than the single dot diameter. However the optical absorption shows confinement which varies only with the dimensions of the individual particles in the aggregates. Films of dots are obtained by drop-casting the TCE CQD solution on interdigitated Pt electrodes on a glass substrate (ABTech IME 1050, 50 periods, 100nm thick, 10 µm width and spacing).

Films

  of HgTe CQD film studied here are a few hundred nanometers thick and we also investigated thinner films made by using more dilute solutions. For all the following measurements, the samples are mounted on the cold finger of a closed cycle He cryostat (Coldedge SDRK 101D) operating between 3 and 300K. Three samples are under investigation in this study with the following cut-off wavelength at room temperature 2.8µm (sample A), 3.4µm (sample B) and 5.3µm (sample C). B. EXPERIMENTAL SETUP I(V) curves are measured using a Keithley 6487 pico-ammeter. The sample is illuminated by a tungsten light-bulb. Its temperature, measured with a pyrometer, is adjusted in the 1400 to 2100°C range, giving an incident power of a few mW on the sample. A home-made Michelson interferometer is used to measure spectral responses. A SiC lamp (equivalent to a 1200°C blackbody) serves as broadband light source and is chopped at a frequency in the kHz range. The detector signal is amplified using a Keithley 6487 and a PAR model 124 lock-in. The transient response of the detector is obtained by illuminating the sample with short pulses (10ps, 25Hz) of a Nd:YAG laser at 1.06µm. The transient photocurrent is recorded using a 100MHz digital oscilloscope (Tektronix, TDS 1012B).

  the hopping time between nanocrystals and the mobilities are related by ps Hopping times faster than recombination times are helpful for efficient operation.

.

  This gives energy barriers of 32 meV and 25 meV for samples B and C respectively. Energy disorder due to size inhomogeneity could easily be a larger effect, and may explain the discrepancy with measured values above.

  Figure 4Figure 4 (b), for the other temperatures. Finally the responsivity as a function of the

  Figure 6Figure 6 (c) for f=10Hz. The number of carriers can be evaluated as the product of the carrier density, which was estimated in table 1, and the volume of the sample. This gives 4×10 10 carriers for sample C. Since the value of N H  is typically 10 -11 for sample C, H  is therefore around 0.4, two decades above the Hooge value. The larger value of the Hooge constant therefore suggests that our CQD samples are possibly about 10 fold noisier

  research was supported by the US National Science Foundation NSF under Grant No DMR-070626 and by DOE under Grant No. DE-FG02-06ER46326. The authors made use of shared facilities supported by the NSF MRSEC Program under DMR-0820054. EL thanks Ecole Polytechnique, Palaiseau, France, for a postdoctoral fellowship.

Figure 1 :

 1 Figure 1: (a) TEM image of the HgTe material. (sample C) Scale bar is 10 nm. The inset shows a

Figure 2 :

 2 Figure 2: Dark current for sample C, as a function of the inverse of the temperature, for three

Figure 3 :

 3 Figure 3: Normalized spectral response for the sample A (top), B (middle) and C (bottom) for

Figure 4 (

 4 Figure 4 (a) Position of the long-wavelength half maximum of the spectral response as a function

Figure 5 :

 5 Figure 5: Responsivity for samples A, B and C as a function of the applied bias for different

Figure 6 :

 6 Figure 6: (a) Current noise spectral density for a 2V bias at temperatures: 70, 110, 150, 190, 230,

Figure 7 :

 7 Figure 7: Map of the specific detectivity as a function of temperature and applied bias for sample

Figure 8 :

 8 Figure 8: (a) Dark current as a function of the applied voltage for three films. (b) Transient

Table 1 :

 1 Activation energies for the dark current of the three samples. This value is compared to

	the half value of the optical gap at room temperature. N0 is the estimated random close packed
	density of CQDs with the range based on a 20% standard deviation of the diameter. The carrier
	density, n, is estimated based on a model for intrinsic thermal activation. .
	Sample λcut-off	λcut-off
	(295	(70K)
	K)
	(µm)

(µm) Ea (meV) EG/2 (meV) From optical measurement N0 (×10 18 cm -3 ) n (×10 16 cm -3 ) α (10 -4 eV.K -1 ) β (K) EG(0K) (eV)

									From
									Varshni's
									fit
	A	2.8	3	273	221	2.6-7.6	0.08-0.3 1.5	500 0.43
				±10	±44
	B	3.4	4.35 201	182	1.5-5	0.2-0.8	7	500 0.33
				±10	±36
	C	5.3	6.6	106	117	0.6-2	1-4	7	500 0.23
				±10	±20

Table 2 :

 2 Properties of the three films based on sample C (λcut-off≈5µm) and of different thicknesses. The time response is the 1/e decay time.

		Thick	Medium film Thin film
		film
	Appearance	Brown	Barely	Colorless
			colored
	Thickness	450	35	15
	(nm)
	Dark Current,	22	0.56	0.031
	at 10V (mA)
	Resistance	0.5	17	320
	(room T)
	at 10V (kΩ)
	Responsivity	1	7%	0.2%
	(normalized
	by thick film
	value)
	Decay time	63	2×10 3	5×10 5
	(ns)