M Thual

P Chanclou

J Lostec

A Laurent

H Poignant

Micro-optics and Bragg Gratings for DWDM Source Array

Keywords:

A collective process is presented to achieve a four channels fibres ribbon with long working distance micro-optics (46  0.7 µm) and UV-induced Bragg Gratings. Four different external cavity single-longitudinal mode sources are obtained with this fibres ribbon in front of a Fabry Perot Laser Diode (FP/LD) at a wavelength around 1300 nm with Side Mode Supression Ratio of 44.35  1.85 dB, 3dB bandwith of 76  1.7 pm (13.7 GHz) and Relative Intensity Noise around -138.6  2.8 dB/Hz. This is promising for assembling multichannel multiwavelengths emitters for Dense Wavelength Division Multiplexing (DWDM) networks.

Introduction

The use of UV-written Bragg reflector in telecommunication SingleMode Fibre (SMF) to obtain single-longitudinal mode external cavity LD has already been demonstrated [START_REF] Gall | 24th European Conference on Optical Communications, ECOC'98[END_REF] [2] [3] [4]. These kinds of sources require highly efficient coupling and low parasitic reflections on LD and fibre [START_REF] Gall | 24th European Conference on Optical Communications, ECOC'98[END_REF]. Elsewhere the interest of collective and passive assembling of LD array and SMF ribbons to reduce the cost and size of emitters at the central office has been shown [5]. We have reported recently highly efficient and long working distance collective micro-optics suitable for those multichannel modules [6]. In this letter, we propose to investigate the association of both UV-writing and micro-optics fabrication technologies on fibre ribbon to achieve a multiwavelengths multichannel external cavity laser diode source (Figure 1) for DWDM applications. These micro-optics are a compromise between the coupling loss inside the cavity and tolerance to misalignment for collective assembling [6]. This device is expected to be less sensitive to temperature than Distributed Bragg Reflector laser diode due to the lower temperature dependence of the Bragg grating in silica, 15 pm.K -1 , compared with 100 pm.K -1

for DBR laser diode.

Fig. 1 Laser Diode array and SMF ribbon with micro-optics and Bragg gratings.

Fabrication process

Bragg reflectors centred on appropriate wavelengths (1=1280, 2=1282.46, 3=1284.76, 4 =1286.81 nm) were first UV-written within each of the standard ribbon including four 250 µm spaced SMF. The UV exposure was achieved using a transverse holographic arrangement (Lloyd mirror) [START_REF] Poignant | European Workshop on Bragg Grating Reliability[END_REF] involving a frequency doubled Argon ion laser (Sabre Fred, from Coherent), operating at P=244 nm. 100 mW output powers were used for this experiment. Prior to the irradiation, the fibre ribbon was loaded with hydrogen for 20 days at room temperature at a pressure of 150 atmospheres, in order to increase the writing efficiency. The Bragg grating wavelength B is fixed by the angle  between the laser beam and the mirror and by the relation :

    sin ). ( n P eff B 
, where neff()

is the fibre core effective index at . The gratings (50% reflectivity, 10 mm long) were written successively by moving the cylindrical lens by means of a micro-translation stage ( see the set-up, Figure 2).



Fig. 2 Bragg gratings writing process.

Secondly micro-optics called GRADISSIMO were made at the end of this fibre ribbon. They are based upon a GRADed Index section, which acts as a graded index (GRIN) lens, aligned and maintained with a coreless Silica section at the appropriate distance from the SIngle MOde fibre (Fig. 1). The fabrication process which has been described elsewhere [6] [8] [9] is briefly sketched in Figure 3. It consists in splicing graded index and silica fibre ribbons with an arc-discharge fibre ribbon splicer.

Then the coreless silica ribbon is cleaved to obtain a length Ls of silica. These silica sections are spliced in ribbon to the SMF ribbon. Finally the GRIN ribbon is cleaved to obtain GRIN sections of the desired length Lg. The process is collective as only ribbons of GRIN, silica and SingleMode Fibres are clived and spliced.

Fig. 3 Micro-optics collective fabrication process.

The 125 µm outer diameter is maintained all along the fibres and the pitch between fibres of the ribbon remains 250 µm. So these ribbons with micro-optics and Bragg reflectors are suitable for hybridisation in V or U grooves on silicon motherboards. The return loss of this micro-optics without any Anti-Reflection (AR) coating, >25 dB, is higher than the return loss of a cleaved SMF (14.7 dB). This relative high return loss associated with a long working distance (around 50 µm) are very interesting features to achieve an external cavity source.

Experiments and results

Each fibre of this non-coated ribbon is then successively aligned in front of a FP/LD operating near 1.3 µm, whose front facet is AR coated (10 -4 parasitic reflection). So, the 18 mm long resonant cavity is composed of the non-coated LD back facet and the Bragg reflector (see Fig. 1). Light emitting half angles (and radius mode) of LD, at 1/e² of the maximum intensity, are 25° (=0.89 µm) and 30° (=0.71 µm) in the parallel and perpendicular directions against the epitaxial surface, respectively. The different characteristics measured are summarised in Table I.

Table I. Measured results for each fibre in front of the LD.

Due to micro-optics, the working distances between the LD and fibres are homogeneous (46  0.7 µm) and long enough to reduce drastically parasitic reflections on fibres compared with SMF butt coupling. The optical powers are also uniform around 3.18  0.14 mW (4.5 dB loss compared with 10.5 dB for butt coupling) for a 60 mA LD bias current at 22°C. Notice that the same micro-optics in front of tapered LD leads to 1.07 dB loss. We obtain a single-longitudinal mode emission in each fibre (Figure 4), with a wavelength spacing between two neighbouring channels around 2.3 nm (421.1 GHz) from 1280 to 1286.81 nm, a Side-Mode-Suppression Ratio (SMSR) of 44.35  1.85 dB, and a 3 dB bandwidth (FWHM) of 76  1.7 pm (13.7 GHz). The Relative Intensity Noise (RIN) [10] is measured between -141.7 and -135.8 dB/Hz for the four fibres. As a comparison we have measured the RIN of a DFB laser diode around -155 dB/Hz. We could reduce the RIN of the external cavity LD by using a more selective Fibre Bragg Grating. The measurement precisions are  2.5 µm for the working distance,  0.1 mW for the power,  20 pm for the wavelength,  0.5 dB for the SMSR,  2 dB/Hz for the RIN and  3 pm for the bandwith.

The experiment was made around 1300 nm because componants were available in the laboratory at this wavelength but the same experiment could be reproduced at a wavelength of 1550 nm.

Conclusions

We have presented four single-longitudinal mode external cavity sources (SMSR= 44.35  1.85 dB, RIN around -140 dB/Hz), about 2.3 nm spaced wavelengths around 1284 nm and uniform optical powers (3.18  0.14 mW), based on a four channels fibres ribbon. Both grating fabrication process and micro-optics arrangement are performed collectively on this fibre ribbon. This method can be applied successfully to hybridisation on silicon motherboards, which is promising to reduce the assembling cost of multiwavelengths sources for DWDM purposes [START_REF] Hunziker | Circuits and Devices[END_REF]

Fig. 4

 4 Fig.4 Spectrum in each channel (span 17 nm).

  .

			Working	Power	 (nm) SMSR	RIN	FWHM
			distance	(mW)		(dB)	(dB/Hz)	(pm)
			(µm)
	Fibre 1	46,7	3,32 1279,97 44,1 -139,9	76
	Fibre 2	45,4	3,24 1282,44 46,2 -141,7	73
	Fibre 3	46,5	3,04 1284,84 45	-135,8	74
	Fibre 4	46,6	3,08 1286,84 42,5	-137	74
		1270	1275	1280		1285	1290	1295
		0		1	2	3	4
	Power (dBm)	-20	S M S R
		-40
		-60
				Wavelength (nm)

Acknowledgments

The authors would like to thank J.F. Bayon for loading ribbon with hydrogen before UV-writing, J. Landreau for anti-reflection coating, N. Devoldere, A. Le Corre, M. Gadonna and D. Pavy for fruitful discussions.


Rémy Müller
email: remy.muller@ircam.fr

Thomas Hélie

FULLY-IMPLICIT ALGEBRO-DIFFERENTIAL PARAMETRIZATION OF CIRCUITS

This paper is concerned with the conception of methods tailored for the numerical simulation of power-balanced systems that are well-posed but implicitly described. The motivation is threefold: some electronic components (such as the ideal diode) can only be implicitly described, arbitrary connection of components can lead to implicit topological constraints, finally stable discretization schemes also lead to implicit algebraic equations.

In this paper we start from the representation of circuits using a power-balanced Kirchhoff-Dirac structure, electronic components are described by a local state that is observed through a pair of power-conjugated algebro-differential operators (V, I) to yield the branch voltages and currents, the arc length is used to parametrize switching and non-Lipschitz components, and a power balanced functional time-discretization is proposed. Finally, the method is illustrated on two simple but non-trivial examples.

INTRODUCTION

Network analysis of circuits and expression of Kirchhoff laws, naturally leads to implicit differential algebraic equations (DAE). Indeed in the most general form, the branch equations are not described by functions but by relations (in the voltage-current plane for algebraic components, voltage-charge for capacitor, currentflux for inductors . . . ). One of the most general approach is the Sparse Tableau analysis [START_REF] Hachtel | The sparse tableau approach to network analysis and design[END_REF] which involves both the nodes and branch variables.

In the study of power-balanced systems, and more generally in the field of geometrical numerical integration, one is not only concerned with the quantitative accuracy of numerical simulations, but also with the qualitative preservation of structural invariants during discretization [START_REF] Celledoni | Energy-preserving and passivity-consistent numerical discretization of port-Hamiltonian systems[END_REF]. It has been shown that the symplectic structure of Hamiltonian systems, responsible for energy preservation, can be generalized to open systems with algebraic constraints by the notion of a Dirac structure [START_REF] Dorfman | Dirac structures of integrable evolution equations[END_REF] [START_REF] Courant | Action hamiltoniennes de groupes[END_REF]. It can even be extended to infinite-dimensional systems such as partial differential equations using a Stokes-Dirac [START_REF] Kotyczka | Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems[END_REF] structure. It has been shown in [START_REF] Van Der Schaft | Port-Hamiltonian systems theory: An introductory overview[END_REF] (see also [START_REF] Falaize | Passive guaranteed simulation of analog audio circuits: A port-Hamiltonian approach[END_REF] [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF]) that Kirchhoff laws generates a Kirchhoff-Dirac structure. Recent work [START_REF] Mehrmann | Structure-preserving discretization for port-Hamiltonian descriptor systems[END_REF] also study the properties and numerical discretization of Port-Hamiltonian DAE systems in descriptor form.

Usually, when possible, DAE are reduced to ordinary differential equations (ODE) or semi-explicit index-1 DAE [START_REF] Holters | A generalized method for the derivation of non-linear state-space models from circuit schematics[END_REF] [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF] for which a rich literature of results from system theory and numerical analysis is available to study stability, conservation laws, attraction points, existence and uniqueness of solutions . . . In these reduction processes, a choice has to be made regarding the variables the system is solved for. Choosing the node voltages leads to the Nodal Analysis (NA) method. But it is not sufficient to represent all systems, adding some branch currents leads to the popular Modified Nodal Analysis (MNA) [START_REF] Ho | The modified nodal approach to network analysis[END_REF]. The importance of state variable choices for computable numerical simulations can be found in [START_REF] Serafini | State variable changes to avoid non computational issues[END_REF]. Similar issues are addressed for wave digital filters in [START_REF] Werner | Generalizing root variable choice in wave digital filters with grouped nonlinearities[END_REF]. A procedure to guide these choices is the Sequential Causality Assignment Procedure (SCAP) in the bondgraph literature [START_REF] Breedveld | A systematic method to derive bond graph models[END_REF]. In the case of switching-circuits, such as those containing ideal diodes or discontinuous laws (see [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]) an approach is to solve for different variables according to the switching state of the system, but the number of such states becomes exponential in the number of switching components.

Since after time discretization, one is left with an algebraic system of (nonlinear) equations which has to be solved by an iterative scheme anyway, the goal of this article, is to propose a structure-preserving power-balanced numerical method capable of dealing with the implicit nature of the network equations.

Section 2 recalls how any electronic circuit can be represented by a Kirchhoff-Dirac structure, uniquely determined by the circuit's incidence matrix. Section 3 describes how to parametrize the (possibly implicit) relation imposed by any circuit component. Power-conjugated voltages and currents (v, i) are obtained by the application of a pair of nonlinear algebro-differential operators (V, I) to a parameter x which stands for the component's local state. In Section 4 arc-length and pseudo arc-length parameterizations 1 are proposed to overcome computational causality problems that arise in switching components and reduce numerical stiffness caused by high Lipschitz constants. In Section 5 a power-balanced and structure preserving time-discretization is presented using a functional framework. This leads to a nonlinear system of algebraic equations which is solved using Newton iteration. Finally two tests circuits are studied in Section 6, a stiff switching diode clipper and a conservative (nonlinear) LCLC circuit with an implicit topological constraint.

KIRCHHOFF-DIRAC STRUCTURES FOR CIRCUIT GRAPHS

From a circuit theory perspective, a Dirac structure is simply a multi-port that doesn't generate or dissipate power i.e.

P = i | v = 0.
Considering components and their interconnections separately, because of Kirchhoff laws, the multi-port connecting all components 1 Curvilinear coordinates for multi-ports are possible but not addressed. (the PCB) is necessarily a Dirac structure. To formalize it for circuits, we borrow and slightly adapt the notations from [START_REF] Van Der Schaft | Port-Hamiltonian systems theory: An introductory overview[END_REF] [5] [START_REF] Mehrmann | Structure-preserving discretization for port-Hamiltonian descriptor systems[END_REF].

DAFx

Circuit Graphs

A directed circuit graph G(N , E) is defined by a set of n nodes N = {η1, . . . , ηn} and a set E = { 1, . . . , m} of m directed edges (links, branches) with no self-loops. Edges are ordered pairs of nodes i = (ηi,0, ηi,1). Over each node (k = 0) and edge (k = 1) 2 , using the receiver convention for both, we define conjugated current and voltages

(i k , v k ) ∈ I k × V k =: B k , k ∈ {0, 1} (1)
where V0 ∼ R n , V1 ∼ R m are the spaces of voltages over the nodes N (resp. the edges E) and I0 = V * 0 , I1 = V * 1 the dual spaces of functionals V0 → R, V1 → R. The spaces B0 and B1 are respectively the spaces of bonds corresponding to the nodes and edges such that power is given by the duality pairings

i k | v k B k := i T k v k , k ∈ {0, 1}. (2)
Note that since the spaces are finite-dimensional, one can identify each space with its dual V0 ∼ I0 = R n , V1 ∼ I0 = R m . Furthermore, the directed graph is uniquely specified by its (reduced) co-incidence matrix D given by

D = [dij] m×n , di,j =      1 i,1 = ηj -1 i,0 = ηj 0 otherwise . (3)
Kirchhoff Current (KCL) and Voltage laws (KVL) 3 can be expressed with an elegant duality (see [START_REF] Chua | Linear and nonlinear circuits[END_REF] p.710) using the incidence and coincidence matrices by

v1 = Dv0, i0 = -D T i1 = 0. (4)
i.e. we have the following diagram.

v0 ∈ V0 v1 ∈ V1 i0 ∈ I0 i1 ∈ I1 D • | • B 0 • | • B 1 -D T (5)
2 This notation is convenient to make the link with automated circuit to Bond-graph algorithms [START_REF] Breedveld | A systematic method to derive bond graph models[END_REF]: 0-junctions (shared voltage, parallel connection) for nodes and 1-junctions for branches (shared current, serial connection) see Figures 5 and6 for examples. It is also a mnemonic to remember that lumped circuit equations arise from the spatial discretization of electro-magnetic 1-forms for branches and 0-forms for nodes. 3 The minus sign in front of i 0 comes from the consistent use of the receiver convention for both nodes and branches: the sum of edge currents i 0 entering each node has to be zero.

Kirchhoff-Dirac structure

Written in matrix form, one obtains the canonical Kirchhoff-Dirac structure D (with a structure very similar to the ones obtained for partial differential equations (PDE) [START_REF] Jacob | Linear port-Hamiltonian systems on infinite-dimensional spaces[END_REF] [START_REF] Kotyczka | Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems[END_REF])

D :   i0 v1   =   0 -D T D 0     v0 i1   , i0 = 0. (6)
i.e. Kirchhoff Current Laws can be interpreted as zero boundary conditions on the node currents, and the co-incidence matrix D as a (lumped) differential operator. Left multiplying by [v0 i1], the duality products and skew-symmetry leads to the power balance

P = i0 | v0 + i1 | v1 = v0 i1   0 -D T D 0     v0 i1   = 0.
Furthermore since we have conservation of charge i0 = 0 on the nodes N , this yields the Tellegen theorem over the edges 4 E

i1 | v1 = ∈E i | v = 0.
We also remark that the node voltages v0 can be interpreted as Lagrange multipliers parametrizing the sub-manifold defined by the linear constraints i0 = 0.

(Reduced) Hybrid Dirac structure

Whereas MNA solves the system for node voltages and branch currents, in Hybrid Analysis [START_REF] Chua | Linear and nonlinear circuits[END_REF] and skew-gradient DAE [START_REF] Falaize | Passive guaranteed simulation of analog audio circuits: A port-Hamiltonian approach[END_REF] [8], the node voltages are eliminated. First a spanning tree T is chosen, this yields a partition of the branch currents and voltages into tree (vT , iT ) and link variables (vL, iL). Partitioning equations according to the spanning tree, Kirchhoff laws (4) are rewritten as

  vT vL   =   DT DL   v0, D T T D T L   iT iL   = 0. (7)
From graph theory, having a spanning tree ensures that the matrix DT ∈ R n×n is invertible. So we can eliminate the node voltages v0 using v0 = D -1 T vT . This yields a reduced Hybrid Dirac structure specified by its link-cutset matrix

C = (DLD -1 T ) T D :   iT vL   =   0 -C C T 0     vT iL   . (8)
Traditionally, the spanning tree is chosen to be a proper tree (i.e. containing all current-driven branches: Voltages Sources, Capacitors, . . . ) such that vT is current-driven by iT (i.e. computable from iT ). However topological constraints such as in example 6.2 may prevent a proper tree to be found. Since the proposed method is fully-implicit by nature, it does not have such a requirement. Either the Kirchhoff-Dirac structure or any reduced Hybrid Dirac structure can be used for simulation.

For a formal definition of Dirac structures in the broader context of multi-physical networks, pleaser refer to [START_REF] Van Der Schaft | Port-Hamiltonian systems theory: An introductory overview[END_REF] and references therein. A generic example of a Dirac structure and its graph, emphasizing the node-edge incidence structure, is shown on Figure 1. Detailed case-study are shown on Figures 5 and6 and studied in Section 6.

ALGEBRO-DIFFERENTIAL PARAMETRIZATION OF COMPONENT LAWS

From now on, for functional discretization purpose, we adopt a Hilbert space viewpoint, and lift Dirac structures over time steps. Consider a time interval Ω ⊂ R, the branch voltage and current spaces are lifted to the dual Hilbert spaces I1 ∼ V1 ⊆ L 2 (Ω) m (L 2 being a pivot space) equipped with the inner (duality) product

u | v := 1 |Ω| Ω u(t) T v(t) dt. (9)
We assume that branch equations can be parametrized locally by a state x ∈ X1 ⊆ L 2 (Ω) m , nonlinear differential-algebraic operators I1 : X1 → I1, V1 : X1 → V1 and a law

F : X1 -→ B1 := I1 × V1 x -→ (I1(x), V1(x)) . ( 10
)
Likewise the KCL node boundary conditions ( 4) can be parametrized by the vector of node voltages λ ∈ X0 ⊆ L 2 (Ω) n and the linear constraint

B : X0 -→ B0 := I0 × V0 λ -→ (I0, V0)(λ) = (0, λ) . ( 11
)
Composing ( 6) with ( 10) [START_REF] Ho | The modified nodal approach to network analysis[END_REF] we obtain the fully implicit algebrodifferential formulation of a Port-Hamiltonian system (PHS)

Σ =    (I0, V0, I1, V1)(x, λ) ∈ B1 × B0; N (x) = 0, ∀(x, λ) ∈ X1 × X0    (12)
defined by the operator

N : X0 × X1 → L 2 (Ω) m+n N (x, λ) =   0 V1(x)   -   0 -D T D 0     λ I1(x)   . ( 13
)
For the reduced Hybrid Dirac structure one gets

Σ = (IT , VT , IC , VC )(x) ∈ B1| N (x) = 0, ∀x ∈ X1 (14) with the algebro-differential operator N : X1 → L 2 (Ω) m N (x) =   IT (x) VC (x)   -   0 -C C T 0     VT (x) IC (x)   . ( 15
)
We note that for differential components, the state space is given by the Sobolev space X ⊆ H 1 (Ω) ⊂ L 2 (Ω) defined by

X = x ∈ L 2 (Ω) ẋ ∈ L 2 (Ω); x(t) = x0 + t 0 ẋ(s) ds , (16
) whereas for algebraic components, no additional smoothness is implied so X ∼ L 2 (Ω).

The differential-algebraic operators corresponding to common electronic components are summarized in Table 1 and the case of implicitly parametrized algebraic components is now further detailed in Section 4.

(PSEUDO) ARC-LENGTH PARAMETRIZATION

We study here implicit arc-length and pseudo arc-length parameterizations of algebraic components whose laws cannot be represented as functions of either current or voltage (or such that unbounded Lipschitz constants may cause numerical problems during simulations). As an example we consider the cases of the ideal diode, a nonlinear resistor and the Shockley diode.

The ideal diode

An ideal diode law is determined by the set (see [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF])

RD =    (v, i) ∈ R 2 v = 0 i ∈ R + , i = 0 v ∈ R -.    (17)
It has the numerical disadvantage of being alternatively voltage and current controlled. In the hybrid formulation, computational causality assignment [START_REF] Breedveld | A systematic method to derive bond graph models[END_REF] would imply that a different Dirac structure such as ( 8) should be used according to the current state of the circuit. Furthermore, when the number of switching components grows, the number of switch configurations of the circuit grows exponentially. A solution around this problem is to consider the parametrization RD : λ → (VD(λ), ID(λ)) with arc-length

λ(v, i) = i/I0 v = 0, i ∈ R + v/V0 i = 0, v ∈ R -, (18)
for arbitrarily chosen positive reference current and voltages I0, V0.

Inverting the relation, one obtains the algebraic operators

VD(λ) = V0 min(λ, 0), ID(λ) = I0 max(λ, 0). (19)
with

V D (λ) = V0 • 1 R -(λ), and
I D (λ) = I0 • 1 R + (λ)
, where 1A(λ) denotes the indicator function of a set A.

Differential x V (x) I(x) H(x) Capacitor q q/C q q 2 /2C Inductor φ φ φ/L φ 2 /2L
Nonlinear Capacitor q ∇H(q) q H(q)

Nonlinear Inductor φ φ ∇H(φ) H(φ) Algebraic x V (x) I(x) P (x) Resistor i Ri i Ri 2 Conductor v v Gv Gv 2 Nonlinear Resistor i z(i) i i • z(i) Nonlinear Conductor v v z(v) v • z(v) Voltage source i V i V • i Current source v v I I • v
Table 1: Differential and Algebraic components. H (energy), P (power), q (charge), φ (flux), z (non linear function).

A Hard Clipping resistor

We now consider the case of a hard clipping resistor (it will be used in example 6.1) whose (v, i) graph is described by the set

RD =        (v, i) ∈ R 2 ;      i ∈ R -\ {0} v ∈ {-1} i ∈ {0} v ∈ (-1, 1) i ∈ R + \ {0} v ∈ {1}        . ( 20
)
We parametrize it continuously using (see Figure 4 page 8)

RD = (v, i) ∈ R 2 ; (v, i) = (V (λ), I(λ)), ∀λ ∈ R (21)
with the voltage and current operators

V (λ) = V0 clip [-1,1] (λ), (22)
I(λ) = I0 min(0, λ + 1) + max(0, λ -1) . ( 23
)
For arbitarily chosen positive reference voltage and currents V0, I0.

The Shockley diode

We finally consider the Shockley diode model 5 .

I(v) = IS exp v VT -1 , (24)
where IS is the saturation current, VT = k b T /qe the thermal voltage, with temperature T , Boltzmann constant k b and electron charge qe. It is C ∞ -continuous, but not globally Lipschitz. For a chosen reference resistance R0, the true arc-length of the graph (v, R0I(v)) is determined by dλ 2 = (1 + (R0I (v)) 2 ) dv 2 but it is not practical to manipulate. Instead, introducing the diode cutoff point (V0, I0) as the point of unit slope

R0I (V0) = 1, I0 = I(V0), (25)
where V0 = VT ln V T R 0 I S , I0 = VT /R0 -IS. Remarking that for v V0, dλ ≈ dv and for v V0, dλ ≈ R0I (v) dv, one can introduce the C 0 pseudo arc-length differential

d λ(v) = dv v < V0 R0I (v) dv v ≥ V0 . ( 26
) Integrating λ(v) := v 0 d λ one obtains the C 1 pseudo-arclength λ(v) = v v < V0 V0 + R0(I(v) -I0), v ≥ V0. ( 27
)
Inverting the relation leads to the algebraic operators

VD(λ) =      λ λ < V0, VT ln 1 + I0 + (λ -V0)/R0 IS λ ≥ V0, (28)
ID(λ) =      I(λ) λ < V0, I0 + λ -V0 R0 λ ≥ V0 . ( 29
)
5 Anti-parallel Shockley diodes will be simulated in example 6.1 such that by construction, Lipschitz constants are unitary (this property is key to deal with convergence and numerical stiffness)

LV = sup λ V D = 1, LI = sup λ R0I D = 1. (30)

FUNCTIONAL DISCRETIZATION AND NUMERICAL SOLVER

We now use the functional framework presented in Section 3 to discretize the system with a finite number of parameters per time step, (see the reference [START_REF] Dragotti | Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets strang-fix[END_REF] for the representation of non bandlimited signals having a finite rate of innovation).

Our time discretisation scheme can be interpreted as an extension of (spectral) time-finite elements methods [START_REF] Tang | Time finite element methods: A unified framework for numerical discretizations of ODEs[END_REF] to DAE. It is based on the following theorem which proves that a weak PHS is preserved over the chosen approximation subspace.

Theorem 5.1 (Weak PHS).

Let Ω be a time step, x ∈ X ⊆ L 2 (Ω) m a functional state, two operators b : X → L 2 (Ω) m , a : X → L 2 (Ω) m and a skew-symmetric matrix J defining the PHS operator

N (x) = b(x) -Ja(x) = 0, J = -J * . ( 31
)
Let P : L 2 (Ω) → R(P ) ⊆ L 2 (Ω) m be a projector (P 2 = P ) satisfying the skew-adjoint commutation P J = JP * , for the L 2 inner product (9), then the projected operator

P • N (x) = 0 (32)
defines a weak PHS which preserves the power balance.

a(x) P b(x) = 0. (33)
Proof. Using (32), taking the inner product with a(x), and using the fact that 1) P 2 = P (idempotence), 2) we have the commutation P J = JP * and 3) P JP * is skew-adjoint, we obtain a(x) P N (x) = 0 ⇐⇒ a(x) P b(x) = a(x) P J a(x)

1 = a(x) P 2 J a(x) 2 = a(x) P JP * a(x) 3 = 0.
Remark (Energy conservation). As an immediate consequence, for a conservative Hamiltonian system given by the operator

N (x) = dx dt -J∇H(x) = 0, J = -J T . ( 34
)
discretized such that ẋ = P J∇H(x), then the Hamiltonian energy H is preserved over a time-step Ω = (t0, t1),

H(x(t1)) -H(x(t0)) = 0. ( 35
)
Indeed, let b = d dt and a = ∇H, from the gradient theorem and using the same arguments as the previous proof, it follows that

H(x(t1)) -H(x(t0)) = ∇H(x) ẋ = ∇H(x) P JP * ∇H(x) = 0.

Piecewise constant and affine polynomial spaces

In this article we will restrict ourselves to constant and affine polynomial spaces P 0 , P 1 for which we have exact closed-form expression of the projected operators. (Higher-order polynomial spaces require the use of approximate quadratures rules [2] [9]). Results are exposed without proof except when the proof is not available elsewhere (see [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF]).

Consider a unit time step Ω = (0, 1), for the normalized time variable τ ∈ (0, 1) and two orthogonal polynomials

0(τ ) = 1, 1(τ ) = τ - 1 2 .
The operator PK :

L 2 (Ω) → P K (Ω) ⊂ L 2 (Ω), K ∈ {0, 1}
defined by

(PK u)(τ ) = K i=0 i(τ ) i | u i | i ( 36
)
is an orthogonal projector. i.e. PK is self-adjoint (PK = P * K ) and idempotent (P 2 K = PK ). For notational simplicity, we define the following notation. Let A : L 2 (Ω) → L 2 (Ω) be an operator, the projected operator ĀK : L 2 (Ω) → P K (Ω) is defined by

ĀK := PK • A, Ā := Ā0. ( 37
)
By extension, for a vectorized projector P := PK ⊗ In, it yields the projected PHS operator

N (x) := P • N (x) (38)
Because of the tensor product construction, we also have the commutation P J = JP = JP * such that P satisfies Theorem 5.1.

For numerical computations, it is necessary to compute the polynomial coefficients of the image of a trajectory through a nonlinear function. This is possible thanks to the following property Property 5.1 (Projected function). Let f : R → R be a semicontinuous function with known antiderivative F and a function

x(τ ) = 0(τ )x + 1(τ )δx ∈ P 1 (Ω), (39)
parametrized by its mean and variation Θ = (x, δx) ∈ R 2 Then the projected function P1 • f • x has the projection coefficients f : R 2 → R 2 defined by

f i := i | f • x / i 2 . ( 40
)
They are given in closed form by

f 0(Θ) =          F (x + δx 2 ) -F (x -δx 2 ) δx δx = 0 f (x + ) + f (x -) 2 δx = 0 (41) f 1(Θ) =      12 δx F (x1) + F (x0) 2 -F0(Θ) δx = 0 0 δx = 0 (42)
where

x1 = x + δx/2, x0 = x -δx/2.
Proof. See Appendix A.

Note that for a scalar (or separable) potential F , using f = ∇F , and x = (x0 + x1)/2, δx = x1 -x0 in property 5.1 yields the Average Discrete Gradient from [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF] (this is also an instance of anti-derivative anti-aliasing)

∇F (x0, x1) := f 0(Θ). (43)
Additional results for linear gradients are given in appendix B.

Newton iteration

For each time step Ω, let Θ denote the unknown parameters of a local state xΘ ∈ (P K (Ω)) m we look for a zero N(Θ ) = 0 of

N(Θ) := i N (xΘ) / i 2 i=0...K (44)
using Newton iteration (line search is not used in this paper)

Θκ+1 = Θκ + ∆Θκ, ∆Θκ = -N (Θκ) -1 N(Θκ). (45)
A detailed convergence analysis for the general case is out of the scope this paper and is left for future work. Please refer to [START_REF] Deuflhard | Newton methods for nonlinear problems: Affine invariance and adaptive algorithms[END_REF] for more details. When N is only semi-smooth which is the case of the ideal and hard clipping diodes, special care should be taken to ensure convergence using semi-smooth Newton methods [START_REF] Hintermüller | Semismooth Newton methods and applications[END_REF].

It should be noted that in piecewise constant spaces (k = 0), algebraic constraints simplifies to V(s) = V (s), I(s) = I(s), and one can compute the Jacobian from the derivative V , I . For affine trajectories (k = 1) one should use the results from properties 5.1 and the following property from [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF] to compute the coefficients and the Jacobian. Property 5.2. Given a potential F ∈ C 2 (R, R), and its discrete gradient ∇F (x0, x1) defined in Equation ( 43), the derivative of the discrete gradient with respect to x1 is

∂ ∇F ∂x 1 =          ∇F (x1) -∇F (x0, x1) x1 -x0 x0 = x1 1 2 ∂ 2 F ∂x 2 (x0) x0 = x1 . (46)
6. EXAMPLES

Diode Clipper

We consider the diode clipper circuit shown in Figure 5. This circuit which is dissipatively stiff because of the diode unbounded Lipschitz constant is commonly used to benchmark numerical schemes.

In this paper the nonlinear resistor D is considered abstract and will be substituted by anti-parallel Shockley and hard clipping diode models from Section 4.

Over each time step it is parametrized by the vector of Legendre coefficients Θ = (iS, iC , vR, xD) ∈ (R K+1 ) 4 , for K ∈ {0, 1} and the functional state

xΘ = [iS, iC , vR, xD] T ∈ (P K (Ω)) 4 such that each element v ∈ P K (Ω) is of the form v(t0 + hτ ) = K n=0 n(τ )v[n].
The projected Dirac structure (where 1 is the identity on R K+1 ) is then given by the operator

N =        iS iC vR VD(xD)        -        . . -1 0 . . 1 -1 1 -1 . . 0 1 . .               VS VC (iC ) IR(vR) ID(xD)        . (47)
Results are shown on Figure 2 for an input vS = V sin(2πf0t) with high input gain V = 10 4 , fundamental frequency f0 = 500 Hz, R = 1 kΩ , C = 10µF, IS = 100 fA, R0 = 0.1Ω, sampling frequency fs = 96 kHz. Anti-parallel Shockley diodes with arc length converge on average in 2 iterations and 4 times reduction of the worst-case iteration count (Newton tolerance r = 10 -5 ), Hard clipping diodes exhibit convergence in one iteration most of the time (2 when switching) even for r = 10 -10 .

LCLC circuit

We study here an LCLC circuit (shown on Figure 6) chosen to demonstrate the proposed method when the circuit is conservative, nonlinear and contains topological constraints (parallel capacitors, serial inductors . . . ). Here the circuit contains two inductors with the implicit topological constraint iL 1 = iL 2 .

In traditional solvers, such constraints usually needs to be detected and eliminated before proceeding to simulation. A possible approach is the use of equivalent macro components (see [22] [23]). In contrast, the proposed approach doesn't require such a preprocessing step, and keeps the modularity and sparsity of the component-based description. To demonstrate energy conservation, the capacitor C2 is chosen first with a linear law VC 2 (q) = q/C2 and an hardening nonlinearity VC 2 (q) = Vα sinh( q C 2 Vα ) with Vα = 1/30 (V).

Using the vector of Legendre coefficients as unknown Θ = (iC 1 , vL 1 , iC 2 , vL 2 ), we have the projected Dirac structure operator

N =        iC1 IL1(vL1) iC2 vL2        -        . . . 1 . . . 1 . . . 1 -1 -1 -1 .               VC1(iC1) vL1 VC2(iC2) IL2(vL2)        (48)
Simulation results are shown for the implicit and nonlinear LCLC circuit on Figure 3 for fs = 88.2 kHz, C1 = 20µF, C2 = 100µF, L1 = 1mH, L2 = 100µH, zero initial conditions and vC 1 (0) = 1V. We observe that both the algebraic constraint iL 1 = iL 2 and the conservation of total energy H are respected. Convergence is reached in 1 iteration for the linear case and between 1 and 2 iterations for the nonlinear one (relative tolerance r = 10 -5 ).

CONCLUSIONS

A new power-balanced, fully implicit component oriented method has been presented with a functional time-discretization. Its main strengths (not necessarily unique to this method) are: a) it retains the topological sparsity and modularity of the network based description, b) it is power-balanced and energy-conserving (including nonlinear components), c) it can deal with implicit topological constraints (capacitor loops, inductor cutsets) without the need of manual substitution of equivalent components, d) it can deal with implicit components including switching components, e) it uses finite-dimensional subspace projection as a unifying discretization tool common to ODE, PDE and DAE. f) Newton iteration converges faster using arc-length description of algebraic components with unbounded Lipschitz constants, Regarding perspectives, a detailed convergence study of the Newton iteration is needed (such as the one in [START_REF] Fontana | Newton-raphson solution of nonlinear delay-free loop filter networks[END_REF]), but has been postponed for future work. Using different and higher order functional approximation spaces is also an obvious perspective provided the projections can be computed exactly and efficiently. In particular, from a generalized sampling theory viewpoint, it would be interesting to perform a comparative analysis of implementation cost and convergence rate (to the true solution) between functional projection and oversampling. There is an apparent computational causality conflict shown in red on subfigure f): the loop current can either be controlled by L1 or L2 but not by both. The circuit has thus an implicit constraint IL 1 (φ1) = IL 2 (φ2). The inductor L1 is said to have a differential causality since vL 1 = φ1, whereas C1, C2, L2 are said to have an integral causality.

DAFx.8

. 1 Figure 1 :

 11 Figure 1: Dirac structure example with edges E = {A, B, C, D}, nodes N = {0, 1, 2} and chosen spanning tree T = {A, C}.

Figure 2 :

 2 Figure 2: Diode clipper: anti-parallel Shockley diodes (top) with V = 10 4 to emphasize Newton iteration differences, Hard clipping diodes (bottom) V = 10 2 to see dynamic and saturation.

Figure 3 :

 3 Figure 3: Conservative LCLC circuit: Linear (top) and Nonlinear (bottom). Notice the periodicity change and conserved energy.

Figure 4 :

 4 Figure 4: (Pseudo) Arc-length parametrization of hard clipping resistor and anti-parallel Shockley diodes.

Figure 5 :

 5 Figure 5: Diode Clipper circuit: From the schematic (a) Kirchhoff laws immediately yield the bond-graph (b) which can be reduced to the bond-graph c). Using the Graph incidence matrix d), one obtains the Kirchhoff-Dirac structure e). Elimination of the node voltages yields the reduced Dirac structure f).

Figure 6 :

 6 Figure 6: Conservative LCLC circuit (a single cell of a transmission line).There is an apparent computational causality conflict shown in red on subfigure f): the loop current can either be controlled by L1 or L2 but not by both. The circuit has thus an implicit constraint IL 1 (φ1) = IL 2 (φ2). The inductor L1 is said to have a differential causality since vL 1 = φ1, whereas C1, C2, L2 are said to have an integral causality.

Indeed (see[START_REF] Chua | Linear and nonlinear circuits[END_REF] p. 30) any two of KCL, KVL and Tellegen theorem implies the third one.

Proceedings of the 23 rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria,September 8-12, 2020

A. PROOF OF PROPERTY 5.1

The proof of Equation ( 41) is available in [START_REF] Muller | Power-balanced modelling of circuits as skew gradient systems[END_REF] and is not reproduced here. To prove its extension to semi-continuous functions, using left and right Taylor series expansion one finds

For the second coefficient, one finds 1 2 = 1/12 and using integration by parts, one gets the recursive relation

Finally, when δx = 0, one finds

B. LINEAR DIFFERENTIAL COMPONENTS

When ∇H(x) = Wx with state ẋ(τ ) ∈ P 1 and coefficients ẋ expressed in the orthonormal Legendre basis {Li}, the projected gradient is

For ẋ(τ ) ∈ P 0 it reduces to the midpoint integration rule ∇H(x0, ẋ) = W x0 + h 2 ẋ .


V Filipe Martins-Da-Rocha

Rafael Mouallem
email: srafaelmouallem@hotmail.com

Complete Markets with Bankruptcy Risk and Pecuniary Default Penalties

, then an equilibrium never exists, even if agents' beliefs are homogeneous.

Introduction

Consider a stochastic, finite horizon economy with a complete set of contingent contracts. It is well known that a necessary condition for the existence of competitive equilibrium is that agents' beliefs are equivalent in the sense that agents agree on the set of events with positive probability. An important question to ask is whether equivalence of beliefs is a relevant property for the existence of equilibrium in infinite horizon economies. This is because, when time extends to infinite, this property implies a rather strong uniform perception of uncertainty, namely that agents have homogeneous posterior beliefs on a set of paths that all assign full measure.

We propose to consider the simple model of an infinite horizon stochastic economy with one consumption good and a complete set of contingent markets at the initial period. Agents are assumed to be expected utility maximizers. In the standard Arrow-Debreu paradigm when bankruptcy and default are not allowed, [START_REF] Riedel | Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist[END_REF] proved that agents need not agree completely about the probabilities of finite-time events to ensure existence of a competitive equilibrium. When bankruptcy is permitted but there is a penalty for it, [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] proved that if agents' posterior beliefs do not become eventually equivalent then an equilibrium without strategic bankruptcy does not exist. In particular, if information about uncertainty is generated by a sequence of independent, identically distributed random variables, then their result states that a necessary condition for equilibrium existence is the homogeneity of beliefs. This analysis was recently extended by [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF] to economies with smooth ambiguity-averse agents. They show that if agents' posterior belief reductions given by their "average probabilistic beliefs" do not become homogeneous then a competitive equilibrium without strategic bankruptcy does not exist.

The convergence results in [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] and [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF] hinge on the possibility of bankruptcy as well as on the specific default punishment considered by these authors: if agent i defaults one unit of consumption good at period t contingent to an event A, then the (discounted) penalty perceived by this agent at the initial period is β t P i (A)λ i where β is a discount factor, P i is agent i's beliefs and λ i is the unitary default penalty. If the beliefs of two agents (i, k) do not eventually converge, then it is possible to find an event A with an arbitrary small probability for agent i but with probability arbitrarily close to 1 for agent k. The price at the initial period of the claim contingent to this event is bounded away from zero because of agent's k beliefs while the cost (in terms of utility) for agent i to default contingent to this event is arbitrarily small. Consequently, agent i chooses to short-sell an arbitrarily large amount of the claim contingent to the event A while agent k is willing to accommodate agent's i position. Since bankruptcy should occur only along out-of-equilibrium paths, non-convergent beliefs and equilibrium are not compatible. [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] analyze a necessary condition on beliefs for existence of an equilibrium but do not provide sufficient conditions. It turns out that even if agents are expected utility maximizers with homogeneous beliefs, existence is not automatic. Agents may still have incentives to risk to be bankrupt provided that the default penalty is not too harsh. We show that a necessary condition for existence of equilibrium is that, in the long run, default penalties are larger than marginal utilities. [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF] also provide sufficient conditions for equilibrium existence when agents are smooth ambiguity-averse. However, how harsh default penalties should be depends on the distribution of agents' characteristics.

The economic relevance of these results depends on the interpretation of default penalties since the question at issue is: are the "real-life" default penalties severe enough to prevent strategic bankruptcy? According to [START_REF] Dubey | Default and punishment in general equilibrium[END_REF], default penalties might be interpreted as the consequence in terms of utility of extra-economic punishments such as prison terms or pangs of conscience. [START_REF] Zame | Efficiency and the role of default when security markets are in[END_REF] proposed to interpret default penalties as the consequences of a non-modeled economic punishment, such as, reputation losses and exclusion from credit markets or garnishing of future income. An infinite horizon model is the appropriate environment to model explicit economic punishments. The natural question is then: what are the explicit economic punishments that may lead to indirect and endogenous default penalties that are harsh enough to preclude strategic bankruptcy? In this paper, we propose to study the most favorable case. We impose the most severe economic punishment by assuming that when an agent defaults on a contract, she has both her assets and private endowments seized, and this is for ever. The striking result is that even under such a harsh default punishment, an equilibrium never exists, independently of the heterogeneity of agents' beliefs.

The model

We consider a stochastic infinite horizon economy with complete contingent markets and bankruptcy risk.

Uncertainty

Time is discrete and continues forever, i.e., t ∈ N 0 := {0, 1, 2, . . .}. Exogenous uncertainty is represented by the realization of some sequence of random variables taking values in a finite set S. We denote by Ω the set of all possible overall histories of states, i.e., Ω = S N where N = {1, 2, . . .}. For every t 1, we let

X t : Ω -→ S (2.1)
be the t-th projection from the space of sequences Ω to S, i.e., X t ((s τ ) τ 1 ) := s t . The history up to time t 1 is given by the algebra F t on Ω defined by

F t = σ(X 1 , . . . , X t ). (2.2)
There is no uncertainty at the initial date and we denote by F 0 the trivial algebra on Ω. We denote by F ∞ the σ-algebra of the overall history defined by

F ∞ = σ{X t : t ∈ N} = σ{F t : t ∈ N}. (2.3)
Given an arbitrary set Z, we denote by A(Z) the space of Z-valued processes (a t ) t 0 adapted to the filtration F = (F t ) t∈N 0 . Since a t is F t -measurable, the term a t (ω) only depends on s t := (s 1 , . . . , s t ) = (X 1 (ω), . . . , X t (ω)). In particular, a t only takes finitely many values. We slightly abuse notations and write a t (s t ) or a(s t ) for a t (ω).

When Z is a subset of R, we let A ∞ (Z) be the subset of bounded processes in A(Z), i.e., for any a = (a t ) t 0 ∈ A(Z), we have

a ∈ A ∞ (Z) ⇐⇒ sup{|a t (s t )| : t ∈ N 0 and s t ∈ S t } < ∞.
We let A 1 (Z) be the subset of summable processes in A(Z), i.e., for any a = (a t ) t 0 ∈

A(Z), we have a ∈ A 1 (Z) ⇐⇒ t 0 s t ∈S t |a t (s t )| < ∞
with the convention that S 0 := {s 0 } and a 0 (s 0 ) := a 0 . Fix an arbitrary probability measure P on (Ω, F ∞ ). For each period t 1, we denote by P t the marginal probability on the space S t of possible states of nature at period t defined by

∀s t = (s 1 , . . . , s t ) ∈ S t , P t (s t ) = P {(X 1 , . . . , X t ) = s t }. 1 (2.4)
Given an overall history of states ω = (s 1 , . . . , s t , . . .) ∈ Ω, the probability P t (s 1 , . . . , s t ) is also denoted by P t (ω). The probability over possible shocks at date t + 1 conditional to current history s t is denoted by P t+1 (•|s t ) and defined by

P t+1 (s|s t ) := P t+1 (s t , s) P t (s t )
when well-defined. For every t 1 and every history of states σ ∈ S t , we denote by 1 t,σ the adapted process defined by

∀τ = t, [1 t,σ ] τ = 0 and ∀ξ ∈ S t , [1 t,σ ] t (ξ) = 1 if ξ = σ 0 if ξ = σ. (2.5)
We denote by 1 0 the adapted process defined by [1 0 ] 0 = 1 and [1 0 ] t = 0 for all t 1.

Endowments and Preferences

There is a single consumption good and a finite set I of agents. Agent i's endowments at period t are given by an F t -measurable function

e i t : Ω -→ (0, +∞). (2.6)
The process of aggregate endowments is denoted by e = (e t ) t 0 where e t = i∈I e i t .

Assumption 2.1. For each agent i, the process e i of endowments is adapted, uniformly bounded from above and takes strictly positive values. Moreover, the process of aggregate endowments is uniformly bounded away from 0. In other words,

∀i ∈ I, e i ∈ A((0, ∞)), (2.7)
and ∃e > 0, ∃e > 0, ∀t 0, e ≤ e t ≤ e.

(2.8)

A consumption plan is an adapted process c = (c t ) t 0 with value in [0, ∞), i.e., c ∈ A(R + ). To define agents' preference relations over consumption plans, we introduce the following notations. Denote by L 0 + (F ∞ ) the space of non-negative and F ∞ -measurable functions. Each agent i ranks consumption plans according to the following function from

A(R + ) to [0, ∞] W i t 0 β t u i (c t )
where

W i : L 0 + (F) → [0, ∞] is an expectation operator, u i : R + → R is agent i's utility function and β ∈ (0, 1) is a time-preference discount factor. 2 Assumption 2.2. For each agent i, the function u i : [0, ∞) → R + is continuous,
concave, strictly increasing, satisfies u i (0) = 0 and the Inada condition at the origin. 3

Assumption 2.3. The function

W i : L 0 + (F ∞ ) → [0, ∞] satisfies the following proper- ties: (a) the domain of W i contains bounded random variables; 4 (b) for every z ∈ R, we have W i (z1 Ω ) = z;
(c) it is continuous with respect to the sup-norm on the subset of bounded random variables, i.e., lim

n→∞ w n -w ∞ = 0 =⇒ lim n→∞ W i (w n ) = W i (w); 5
(d) it is strictly increasing in the sense that for every ε > 0,

W i (w) < ∞ and w w + ε1 Ω =⇒ W i (w ) > W i (w).
Observe that under the above properties, the function W i is also increasing in the sense that W i (w ) W i (w) for any w w.

Example 2.1. The standard example is the expected utility model considered by [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] where

∀w ∈ L 0 + (F ∞ ), W i (w) = E P i (w)
2 We assume that β is homogeneous to simplify the presentation. 3 In the sense that lim x→0 (u i (x)

-u i (0))/x = ∞. 4 The domain of W i is the set of random variables w ∈ L 0 + (F ∞ ) satisfying W i (w) ∈ R. 5 For every w ∈ L 0 (R), we let w ∞ := sup{|w(ω)| : ω ∈ Ω}.
for some probability measure P i on F ∞ representing agent i's subjective beliefs.

Example 2.2. One may also follow [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF] and consider the expectation operator associated to the smooth ambiguity averse model of [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF] (see also [START_REF] Klibanoff | Recursive smooth ambiguity preferences[END_REF]):

∀w ∈ L 0 + (F ∞ ), W i (w) = (φ i ) -1 ∆(Ω) φ i (E P (w)) µ i (dP )
where ∆(Ω) is the space of probability measures on (Ω, F ∞ ), µ i is a probability measure on ∆(Ω) and φ i : R → R is strictly increasing and concave.

Investment Strategies

Trade occurs only at the initial period t = 0. There exists a complete market of claims contingent on future histories of states of nature. Contingent to each possible date-t history s t ∈ S t , each agent chooses a claim θ t (s t ) of units of consumption at unitary price q t (s t ) (in units of the consumption good at t = 0). Short-sales are allowed and a negative claim means a promise to deliver units of the consumption good. As usual we make the following abuse of notation:

∀(t, ω) ∈ N × Ω, θ t (ω) = θ t (X 1 (ω), . . . , X t (ω)).
(2.9)

By convention, we pose θ 0 = 0. Observe that θ = (θ t ) t 0 belongs to subspace A 0 (R) of all adapted processes a = (a t ) t 0 ∈ A satisfying a 0 = 0. A vector in A 0 (R) is called an investment strategy (or a portfolio).

Since we want to formally define the cost of a portfolio θ ∈ A 0 (R), we restrict investment strategies to satisfy the following property

q(θ) := lim τ →∞ τ t=1 q t (s t )θ t (s t ) ∈ R.
(2.10)

We denote by Θ the subset of all "admissible" portfolios θ ∈ A 0 (R) satisfying the restriction (2.10).

Remark 2.1. In [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF], agents are restricted to choose portfolios in the set A ∞ (R) of bounded processes. Since they also assume that asset prices q = (q t ) t 0 are summable, then (2.10) is automatically satisfied.

Competitive Equilibrium with Risk of Bankruptcy

At the initial period t = 0, each agent i chooses a consumption c i 0 0 and an investment strategy θ i ∈ Θ satisfying the budget restriction

c i 0 + q(θ i ) ≤ e i 0 .
(3.1) At period t, agent i has an endowment e i t and, either receives the payment θ i t of his investment if θ i t 0, or has to honor his debt by paying -θ i t units of the consumption good if θ i t < 0. Bankruptcy is allowed in the sense that agent i is not restricted at t = 0 to choose and investment strategy θ i satisfying the solvency constraint

∀t 1, θ i t + e i t 0. (3.2)
If agent i is solvent at event s t , i.e., θ i t + e i t 0 then he consumes θ i t + e i t units of the good and gets the utility u i (θ i t + e i t ). If agent i is not solvent, i.e., θ i t + e i t < 0 then he goes bankrupt.

We should now specify the punishment for default.

Default Penalties

Following [START_REF] Zame | Efficiency and the role of default when security markets are in[END_REF] and [START_REF] Dubey | Default and punishment in general equilibrium[END_REF], Araujo and Sandroni (1999) and Araujo, da Silva and Faro (2016) assume that in case of bankrupcty, an agent does not consume and suffers a penalty λ i t [θ i t + e i t ] -proportional to his default where λ i t (ω) 0 is the unitary default penalty at period t contingent to the history ω. There is no other punishment. In particular, there is no garnishment of assets or future income. Under this assumption, the indirect utility associated to an investment strategy θ i ∈ Θ is given by

V i (θ i ) := W i u i (e i 0 -q(θ i )) + ∞ t=1 β t v i (e i t + θ i t )
where v i is the felicity function defined on R by

∀z ∈ R, v i (z) = u i (z) if z 0 λ i z if z < 0. (3.3)
Following [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] and Araujo, da Silva and Faro (2016), we introduce the following concept of competitive equilibrium with risk of bankruptcy and default penalties. Definition 3.1. A competitive equilibrium with risk of bankruptcy and default penalties is a family (q, (θ i ) i∈I ) of asset prices q = (q t ) t 1 and an allocation (θ i ) i∈I of investment strategies θ i ∈ Θ such (a) the investment strategy is optimal, i.e., θ i ∈ argmax V i (θ) : θ ∈ Θ and e i 0 -q(θ) 0 , (3.4) (b) asset markets clear at t = 0, i.e., i∈I θ i = 0, (3.5) (c) for every period t 1, there is no bankruptcy, i.e., ∀i ∈ I, θ i t + e i t 0.

(3.6)

Remark 3.1. When bankruptcy is allowed, agents have the option to take the risk to be bankrupt on some states. [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] and Araujo, da Silva and Faro (2016) look for equilibrium solutions for which agents optimally opt to be solvent at any contingency. This is in sharp contrast with equilibrium concepts analyzed in [START_REF] Zame | Efficiency and the role of default when security markets are in[END_REF] and [START_REF] Dubey | Default and punishment in general equilibrium[END_REF] where default may occur along the equilibrium path. Naturally, default penalties should be large enough to discourage agents from making promises that they know in advance they will not be able to honor.

Implicit Assumptions

Two important assumptions are implicitly imposed for the validity of the model and the equilibrium concept. The first one concerns the ex-ante information available to agents.

In sharp contrast with [START_REF] Kehoe | Debt-constrained asset markets[END_REF] (see also [START_REF] Alvarez | Efficiency, equilibrium, and asset pricing with risk of default[END_REF]), it is assumed that the information about agents' endowments and utility functions is not common knowledge at t = 0. Indeed, if the market knows at t = 0 that agent i's endowment at period t contingent to the history s t ∈ S t is e i t (s t ), then agent i should be constrained to choose an affordable claim θ i t (s t ) -e i t (s t ), as it is the case in the standard Arrow-Debreu paradigm (see for instance [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF], [START_REF] Dana | Existence and uniqueness of equilibria when preferences are additively separable[END_REF] and [START_REF] Riedel | Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist[END_REF]).

The second assumption concerns ex-post enforcement. In [START_REF] Dubey | Liquidity and bankruptcy with incomplete markets[END_REF], [START_REF] Zame | Efficiency and the role of default when security markets are in[END_REF] and [START_REF] Dubey | Default and punishment in general equilibrium[END_REF], each agent chooses whether to repay his debt or to bear the penalty for defaulting. In particular an agent may decide not to meet at all his liabilities even if he could afford them. In [START_REF] Kehoe | Debt-constrained asset markets[END_REF] and [START_REF] Alvarez | Efficiency, equilibrium, and asset pricing with risk of default[END_REF] it is assumed that assets can be seized but endowments are non-recourse. In [START_REF] Geanakoplos | The Economy as an Evolving Complex System II, Addison-Wesley, chapter Promises[END_REF], [START_REF] Geanakoplos | Collateral and the enforcement of intertemporal contracts[END_REF], [START_REF] Araujo | Collateral avoids Ponzi schemes in incomplete markets[END_REF] and [START_REF] Kubler | Stationary equilibria in asset-pricing models with incomplete markets and collateral[END_REF], the only enforcement mechanism is the seizure of collateral bundles. In the model of [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] and [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF], it is implicitly assumed that in case of bankruptcy, endowments can be seized but assets cannot.

Default Penalties Need to Be Large Enough

Since agents have the option to go bankrupt but optimally decides not to do so, default penalties must be sufficiently large, even if agents are expected utility maximizers with homogeneous beliefs. We show below that default penalties have to be larger than equilibrium marginal utilities.

Proposition 3.1. Assume that there exists a probability measure P ∈ ∆(Ω) with full support such that W i (w) = E P (w). 6 If there is competitive equilibrium with risk of bankruptcy and default penalties, then, in the long run and for every agent, default penalties must be larger than marginal utilities. More precisely, we must have

∀i ∈ I, ∀ω ∈ Ω, lim sup t→∞ ∇u i (c i t (ω)) -λ i t (ω) ≤ 0 (3.7)
where c i t := e i + θ i t is the equilibrium consumption.

Proof of Proposition 3.1. Consider a competitive equilibrium (q, (θ i ) i∈i ). Let c i t := e i + θ i t and recall that c i t 0. Since u i satisfies Inada's condition, we have that c i t > 0. In particular, we can define the marginal utility ∇u i (c i t (ω)) for every t 0.

Euler equations associated to the optimal choice of θ i imply that

∀t 1, ∀ω ∈ Ω, q t (ω) = β t P t (ω) ∇u i (c i t (ω)) ∇u i (c i 0 )
.

Observe that we always have c i t (ω) ≤ e and c i 0 ≤ e. Moreover, since the allocation (c i ) i∈I is feasible, there exists i 1 ∈ I such that c i 1 0 e and for each t 1 and history of states σ ∈ S t , there exists i 2 ∈ I such that c i 2 t (ω) e. Since utility functions are concave, we get

0 < m := min k∈I ∇u k (e) ∇u k (e) ≤ ∇u i (c i t (ω)) ∇u i (c i 0 ) ≤ max k∈I ∇u k (e) ∇u k (e) := M < ∞.
Assume by way of contradiction that there exists an agent i ∈ I and an history of states ω ∈ Ω such that lim sup

t→∞ ∇u i (c i t (ω)) -λ i t (ω) > 0.
Fix arbitrarily α > 0 and consider the investment strategy η i (α) ∈ A 0 (R) defined by

η i (α) := θ i -(α + θ i t (ω))1 t,ω .
With the portfolio η i (α), agent i is bankrupt at date t contingent to state ω. The indirect utility is then

V i (η i (α)) = V i (θ i ) + u i (c i 0 + f i t (α)) -u i (c i 0 ) -αλ i t (ω) + u i (c i t (ω)) β t P t (ω)
where

f i t (α) = αβ t P t (ω) ∇u i (c i t (ω)) ∇u i (c i 0 )
.

We then get

V i (η i (α)) -V i (θ i ) ∇u i (c i 0 + f i t (α))f i t (α) -αλ i t (ω) + u i (e) β t P t (ω).
Now fix arbitrarily γ > 0 and choose α > 0 such that

αβ t P t (ω) = γ.
It follows that

V i (η i (α)) -V i (θ i ) ∇u i (c i 0 + γM )γ ∇u i (c i t (ω)) ∇u i (c i 0 ) -γλ i t (ω) + u i (e)β t P t (ω) implying that lim sup t→∞ V i (η i (α)) -V i (θ i ) γ ∇u i (c i 0 + γM ) -∇u i (c i 0 ) m + lim sup t→∞ ∇u i (c i t (ω)) -λ i t (ω) .
Choosing t large enough and γ > 0 small enough, we get that V i (d i (α)) > V i (c i ). This contradicts the optimality of θ i .

Proposition 3.1 provides a necessary condition relating unitary default penalties which are primitives of the model and marginal utilities at equilibrium which are endogenous variables. However, we can easily provide sufficient conditions for nonexistence, only in terms of primitives: if there exists an every overall history of states ω ∈ Ω such that either ∀i ∈ I, lim inf t→∞ λ i t (ω) < ∇u i (e) (3.8) or ∃i ∈ I, lim inf t→∞ λ i t (ω) < ∇u i (e) (3.9) then there does not exist a competitive equilibrium without default.

Explicit Economic Punishment

We have seen that default penalties should be large enough to ensure existence of a competitive equilibrium even when agents are expected utility maximizers with homogeneous beliefs. [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF] show that when agents are smooth ambiguity averse, then under a strong compatibility condition on second order beliefs, it is possible to identify large enough default penalties for which there exists a competitive equilibrium.

Since the level of default penalties matters, we should investigate what exactly are these default penalties. According to [START_REF] Dubey | Default and punishment in general equilibrium[END_REF], default penalties might be interpreted as the consequence in terms of utility of extra-economic punishment such as prison terms or pangs of conscience. Default penalties may also be interpreted as the consequence in terms of utility of a non-modeled economic punishment such as reputation losses and exclusion from credit markets or garnishing of future income. We quote hereafter [START_REF] Zame | Efficiency and the role of default when security markets are in[END_REF]:

In the real world, such default may entail many and varied consequences: creditors might be able to seize assets and be awarded judgments against future earnings, defaulters might be barred from future credit markets, and so on. Rather than attempt to model such institutional details, I assume that the only consequences of default are penalties assessed against the defaulters, and these penalties are assessed directly in terms of utility.

Since existence of a competitive equilibrium depends on the level of the default penalty, the natural question is then: what are the explicit economic punishments that lead to endogenously determined default penalties that are large enough to preclude strategic bankruptcy? Is there a sufficiently severe economic punishment such that, independently of the distribution of agents' characteristics, strategic bankruptcy is precluded?

To answer this question, we consider the most severe economic punishment. As in [START_REF] Araujo | On the convergence to homogeneous expectations when markets are complete[END_REF] and [START_REF] Araujo | Ambiguity aversion in the long run: "to disagree, we must also agree[END_REF], we assume that upon default, current endowment is seized. In addition, in the line of [START_REF] Kehoe | Debt-constrained asset markets[END_REF] and [START_REF] Alvarez | Efficiency, equilibrium, and asset pricing with risk of default[END_REF], we assume that assets are also seized. Moreover, not only current endowment is seized, but all future endowments are seized. This means that if agent i chooses an investment strategy θ i such that for some history ω and date t, we have θ i t (ω) > -e i t (ω), then for every τ t the agent consumption is c i τ (ω) = 0. Since we assumed that u i (0) = 0, the indirect utility of an investment strategy θ is now defined by

V i (θ) := W i   u i (e i 0 -q(θ)) + τ i (θ) t=1 β t u i (e i t + θ t )   (4.1)
where the stopping time τ i (θ) : Ω → N is defined by

τ i (θ)(ω) = inf{t 1 : θ t (ω) + e i t (ω) < 0} (4.2)
with the convention that inf ∅ = ∞ and 0 t=1 = 0. Even when the most severe explicit economic punishment is imposed, a competitive equilibrium without default never exists.

Theorem 4.1. Assume that the default punishment is complete garnishment of current and future income (assets and endowments). Then, a competitive equilibrium with risk of bankruptcy does not exist.

Proof. Assume, by way of contradiction, that there exists a competitive equilibrium (q, (θ i ) i∈I ) with risk of bankruptcy. Fix an agent i and a date ξ 1. Consider the alternative investment strategy η i,ξ defined by

η i,ξ :=    θ i t , if t = ξ + 1 θ i ξ+1 -α ξ+1 , otherwise,
where α ξ+1 > 0 is such that s ξ+1 ∈S ξ+1 q ξ+1 (s ξ+1 )α ξ+1 = 1.

In the new strategy η i,ξ , debt is increased uniformly at all events at date ξ +1 such that the corresponding amount borrowed at date 0 is exactly 1. Recall that in a competitive equilibrium with risk of bankruptcy, the equilibrium portfolio θ i is such that agent i is never bankrupt, i.e., e i t + θ i 0, ∀t 1.

When ξ tends to infinite, the unit price q ξ+1 (s ξ+1 ) may converge to zero. Therefore, the new debt α ξ+1 may tend to infinite. This implies that we may have e i ξ+1 (ω)+η i,ξ ξ+1 (ω) < 0 for some history ω. Therefore, we get the following lower bound of the indirect utility associated to the new investment strategy V i (η i,ξ ) W i u i (e i 0 -q(θ) + 1) + ξ t=1 β t u i (e i t + θ i t ) . (4.3)

Comparing the two indirect utilities, we get

V i (η i,ξ ) -V i (θ i ) W i       
u i (e i 0 -q(θ) + 1) + ξ t=1 β t u i (e i t + θ i t )

=:

U i,ξ        -W i        u i (e i 0 -q(θ)) + ∞ t=1 β t u i (e i t + θ i t ) =: U i       
.

Observe that

U i,ξ = U i + δ i 0 1 Ω - ∞ t=ξ+1
β t u(e i t + θ i t ) with δ i 0 := u i (e i 0 + q(θ i ) + 1) -u i (e i 0 + q(θ i )).

Since markets clear at equilibrium, we have e i t + θ i t ≤ e for every t 1. This implies that 0 ≤ ∞ t ξ+1 β t u(e i t + θ i t ) ≤ β ξ+1 1 -β u(e).

We then get that lim

ξ→∞ U i,ξ -(U i + δ i 0 1 Ω ) ∞ = 0.
By sup-norm continuity of W i , we deduce that

lim ξ→∞ V i (η i,ξ ) -V i (θ i ) = W i (U i + δ i 0 1 Ω ) -W i (U i )
which is strictly positive since W i is strictly increasing. This contradicts the optimality of θ i .

We slightly abuse notation since we should write P t {s t } instead of P t (s t ).

P has full support means that P t (s t ) > 0 for every date-t event s t ∈ S t .

complete, American Economic Review 83, 1142-1164.


Emmanuel Lhuillier

Sean Keuleyan

Pavlo Zolotavin

Philippe Guyot-Sionnest

Mid-Infrared HgTe/As 2 S 3 FETs and photodetectors

Keywords: HgTe, As 2 S 3, colloidal quantum dot, inorganic matrix, FET, mid-IR, photodetection

Colloidal quantum dots (CQDs) with mid-infrared (IR) tunable bandgaps present a new paradigm for mid-IR materials, but advances in charge transport and collection are needed for practical use in electronic applications. By replacing the organic ligands in films of HgTe CQDs with As 2 S 3 , a problem of oxidation in air is avoided and charge carrier mobilities are improved 100 fold compared to the standard organic ligand exchange. The composite inorganic HgTe/As 2 S 3 material allows for ambipolar field effect transistors with on-off ratios up to 10 7 , and photodetectors with high sensitivity, reaching 3x10 10 Jones at 230 K with a 3.5 micron cutoff wavelength.

dominate mid-infrared applications, the development of CQDs as a low-cost technology could open new markets.

Quantum dots of HgTe, a semi-metal in the bulk, could in principle have bandgaps tunable through the full infrared range by controlling the size. The synthesis [8] and optical properties [9] of HgTe CQDs with bandgaps tunable through the mid-infrared range of 3-5 µm (0.2 to 0.5eV) were recently reported. Particles with diameters between 6 and 12 nm have been prepared and the monodisperse samples show sharp absorption and photocurrent spectral edges, as shown in Figure 1(a). Competitive photodetection sensitivities, however, will require the elimination of the mid-IR absorption from the organic ligands, as along with improvements in charge transport and collection as well as reduced noise. [10] Initial strategies for improving mobilities relied on organic ligand exchange of preformed films of CQDs [11,12,13] To produce fully-inorganic composite materials, a number of reports have been published on the use of solution phase inorganic ligands, providing a variety of choices for passivation and barrier control. [14,15,16,17,18] Arsenic sulfide, in addition to its use as an inorganic ligand [19,20] has been used as an infrared-transparent inorganic matrix, with advantages for the near-infrared photoluminescence of PbS/CdS core/shell quantum dots. [21] In this paper, we use a ligand exchange procedure with arsenic sulfide on films of monodisperse HgTe CQDs and compare the results with organic ligand exchange.

Simply dried films of the monodisperse HgTe CQDs are highly insulating, due to the long alkane ligands, which include octadecylamine, trioctylphosphine, and dodecanethiol. After ligand exchange by dipping the films in a solution of ethanedithiol in ethanol, the films become photoconductive. The photoresponse and optical properties of films of these organicinorganic composite materials, have been reported. [9] Exchange peformed in air with other ligands is also explored, including thiols (1,2 ethanedithiol, 1,4-benzenedithiol, 1,4-butanedithiol) and amines (propylamine, butylamine, butanediamine, 1,7-heptanediamine). While ethanedithiol provides the best response, all ligands tested lead to photoconductive films. However the dark current I(T) curves show that all films exhibit a non-monotonic current as a function of temperature with a more or less prominent bump around 200K, as seen in Figure 1 (b) and (c). Such behavior had not been observed for the films made with aggregated HgTe CQDs [22] and since both thiol and amine ligands lead to a similar feature in the I(T) curve, the feature is not specific to the ligands. The responsivity of the films to infrared illumination is low, ranging from 10µA.W -1 to 5mA.W -1 (Figure 1(d)). Although a photocurrent spectral response was measurable with a standard scanning FTIR (Figure 1 (a)), the responsivity is two decades lower than with the previous aggregated HgTe CQDs.

While performing experiments with a sample in the vacuum of the cryostat, the bump was observed to decrease in magnitude over a period of days but then rapidly recovered upon exposure to air. Suspecting oxidation, [23] samples were prepared in a nitrogen-filled glove box and transferred in a closed cell to the cryostat. With such air-free preparation and transfer, the I(T) curve is monotonic, with an Arrhenius activation energy close to half of the optical band gap in the higher temperature range, see (Figure 1 It was then concluded that the films made from the monodisperse HgTe CQDs need to be processed and kept in air-free conditions. Presumably, the surfaces are more exposed than with the aggregated HgTe CQDs reported earlier, leading to increased sensitivity to oxidation which then leads to the formation of acceptor levels. It is also noted that although the air-free process with EDT decreased the dark current, it did not improve the responsivity significantly.

We then explored the inorganic ligand exchange route. As 2 S 3 is an infrared glass that can be easily dissolved in the low boiling point propylamine and is used to make infrared transparent sol/gel films. [24] Recently Kovalenko et al showed that the organic ligands on PbS/CdS nanoparticles could be replaced by such an As 2 S 3 solution for transparent As 2 S 3 films with dispersed fluorescent PbS/CdS particles. 21 While this is advantageous for making luminescent films, the dilute dispersion of the particles in the matrix reduces the optical absorption of the films, and the wide bandgap of As 2 S 3 renders the films insulating.

To make photoconductive films, the strategy pursued here mimics organic ligand exchange procedures. The drop-cast HgTe QD films are dipped in a dilute solution of As 2 S 3 and propylamine in ethanol and then dried at room temperature. The concentration of As 2 S 3 and propylamine in ethanol needs to be optimized, as at high concentrations, the resulting films are very conductive but without photoresponse, while saturated As 2 S 3 in propylamine (without ethanol) delaminates the film.

The C-H stretch absorption band is strongly reduced by this process, showing that most of the organic ligands are replaced (see Figure 1

(f)).
When performed in air, this process leads to a large dark current with an activation energy of 50 to 60meV which is much smaller than half the optical band gap, see Figure 1 (e). The photoresponse is also very slow (> seconds, see supplementary information), which can be assigned to long lived traps. When the films of HgTe CQD are processed with As 2 S 3 in airfree conditions, the I(T) curve indicates that the intrinsic behavior is recovered, just like for the air-free ligand exchange with EDT. As shown in Figure 2 (a), the I(T) curve is monotonic and the activation energy is close to half the optical band gap.

(g) (f) (d) (e) (c) (b) EDT 1000/T (K -1 ) As 2 S 3 V=1V (a) AS 2 S 3 V DS =10V T=45K T=100K T=300K V GS (V) EDT V DS =10V T=300K T=200K T=100K Current (nA) V GS (V) Current (nA) Current (A) V GS =0V V GS =10V V GS =20V As 2 S 3 -T=100K V GS =30V V GS =0V V GS =-10V V GS =-20V Current (µA) V DS (V) As 2 S 3 -T=100K V GS =-30V Current (µA) V DS (V) electron hole EDT Mobility (cm 2 .V -1 s -1 ) Temperature (K) As 2 S 3 Photocurrent (a.u.) AS 2 S 3 T=300K -V DS =10V
Frequency(Hz) To confirm the intrinsic character of the films of HgTe CQDs and to determine carrier mobilities, field effect transistor (FET) devices were made. Kim and coworkers previously reported FETs with HgTe CQDs [25,26,] . Thin films of HgTe CQD annealed at 150°C, were reported to be p-type, with on-off ratios up to 10 3 and mobility up to 1cm 2 V -1 s -1 . However no study of their infrared optical absorption edge or photoresponse were reported. 27 . Our FETs are made from the HgTe CQD films and processed at room temperature to preserve the optical absorption edge. When films are processed air-free, they are ambipolar, consistent with their intrinsic behavior seen in the I(T) curves. At low temperatures, the FET show high on-off ratios up to 10 7 for As 2 S 3 -processed films, and 10 6 for EDT processed films, see Figure 2 (b) (c) (d) and (e). At room temperature, the on-off ratio is much smaller, and this is attributed to thermally activated carriers.

Carrier mobilities in Figure 2 (g) are obtained from the I DS (V GS ) curves using the relation

DS V GS DS DS FET n V I V WC L     ∂ ∂ = Σ µ
Where L is the electrode spacing (10µm), W is the electrode length (25×2.5mm), Σ C the surface capacitance equal to 34.5µF.m -2 for the 1µm thick SiO 2 and DS V the drain source bias.

Hole and electron mobilities with EDT are similar to previous results where carriers were injected electrochemically [28] . For the film processed with As 2 S 3, µ~ 10 -2 cm 2 V -1 s -1 while with EDT µ~10 -4 cm 2 V -1 s -1 . This 100 fold improvement in mobility readily explains the improved From the dark noise measurements, the As 2 S 3 films also show also much improvement in the detectivity with a value higher than 10 10 jones at 1kHz for a film with a 3.5 µm cut-off at 230 K, shown in Figure 3 (b) and (d). This value of the detectivity is 30 times higher than with the aggregated material [10] . With further optimization, the As 2 S 3 ligand exchange approach is likely to lead to improvements for detector materials at longer wavelengths as well. Moreover, the photoconductivity of the film processed with As 2 S 3 remains fast, with a flat frequency response between 10Hz and 100kHz, as shown in Figure 2 (f).

With the HgTe CQD films, processed air-free with As 2 S, the ambipolar behavior remains even when the film is exposed to air (see supplementary information). Thus, HgTe/As 2 S 3 is more stable to oxidation compared to all tested organic ligands. It is expected that the glassy As 2 S 3 matrix slows down the diffusion of O 2 or H 2 O compared to the organic ligands. This is an additional significant benefit of the inorganic As 2 S 3 ligand exchange. In addition, a thicker encapsulating layer of As 2 S 3 can be deposited on top of the CQD films to provide further protection of the films. 29 At present the films have been prepared at room temperature by slow drying of the As 2 S 3 sol-gel. It is likely that further improvement in the detector properties will be achieved with a careful annealing procedure.

In summary, monodisperse HgTe CQDs have a well defined optical response in the mid-IR but conductive films required an appropriate ligand exchange. Different organic ligands were investigated, where it was found that ligand exchange in air leads to p-type conduction and larger dark current, attributed to acceptor levels arising from oxidation. Intrinsic behavior is obtained when the samples are processed under nitrogen. The best organic ligand, ethanedithiol, leads to only moderate performances with detectivity of 10 8 jones, and this can partly be assigned to the low mobility of 10 -4 cm 2 /Vs. Using air-free ligand exchange in solutions of As 2 S 3 leads to intrinsic material and the extensive removal of organics. FET measurements show ambipolar mobility of ~ 10 -2 cm 2 /Vs with high on-off ratios at low temperatures. Once encapsulated in As 2 S 3 , the films are stable against oxidation in air.

Responsivity greater than 100 mA.W -1 and a detectivity of 3.5x10 10 jones have been obtained at 3.5 microns and 230K. Inorganic ligand exchange and encapsulation is a promising method for improving both the electrical and optical properties of the quantum dot films. These results with HgTe/As 2 S 3 motivate further work on infrared transparent inorganic matrices, improved detectors, and possibly emitters at longer wavelengths.

EXPERIMENTAL SECTION

• Material synthesis

HgTe CQDs were prepared following ref.

• Film preparation and characterization

Films are prepared by drop-casting the hexane-octane solution of CQDs onto interdigitated electrodes. Different electrodes are used for the photoconductive and the FET measurements.

For photoconductive measurements, electrodes (ABTech IME 1050) with 50 periods of Pt interdigitated electrodes (5 mm long and 10 µm spacing) are used. The spectral response is acquired using a FTIR Nicolet magna IR 550, while the device is biased and the signal amplified using a femto DLCPA 200 amplifier.

For FET measurements, gold interdigitated electrodes were deposited on a doped silicon substrate with a 1 µm thick oxide layer. Films are mounted on the cold finger of a closed cycle helium cryostat (ColdEdge SDRK-101D) with a CryoCon 32B temperature controller.

Transport measurements are made using a Keithley 6487 picoammeter. For FET measurements, a National Instruments acquisition board (BNC 2110), amplified 10 times, is used to apply the gate bias.

Responsivity is measured using a blackbody (Omega BB-4A operating at 1255K) as a broadband photon source. The light is chopped at 25Hz and a Ge plate cuts the illumination below 1.7µm. The device is biased and the signal is amplified using a femto DLCPA 200 amplifier. The modulation is monitored on a Tektronix, TDS 1012B oscilloscope. The frequency response of the photocurrent is measured by illuminating the sample with a 800nm laser diode modulated with a signal generator up to 100 kHz.

Dark noise measurements are made by biasing the sample and amplifying the signal with a femto DLCPA 200 amplifier. The output signal is high-pass filtered (with a 0.1Hz cut-off frequency RC filter) and amplified ten times using an operational amplifier. Finally the current spectral density is obtained with a Stanford research SR 760 spectrum analyzer.

Effect of As 2 S 3 in air

Encapsulating the HgTe CQD in As 2 S 3 matrix even in air leads to improved stability with monotonic I(T) curve and improved conductance. This can be explained by a better interparticle coupling. The spectrum is redshifted as the film is dipped for longer times into

  (b)).

Figure 1 (

 1 Figure 1 (a) Typical photocurrent spectra of HgTe CQDs after ligand exchange, the reddest

Figure 2 (

 2 Figure 2 (a) Current as a function of the temperature for a film processed air-free after

2 ≈

 2 properties of the As 2 S 3 treated films. It is instructive to consider the associated hopping time given by (with a the interparticle center to center distance) which at room temperature is 0.65 ns with As 2 S 3 and 65 ns with EDT. The competition between the hopping time and the nonradiative lifetime is key to efficient charge separation. Although the nonradiative lifetime is unknown at present, the 100-fold increased responsivity with As 2 S 3 shown in Figure3(a) and (c), which tracks the improvement in the mobility, suggests that a faster hopping time would still be beneficial.

Figure 3 (

 3 Figure 3 (a) Responsivity map as a function of bias and temperature for a film of HgTe CQD

8 .

 8 Octadecylamine (ODA) was used in place of oleylamine, as some batches of the technical grade (70%) oleylamine were found to result in larger particle size distributions. A 1M solution of Te in trioctylphosphine (TOP) is prepared ahead of time by agitating at room temperature for several days in a glove box. In a typical synthesis, 135 mg of HgCl 2 and 7.4g of ODA are added to a three-neck flask, flushed several times with Ar, heated under vacuum to 120°C for 1 hour then cooled under Ar to 80°C. 0.5mL of the 1M Te in TOP is then quickly injected. The solution darkens over a period of about 20 s, indicating the formation of HgTe. After an appropriate growth time, a solution of 10% (vol.) dodecanethiol in TCE is quickly injected and heating is removed. The reaction mixture is stirred several minutes before precipitating with methanol. After centrifugation, the precipitate is redispersed in toluene and precipitated at least three additional times. Finally the nanoparticles are stored in TCE or in a 9:1 in volume mixture of hexane:octane, in the glovebox.

2 .

 2 Figure S 1(a) shows a transmission electron microscopy image (TEM). The structure of the nanoparticle is zinc blende (see X ray diagram on Figure S 1 (b) and their photocurrent spectra show a rather sharp absorption edge, see Figure S 1 (c).

Figure S 1 3 .

 13 Figure S 1 (a) Transmission electron microscopy picture of HgTe CQD. (b) X ray diffraction diagram of HgTe CQD film. (c) Photocurrent spectra of HgTe CQD films in this study.

Figure S 2 : 4 . Effect of AsCl 3

 243 Figure S 2 : setup used for the noise measurements

Figure S 3

 3 Figure S 3 (a) resistance of a film of HgTe CQD as a function of the process, different ions are tested. (b) Current as a function of the inverse of the temperature of a film of HgTe CQD processed in air with AsCl 3 . (c) and (d) are the responsivity and detectivity map (as a function of bias and temperature) of a a film of HgTe CQD processed in air with AsCl 3.

As 2 S 3 ,

 23 see Figure S 4 (a). Samples processed in air show extremely slow response time compared to the EDT ligand exchange, see Figure S 4 (c) and (d). The HgTe films can be capped with a layer of pure As 2 S 3 , see Figure S 4 (b), which role is to prevent oxidation of the nanoparticle film.

Figure S 4 6 .Figure S 5 :

 465 Figure S 4 (a) Photocurrent spectra of a film of HgTe CQD at room temperature tretaed with ethandithiol and with As 2 S 3 with different duration. . (b) scanning electron microscopy picture of a film oh HgTe CQD treated with As 2 S 3 with its capping layer of pure As 2 S 3 . (c) Photoreponse of a film of HgTe CQD processed in air with ethandithiol during and after its illumination with a 800nm laser. (d) Photoresponse of a film of HgTe CQD processed in air with As 2 S 3 during to a 800nm laser.

7 .

 7 Responsivity and detectivity map at 2 µmWe also investigate the effect of the matrix on different batch of HgTe CQD with cut-off wavelength at 2µm. The associated responsivity and detectivity map are shown on Figure S 6.

Figure S 6

 6 Figure S 6 (a) and (b) are the responsivity and detectivity map as a function of bias and temperature of a film of HgTe CQD treated with As 2 S 3 and a cut-off wavelength at 2µm.

ACKNOWLEDGEMENTS

This work is supported by the DARPA COMPASS program through a grant from ARO. The authors made use of shared facilities supported by the NSF MRSEC Program under DMR-0820054.