F Hamelin

A Iggidr

A Rapaport
email: alain.rapaport@inrae.fr

G Sallet

M Souza
email: maxsouza@id.uff.br

About the identifiability and observability of the SIR epidemic model with quarantine

Keywords: Parameters estimation, identifiability, observability, parameters identification

We analyze the identifiability and observability of the well-known SIR epidemic model with an additional compartment Q of the sub-population of infected individuals that are placed in quarantine (SIQR model), considering that the flow of individuals placed in quarantine and the size of the quarantine population are known at any time. Then, we focus on the problem of identification of the model parameters and review different techniques.

INTRODUCTION

Many papers in epidemiology proposing a mathematical model using dynamical systems, face the problem of parameter estimation. In general, some parameters are given, extracted from the literature, while the remaining unknown parameters are estimated by fitting the model to some observed data, usually by means of an optimization algorithm based on least squares or maximum likelihood methods. Nevertheless, there is relatively few studies about the intrinsic property of a model to admit a unique set of parameters values for a given choice of measured variables. On the other hand, this is a question that is well known in automatic control. Investigation of identifiability in mathematical epidemiology is relatively recent [START_REF] Tuncer | Structural and practical identifiability issues of immuno-epidemiological vector-host models with application to Rift Valley Fever[END_REF]; [START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF]; [START_REF] Xia | Identifiability of nonlinear systems with application to HIV/AIDS models[END_REF]; [START_REF] Saccomani | An effective automatic procedure for testing parameter identifiability of HIV/AIDS models[END_REF]; [START_REF] Miao | On identifiability of nonlinear ODE models and applications in viral dynamics[END_REF]; [START_REF] Eisenberg | Identifiability and estimation of multiple transmission pathways in Cholera and waterborne disease[END_REF]; [START_REF] Evans | The structural identifiability of the susceptible infected recovered model with seasonal forcing[END_REF]. Indeed, to the best of our knowledge, the first paper considering the identifiability problem in a epidemic model is the paper of [START_REF] Xia | Identifiability of nonlinear systems with application to HIV/AIDS models[END_REF]. It is also surprising that the observability and identifiability of the original Kermack-Mckendrick model has not been more studied, since this system has been widely used to model an outbreak of an infection. The observability and identifiability of the classical model SIR, with demography and constant population, was first studied in [START_REF] Evans | The structural identifiability of the susceptible infected recovered model with seasonal forcing[END_REF][START_REF] Evans | The structural identifiability of the susceptible infected recovered model with seasonal forcing[END_REF]. For practical parameters identification with state reconstruction, different approaches can be considered, most of them using observers [START_REF] Bichara | On the estimation of sequestered infected erythrocytes in plasmodium falciparum malaria patients[END_REF]; [START_REF] Diaby | Observer design for a schistosomiasis model[END_REF]; [START_REF] Degue | Interval Observer Design for Sequestered Erythrocytes Concentration Estimation in Severe Malaria Patients[END_REF]; [START_REF] Iggidr | State estimators for some epidemiological systems[END_REF]; [START_REF] Hamelin | Observability, identifiability and epidemiology a survey[END_REF]; [START_REF] Fang | A class of nonlinear state observers for an SIS system counting primo-infections[END_REF] and references therein) but these approaches are rarely considered in the literature compared to least-square methods, differently to other applications domains, such as bio-processing or manufacturing industries.

The global health crisis of COVID-19 outbreak has led to a spectacular resurgence of interest in this type of models, but with specificities related to the detection and isolation of infected individuals. This is why we revisit here the issues of identification and observability for an extended 'SIQR' model [START_REF] Hethcote | Effects of quarantine in six endemic models for infectious diseases[END_REF]) for which such an analysis has not yet been performed (to the best of our knowledge). Once the question of identification has been settled (Sections 3,4,5), we tackle in Section 6 the task of proposing a practical strategy for reconstructing the values of the parameters.

THE MODELS

Inspired by [START_REF] Hethcote | Effects of quarantine in six endemic models for infectious diseases[END_REF], we consider the classical SIR model, where S, I, R denote the size of the populations of respectively susceptible, infected and recovered individuals, with an additive compartment where Q denotes the size of the population of identified and isolated infectious individuals that have been removed from the infected population and placed in quarantine:

   Ṡ = -βS I N -Q , İ = βS I N -Q -(ρ + α)I Q = αI -ρQ, Ṙ = ρI + ρQ.
(1)

When the size of the total population N is large and the size of the population placed in quarantine remains small compared to N during the considered interval of time, one can consider a simplified model:

Ṡ = -βS I N , İ = βS I N -(ρ + α)I, Q = αI -ρQ, Ṙ = ρI + ρQ. (2)
For both models, one has S(t)

+ I(t) + Q(t) + R(t) = N, ∀t ≥ 0 . Since S(t) + I(t) + Q(t) + R(t) is constant
and R has no influence on the first three equations, it is sufficient to consider the following system

Ṡ = -βS I N , İ = βS I N -(ρ + α)I, Q = αI -ρQ, (3)
It can be proved that the open set Ω = {S > 0, I > 0, Q > 0, S + I + Q < N } is positively invariant.

These models have three parameters: the infectivity parameter β, the recovery rate ρ, that we assume to be identical for the infected populations placed in quarantine or not, and the rate of placement in quarantine α. Theses parameters are unknown but we assume the following hypothesis.

Assumption 1. The reproduction number R 0 verifies

R 0 := β ρ + α > 1 .
This assumption implies that the epidemic can spread in the population i.e. at initial time with S(0) = N -I(0) close to N one has İ(0) > 0.

THE IDENTIFICATION PROBLEM

We assume that

• the flow αI(t) of infected people placed in quarantine is known at any time t ≥ 0 • the size Q(t) of the population placed in quarantine is perfectly known at any time t ≥ 0 • the size N of the total population is known • at initial time 0, one has S(0) = N -ε, I(0) = ε, Q(0) = 0, R(0) = 0 with ε ∈ (0, N ).

We consider then the observation function

y(t) = y 1 (t) y 2 (t) := αI(t) Q(t) (4)
and follow the usual definitions of identifiability and observability of systems [START_REF] Walter | Identification of parametric models[END_REF]; [START_REF] Gauthier | Observability for systems with more outputs than inputs and asymptotic observers[END_REF]. However, note that when Q = 0 or I = 0 the system is not infinitesimally identifiable: the knowledge of the outputs and all its derivative do not allow to determine formally ρ. We adopt the following definition of identifiability for these models. Definition 2. Given N > 0 and ε ∈ (0, N ), we shall say that system (1) resp. ( 2) is identifiable for the observation (4) if there exists t > 0 such that the map

( α, β, ρ ) ∈ R ⋆ + 3 -→ y(•) ∈ C ∞ ([0, t], R 2 + ) is injective, where (S(•), I(•), Q(•), R(•)) is solution of the Cauchy problem for the differential system (1) resp. (2) with S(0) = N -ε, I(0) = ε, Q(0) = 0 and R(0) = 0.
If moreover the map

( α, β, ρ, ε ) ∈ R ⋆ + 3 × (0, N ) -→ y(•) ∈ C ∞ ([0, t], R 2 + )
is injective, then the system (1) resp. ( 2) is identifiable and observable for the observation (4).

ANALYSIS OF THE FIRST MODEL

Proposition 3. System (1) is identifiable and observable for the observation (4), in the sense of Definition 2.

Proof. It consists in showing that parameters and unmeasured state variables S, I and R can be expressed as functions of the successive derivatives of the output vector y. As the variable I cannot reach 0 in finite time, we shall assume I ̸ = 0 in the following.

Note first that with Q(0) = 0 one has Q(0) > 0 and then y 2 (t) = Q(t) > 0 for any t > 0. The dynamics of Q gives directly the expression of the parameter ρ as:

ρ = y 1 (t) -ẏ2 (t) y 2 (t) , t > 0. ( 5
)
Posit h 1 := ẏ1 y 1 . One has from the dynamics of

I h 1 = β S N -Q -α -ρ. (6)
and then

(N -Q) ḣ1 = β S N -Q -β I + Q . ( 7
)
Using the equality 6), one obtains from ( 7)

β S N -Q = α + h 1 + ρ from (
h 2 := (N -y 2 ) ḣ1 = (h 1 + α + ρ) (-β I + Q). (8)
Let us write the derivative of h 2 : ḣ2 = ḣ1 (-β

I + Q) +(h 1 + α + ρ) -β β S N -Q I + β (α + ρ) I + Q
which can be also expressed as ḣ2 = ḣ1 (-β

I + Q) + (h 1 + α + ρ) (-h 1 β I + Q).
Then, using relation (8), one obtains the expression ḣ2 = ḣ1 (-β

I + Q) + h 2 (-β I + Q) h 1 (-β I + Q) -h 1 Q + Q which implies ḣ1 (-β I + Q) 2 + (h 2 h 1 -ḣ2 )(-β I + Q) +h 2 (-h 1 Q + Q) = 0
Observe that this last equation is a second order polynomial in the variable

X = -β I + Q. Since I(0) = ϵ, R(0) = Q(0) = 0 and S(0) = N -ϵ.
From this and R 0 > 1, one has (using expression (7))

ḣ1 (0) = β ε N (-β + α) 1 - ε N < 0 (9)
and this allows us to show that one also has

h 2 (0) -h 1 (0) Q(0) + Q(0) = α β ρ ε 2 (β -α) 1 - ε N > 0 (10)
Observe also that one has X(0) = -β I(0) + Q(0) = ε (-β +α) < 0. Therefore, by continuity w.r.t. t, we obtain that for t > 0 small enough, X is the unique negative solution of ḣ1

X 2 + (h 2 h 1 -ḣ2 ) X + h 2 (-h 1 Q + Q) = 0.
that is

X = -(h 2 h 1 -ḣ2 ) -(h 2 h 1 -ḣ2 ) 2 + 4 ḣ1 (h 1 ẏ2 -ÿ2 ) 2 ḣ1 .
The parameter α can be then obtained from equations ( 6)-( 7)

α = (N -y 2 ) ḣ1 X -h 1 -ρ
where ρ is given by ( 5). The initial condition ε is simply reconstructed by ε = y 1 (0)/α and finally one obtains the parameter β = α -X(0)/ε.

ANALYSIS OF THE SIMPLIFIED MODEL

Proposition 4. System (2) is identifiable and observable for the observation (4), in the sense of Definition 2.

Proof. The proof is somehow simpler than the precedent.

Using similar arguments we show that the parameters can be expressed as follows:

             ρ = y 1 (t) -ẏ2 (t) y 2 (t) , t > 0, α = - N ġ1 (t) βI(t) -g 1 (t) = - N ġ1 (t) g 2 (t) -g 1 (t) := g 3 (t), β = α βI(t) y 1 (t) = g 3 (t) g 2 (t) y 1 (t) = -N ġ1 (t) -g 1 (t)g 2 (t) y 1 (t) .
Where

               g 1 (t) := ẏ1 (t) y 1 (t) + ρ, g 2 (t) := -N g1 (t) ġ1 (t) -g 1 (t) + ρ = βI(t), g 3 (t) := - N ġ1 (t) g 2 (t) -g 1 (t).
At last, the initial condition is recovered as ε = y 1 (0)/α.

PRACTICAL ESTIMATION OF THE PARAMETERS

Till now we have studied structural observability / identifiability. While structural identifiability is a property of the model structure, given a set of outputs, practical identifiability is related to the experimental data. A model can be structurally identifiable, but still be practically unidentifiable due to poor data quality, e.g., bad signal-tonoise ratio, errors in measurement or sparse sampling [START_REF] Raue | Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood[END_REF]. This means parameters are identifiable with ideal data (in continuous time, noise-free data). However it does not guarantee that they will be practically identifiable with a finite number of noisy data points. Moreover estimation of parameters will use numerical algorithm and the distance, for the problem considered, to the nearest illposed problem [START_REF] Demmel | On condition numbers and the distance to the nearest ill-posed problem[END_REF]). Particularly this is the case when the system is near a non-observable equilibrium [START_REF] Banks | Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls[END_REF]. The main difficulty with epidemic models is that often parameters need to be reconstructed as early as possible at the beginning of the epidemic, in other words in the neighborhood of the Disease Free Equilibrium (DFE), which is a non-observable equilibrium.

Here after, we assume that the initial condition of model ( 1) or ( 2) is (N -ε, ε, 0, 0) with unknown small ε > 0.

Table 1 gives the parameters values used for data generation. We have considered daily measurements of variables 2) are almost indistinguishable over this time period).

The classical least-square method

We have first tested the classical least-square method on the perfect data (without noise) over increasing time intervals [0, T ] with T = 2, • • • , 30, using the Levenberg-Marquardt algorithm (lsqrsolve function in Scilab software). Fig. 1 shows that parameters values found by the algorithm are quite unstable depending on the time interval, making this method practically unreliable. Yet, one can see on Fig. 2 that the variables y 1 and y 2 are quite well reproduced (apart a few unrealistic points that are not represented).

Fig. 1. Application of the least-square method with perfect data on increasing time intervals.

An alternative method

Here, we aim at exploiting the structure of the model (3) by decoupling the reconstruction of the parameter ρ from the estimations of α and β, and using an approximation of the dynamics of the variable I that is valid as long as S stays very large compared to I, for the reconstruction of α and β once ρ has been estimated. Reconstruction of parameter ρ. Following the identifiability analysis, parameter ρ can be easily identified integrating expression (5). Thus, the parameter ρ can be obtained by linear regression using the available values of the two outputs at the measurement times t i by

ti+1 ti y 1 (τ )dτ + y 2 (t i ) -y 2 (t i+1 ) = ρ ti+1 ti y 2 (τ )dτ
Assuming that measurements are done daily (the unit of time here is a day) the latter can be approximated by the expression y 1 (t + 1) + y 1 (t) 2 + y 2 (t) -y 2 (t + 1) = ρ y 2 (t + 1) + y 2 (t) 2 for t = 0, 1, ..., n.

Reconstruction of parameters α and β. Here, we suppose that ρ is known, i.e. we use the approximation of ρ given before. Posit

η := β N α .
Observe that the solution S(•) of model (2) satisfies

S(t) = S(0)e -η t 0 y1(τ )dτ , t > 0
and thus the solution

I(•) fulfills İ(t) = β S(0) N e -η t 0 y1(τ )dτ -(α + ρ) I(t), t > 0.
Under the assumption that I(0) = ε is small compared to N , the dynamics of y 1 (•) can then be approximated by the expression

ẏ1 (t) = βe -η t 0 y1(τ )dτ -(α + ρ) y 1 (t) (11)
for small time t > 0, from which one can straightforwardly check that one has also

log d dt log y 1 (t) + α + ρ = log(β) -η t 0 y 1 (τ )dτ.
Therefore, the parameter α possesses the property that the variables

Y (t, α) := log d dt log y 1 (t) + α + ρ , X(t) := t 0 y 1 (τ )dτ fulfill a linear dependency: Y (t, α) = log(β) -ηX(t).
Consequently, parameter α can be determined as the value that gives a linear regression between variables Y (t) and X(t) for a set of values t close to 0. Then parameter β can be determined from the Y -intercept of the regression line.

Practically, one filters the data log y 1 (•) to estimate the derivative of log y 1 (•) (for instance with moving average or polynomial adjustment methods) and then α is sought to minimizing the residual sum of squares in the linear regression of the {Y (•), X(•)} data (see for instance [START_REF] Walter | Identification of parametric models[END_REF]). The alternative method has been applied first on the same "perfect" data set (without noise) as before, and the parameters estimation is given on Fig. 3. As expected, the estimation of ρ is quite faithful and Fig. 3. Application of the alternative method on perfect data (without noise).

fast, while the estimation of α, β follows in sequence. We have also tested this method on the same data set but corrupted by an additive noise with a normal distribution of null mean value and standard deviation of 0.5 and 5 (see Fig. 4 and5). In all cases, data have been filtered with a mobile average followed by a Lagrange interpolation. Time derivatives have been then computed analytically on the interpolated polynomial. One can see that the estimation of ρ remains always reliable, but requires a longer time to stabilize, which impacts the reconstruction of the other parameters, especially under large noise. However, the reproduction number R 0 is estimated with a relatively good accuracy. Indeed, after 30 days, we move away from the validity of the approximation (11), which explains the bias obtained on α and β under large noise.

Numerical differentiation

For larger time windows, when the approximation (11) becomes less accurate, we investigate another approach based on numerical differentiation without any approximation of the system. Those techniques become today more Fig. 4. Application of the alternative method on data with small noise.

Fig. 5. Application of the alternative method on data with large noise.

prominent due to increasing interest in data driven models -cf. [START_REF] Kaiser | Sparse identification of nonlinear dynamics for model predictive control in the low-data limit[END_REF] and references therein. Indeed, when the signals y 1 (•) and y 2 (•) become significantly larger (after 30 days in our simulations), we expect their time derivatives to be better estimated despite some noise.

Here we shall use a Total Variation Regularization (TVR) method for numerical differentiation firstly presented in Chartrand (2011) (see also [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF]) for the original development of TVR applied to noise reduction in images. In what follows, we will refer to this approach to numerical differentiation as TVRD.

A brief review of TVR and numerical differentiation The use of Total Variation Regularization, as its name itself indicates, is a regularization approach of the following type:

F (u) = κR(u) + M (Au -f ),
where R is a functional that penalizes irregularity, A is the antiderivative operator and M is a metric functional that penalizes discrepancy between Au and f -κ is the regularization parameter that weighs both penalties.

For TVR, as presented in [START_REF] Chartrand | Numerical differentiation of noisy, nonsmooth data[END_REF], the metric functional is the L 2 norm, whereas the regularizing one is the total variation norm -namely, we have

F (u) = κ L 0 |u ′ | dx + 1 2 L 0 |Au -f | 2 dx. ( 12
)
For the numerical implementation of ( 12) we follow the original presentation in [START_REF] Chartrand | Numerical differentiation of noisy, nonsmooth data[END_REF], for which we refer the reader for further details. In addition, we also use the selection framework developed in [START_REF] Van Breugel | Numerical differentiation of noisy data: A unifying multi-objective optimization framework[END_REF].

Estimation of ρ. We have

ρy 2 = y 1 -ẏ2 (13)
The right hand side of ( 13) is computed by applying the TVRD to measurement y 2 . An estimate ρ is then obtained by performing a linear regression.

Estimation of α and β. These parameters require a more involved estimation. We use the derivative of h 1 written in a convenient way as follows

-ḣ1 = ηy 1 (h 1 + α) , η = β N α . ( 14
)
We then estimate η and α using a non linear least squares fitting. This estimation is usually accurate for η but not always for α. Thus we further estimate α by first computing

y 3 = ηy 1 ,
which is an estimation of βI N . In the sequel we then use a linear regression to estimate α using (14) in the following form

-ḣ1 + y 3 h 1 = αy 3 (15)
in order to obtain α. The absolute value is applied to ensure that the left hand side is non-negative and it works as a kind of filter for the errors introduced in the numerical differentiation.

Some numerical examples

We simulated System (3) using the parameters ρ = 0.1, α = 0.07, β = 0.4 and N = 10 5 . The initial condition is i 0 = 150N -1 , r 0 = 0 and s 0 = 1 -i 0 . This will be referred to as synthetic data. From this data we obtain the measurements y 1 and y 2 . We also perturbed these measurements in two ways: a deterministic high frequency sinusoidal perturbation and a random one.

With synthetic data In this case, we plot the power spectra density of the data to be differentiated together with the cut off frequency used in each case. For data with slow decaying the power spectra we used half of the frequency of the grid, unless a sharper transition was identified. For data with fast decaying -typically ln y 1 and h 1 we used either 10 or 20 times the grid frequency depending on how fast it is decaying. For these choices, we get the following estimates ρ = 0.107, α = 0.0748, and β = 0.413 With deterministic perturbation In this case we obtain, using a time sample of 40 days, the following estimates: ρ = 0.0972, α = 0.0646, and β = 0.375

In this case, the cutting frequency was first estimated as above, and if the result was still excessive wiggling (oscillations considerably greater than the average data magnitude) it was reduced until these oscillations were brought to this typical size.

With random perturbation. Selection of the cutting frequency was done in the same way as in the case of the deterministic perturbation. Further remarks

• The TVR method does not use any linear approximation and it is already in discrete form; • TVR method can be improved by using TV of jerk (third derivative) which would give a smother differentiation; • Integral version of the procedure can be also implemented -h 1 still need to be computed by differentiation however.

CONCLUSION

In this work, we have established the structural identifiability and observability of the SIR model with a quarantine compartment. We have also derived several methods to practically estimate the various parameters of the model.

In particular, we have developed two complementary approaches adapted to the beginning of an epidemic or to its later development. In a forthcoming work, we shall investigate the practical estimation of these parameters using observers with application to some real data of COVID epidemics provided by various territories.

Fig. 2 .

 2 Fig.2. Observations predicted by the least square estimation (output variables of the system in plain lines).

Table 1 .

 1 Parameters used for the simulations

	N	α	β	ρ	I(0)
	10 6	0.1	0.4 0.1	20
	y 1 and y 2 over a period of 30 days, first as "perfect" data
	(i.e. without noise) generated by the integration of model
	(3) (indeed data produced by model (1) or (


1

Theoretical demonstration of a hot-carrier operation in ultra-thin solar-cell

Nicolas Cavassilas 1* , Imam Makhfudz 1 , Anne-Marie Daré 1 , Michel Lannoo 1 , Guillaume Dangoisse 2,3 , Marc Bescond 1,2 and Fabienne Michelini 1 1 Aix Marseille Université, CNRS, Université de Toulon, IM2NP UMR 7334, 13397, Marseille, France 2 LIMMS-CNRS, IRL 2820, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan [START_REF] Humphrey | Reversible Quantum Brownian Heat Engines for Electrons[END_REF] Ecole Normale Supérieure de Paris, Université PSL, Sorbonne Université, Université de Paris, 75005 Paris, France, 24 rue Lhomond F75005 Paris, France *corresponding author: nicolas.cavassilas@im2np.fr Based on a quantum modeling of the electronic transport, this work shows that ultra-thin solar cells can exhibit an improved open-circuit voltage 𝑉 𝑜𝑐 , without current reduction. This improvement is obtained when an energy-selective contact is considered between the absorber and the reservoir, and is attributed to a hot-carrier effect. While extraction with a non-selective contact does not generate hotcarriers, the use of energy-selective contact induces an increase of carrier temperature up to 130 K and a corresponding 𝑉 𝑜𝑐 enhancement of 41 meV, considering an InGaAs absorber. This enhancement agrees with a simple and general expression formulated in the quantum thermal machine field. Concerning the current, we show that current through an energy-selective contact is of the same order of magnitude as the one obtained without selectivity. This remarkable behavior, which is explained by the hybridization of states in the absorber with the state of the contact, requires a quantum confinement and thus an ultra-thin absorber. Finally, we propose a simple rate-model enabling an intuitive interpretation of the numerical results.

I INTRODUCTION

The hot-carrier solar cell (HCSC) is an elegant concept 1 allowing, in theory, to exceed the limit of Shockley-Queisser (SQ) 2 . The idea, as shown in Fig. 1, is that the carriers remain hot in the absorber where they have been photo-generated by a hot source (the sun). Extracting these carriers through energy-selective contacts (for instance at given energies E 𝑛 and 𝐸 𝑝 for electrons and holes respectively) we can theoretically recover this excess of thermal energy in the form of voltage in the reservoirs. The open-circuit voltage can be formally obtained by a simple expression by assuming two different local thermal equilibriums for the reservoirs and the absorber, and sharp energy filters between these equilibriums. To do so, we consider the entropy balance [START_REF] Humphrey | Reversible Quantum Brownian Heat Engines for Electrons[END_REF] of the transfer of an electron at energy 𝐸 𝑛 from the absorber (with a Fermi level 𝜇 𝑛𝑎 and a temperature 𝑇 𝑎 ) to the n-reservoir (with a Fermi level 𝜇 𝑛 and at lattice temperature 𝑇 𝐿 )

∆𝑆 = - 𝐸 𝑛 -𝜇 𝑛𝑎 𝑇 𝑎 + 𝐸 𝑛 -𝜇 𝑛 𝑇 𝐿 . ( 1
)
The perfect energy selectivity enables an isentropic transfer with ∆𝑆= 0. In this case Eq.(1) shows that both electronic distributions in the absorber and in the n-reservoir must have the same value at the transfer energy 𝐸 𝑛 . [START_REF] Humphrey | Reversible Quantum Brownian Heat Engines for Electrons[END_REF] Doing the same analysis for holes at the transfer energy 𝐸 𝑝 , and summing the two expressions, we finally get 1,4

𝑞𝑉 𝑜𝑐 = (1 - 𝑇 𝐿 𝑇 𝑎 ) 𝐸 𝑒𝑥𝑡 + ∆μ 𝑎 𝑇 𝐿 𝑇 𝑎 (2)
with 𝑞𝑉 𝑜𝑐 = 𝜇 𝑛 -𝜇 𝑝 , the open-circuit voltage, 𝐸 𝑒𝑥𝑡 = 𝐸 𝑛 -𝐸 𝑝 and ∆𝜇 𝑎 = 𝜇 𝑛𝑎 -𝜇 𝑝𝑎 . This expression was derived in the context of quantum thermal machine [START_REF] Humphrey | Reversible Quantum Brownian Heat Engines for Electrons[END_REF] and extrapolated to the solar cell field 1,4-6 . This equation shows that if 𝑇 𝑎 = 𝑇 𝐿 (no hot-carriers), we obtain 𝑞𝑉 𝑜𝑐 = ∆𝜇 𝑎 = ∆𝜇 𝐿 , i.e. the Fermi level splitting when the carriers in the absorber are at the lattice temperature. In this case, which corresponds to a conventional solar cell, 𝑉 𝑜𝑐 is directly related to the rate of the radiative generation and recombination in the absorber and therefore, to its energy bandgap 𝐸 𝑔 , leading to the SQ limit. On the other hand, if 𝑇 𝑎 ≫ 𝑇 𝐿 and if electrons and holes are close to equilibrium (for example due to impact ionization) [START_REF] Würfel | Solar energy conversion with hot impact ionization[END_REF] , we then have ∆𝜇 𝑎 ≈ 0 and consequently 𝑞𝑉 𝑜𝑐 ≈ 𝐸 𝑒𝑥𝑡 . It is then possible to consider a material for the absorber with a small 𝐸 𝑔 , to have an significant short circuit current, while considering a contact offering a large 𝐸 𝑒𝑥𝑡 , to have a large 𝑉 𝑜𝑐 . We are therefore freed from the trade-off on 𝐸 𝑔 at the origin of the SQ limit.

Figure 1. Schematic representation of a hot-carrier solar cell. In the absorber the electron-and hole-distributions are defined by the Fermi levels 𝜇 𝑛𝑎 and 𝜇 𝑝𝑎 and by the temperature 𝑇 𝑎 . In both reservoirs the temperature is given by the lattice temperature 𝑇 𝐿 . The electron-distribution in n-reservoir is defined by 𝜇 𝑛 while the hole-distribution in p-reservoir is defined by 𝜇 𝑝 . Carriers can transit from the absorber into the reservoirs through contacts at energies 𝐸 𝑛 and 𝐸 𝑝 , for electrons and holes, respectively.

In practice, the main difficulty in the development of such solar cells is to obtain a hot-carrier effect, generating a 𝑉 𝑜𝑐 improvement, without degrading the extraction of carriers to preserve the current. Indeed, in isolated system (without contact), hot-carrier distributions have been observed under light illumination in steady-states. [START_REF] Nguyen | Quantitative experimental assessment of hot carrier-enhanced solar cells at room temperature[END_REF][START_REF] Hirst | Hot carriers in quantum wells for photovoltaic efficiency enhancement[END_REF][START_REF] Giteau | Identification of surface and volume hot-carrier thermalization mechanisms in ultrathin GaAs layers[END_REF] In such systems the carriers are generated at an average energy 𝐸 𝑔𝑒𝑛 = 𝑘 𝐵 𝑇 𝑔𝑒𝑛 strongly larger than 𝑘 𝐵 𝑇 𝐿 . The carriers cool down in a two-step process: i) they lose energy by optical-phonon emission, and ii) each optical-phonon splits into two acoustic phonons. Assuming a reservoir of acoustic-phonons at the lattice temperature 𝑇 𝐿 , generated carriers thermalize at an energy 𝑇 𝑐 located between 𝑇 𝑔𝑒𝑛 and 𝑇 𝐿 . If the cooling rate is larger than the generation rate, 𝑇 𝑐 is close to 𝑇 𝐿 , while if the generation rate is larger than the cooling rate, 𝑇 𝑐 is close to 𝑇 𝑔𝑒𝑛 . [START_REF] Tsai | Theoretical model and simulation of carrier heating with effects of nonequilibrium hot phonons in semiconductor photovoltaic devices[END_REF] The goal is thus to increase the generation rate with a large power illumination, and to reduce the cooling rate. In case where the cooling is limited by the optical-acoustic rate, hot-carriers emit optical phonons which are preferentially reabsorbed by the carriers rather than split into two acoustic phonons. This behavior seems to be at the origin of hot-carrier in bulk [START_REF] Giteau | Identification of surface and volume hot-carrier thermalization mechanisms in ultrathin GaAs layers[END_REF][START_REF] Tsai | Theoretical model and simulation of carrier heating with effects of nonequilibrium hot phonons in semiconductor photovoltaic devices[END_REF] but could also explains observations made in quantum wells. [START_REF] Zhang | Quantitative study on the mechanisms underlying the phonon bottleneck effect in InN/InGaN multiple quantum wells[END_REF] Moreover, in quantum well, due to more restrictive selection rules, the carrier-optical phonon rate can also be reduced. In all cases, these strategies to reduce the cooling rate are called the phonon bottleneck.

In opposition to an isolated system, we can consider a system in which the absorber is perfectly connected to a reservoir. This means that electronic states of the absorber are totally delocalized in the reservoir and vice versa. A priori, the most favorable case is when a generated hot-carrier can ballistically reach the reservoir, before experiencing an optical phonon emission. However, once in the reservoir, such an electron cools down. Moreover, at 𝑉 𝑜𝑐 , the ballistic carrier is replaced by a cold carrier coming from the reservoir. Hence, the electronic distribution in the absorber is almost equal to that of the reservoir and hot-carrier effect cannot be observed.

A practical HCSC has to maintain the carriers hot by isolating the absorber from the cold reservoir, and on the other hand, it has to enable an efficient extraction of carriers. Those two requirements might be seen as contradictory, but quantum mechanism can break this paradox with a resonant tunneling behavior. 13,[START_REF] Dimmock | A metallic hot-carrier photovoltaic device[END_REF] As we have already investigated, [START_REF] Cavassilas | Hot-carrier solar cell NEGF-based simulations[END_REF] if the absorber is a quantum well, one of the confined states hybridizes with the resonant state in the energy-selective contact. It results an efficient extraction of the photo-generated carriers than can be assisted by phonon scattering. This hybridization is equivalent to delocalize the absorber states into the reservoir, but only at a chosen energy-window. For the other energies, and particularly when the electronic distribution in the reservoir is larger than that in the absorber (close to the band edge), the isolation has to be severely preserved.

In this theoretical article, we confirm that the use of an ultra-thin absorber enables to implement the hot electron strategy. In such a structure, considering an electron selective contact, consisting of a double barrier and a quantum dot (QD) or a quantum well (QW), our numerical calculations show that 𝑉 𝑜𝑐 varies with the extraction energy 𝐸 𝑛 . Since such variations of 𝑉 𝑜𝑐 are consistent with equation (2), they are attributed to the hot-carrier behavior. Moreover, we confirm that the absorption current is preserved by the energy-selective contact. Advances in the fabrication of ultra-thin solar cells, based both on the control of contacts and on the photonic environment, could therefore result in HCSC developpment. [START_REF] Massiot | Progress and prospects for ultrathin solar cells[END_REF]

II. METHODOLOGY

To carry out this study, we model ultra-thin cells with the non-equilibrium Green's functions (NEGF) formalism in a self-consistent framework with the Poisson equation. This physical model, which includes the effects of interactions in a picture of quantum transport, is widely used in the field of semiconductor quantum devices. [START_REF] Cavassilas | Modeling of nanoscale solar cells: The Green's function formalism[END_REF][START_REF] Yangui | Evaporative electron cooling in asymmetric double barrier semiconductor heterostructures[END_REF][19] It indeed enables to accurately consider behaviors such as quantum confinement, tunneling, electron-phonon and electron-photon scatterings and to model the effects of the semi-infinite reservoirs.

Here, we consider a multi-unidimensional model, [START_REF] Cavassilas | Modeling of nanoscale solar cells: The Green's function formalism[END_REF] meaning that we consider a unidimensional potential along the transport direction, and an invariant potential in transverse plane. In this plane, the dispersions of electron and hole are reproduced by a discretization of the transverse wave-vector 𝑘 𝑡 . For these dispersions we assume the effective mass approximation which is also considered in the transport direction. For QD, we do not consider such a transverse dispersion, equivalent to a null transverse kinetic energy. In the present model, this is the only difference between QD and QW. Consequently, the energy position of the confined state is controlled in both cases by the distance between the two confining barriers, while for three-dimensional QD it should be controlled by its shape. This approach is suitable for QD offering an oblate spheroid shape. It also enables to treat any other cases considering that the distance between the two confining barriers is effective. At the boundaries of the devices, in the transport direction, we assume semi-infinite reservoirs meaning that the potential in the reservoir is invariant on semi-infinite length and that electrons follow a thermal distribution at 300 K inside these reservoirs.

In the present study, we consider electron-photon, electron-optical phonon and electron-acoustic phonon scatterings. For scattering with photons, in order to simulate conditions close to the experimental ones, [START_REF] Esmaielpour | Investigation of the spatial distribution of hot carriers in quantum-well structures via hyperspectral luminescence imaging[END_REF] we suppose that the cells are illuminated by a 954 nm laser with a power of 1 kW.cm -2 (power equivalent to 10 4 suns). On the other hand, the cells can emit photons at any energy, such an emission being due to the interband radiative recombination. Note that we assume the radiative approximation, i.e. non-radiative recombinations are neglected.

For scattering with phonons, we consider that both optical and acoustic phonons remain at 𝑇 𝐿 =300 K. Indeed, we only calculate the Green functions of the electrons while phonons are considered at thermodynamical equilibrium. We are aware that this approximation is a limitation of our model. But, this limitation reduces the observed hot-carrier effect. Thus, our model cannot overestimate the conclusions of the present work.

Another substantial approximation is that we do not consider the electron-electron interaction in the active region. This scattering does not diffuse energy and therefore does not participate to the cooling. But it favors the transition from an out-of-equilibrium distribution to a Fermi distribution. This approximation explains why the distribution obtained in the present work are not well described by a Fermi function. Note that this interaction is intrinsically taken into account in the reservoir in which Fermi distributions at room temperature are imposed.

The model we employ is fully described in Ref. [17], but we summarize below the principal lines of the NEGF framework, using matrix representation and considering that all constants equal 1 (the Planck constant ħ, the elementary charge q, the transverse surface of the cell, the position mesh along the transport direction and π). For each transverse wave-vector 𝑘 𝑡 , we define a Hamiltonian matrix 𝐻 𝑐,𝑣 for the conduction band (CB) and the valence band (VB). Diagonal matrices α represent the potential on each space-point along the transport direction, while the off-diagonal β matrices represent the hopping between two successive space-points. The potential is self-consistently calculated with the Poisson equation. The retarded component of the Green's function (function of 𝐸, the electronic energy) is determined from this Hamiltonian matrix following

𝐺 𝑐,𝑣 𝑟 = [𝐸 -𝐻 𝑐,𝑣 -Σ 𝑟 ] -1 (3)
where Σ 𝑟 is the sum of the self-energies describing the influence of reservoirs and the different scatterings taken into account. The imaginary part of the diagonal terms of 𝐺 𝑐,𝑣 𝑟 provides the local density-of-states (LDOS) defined versus both position and energy. Then, the lesser and greater components of the Green's function are calculated from the retarded Green function

𝐺 𝑐,𝑣 ≶ = 𝐺 𝑐,𝑣 𝑟 Σ 𝑐,𝑣 ≶ 𝐺 𝑐,𝑣 𝑟 † (4)
where Σ 𝑐,𝑣 ≶ are the lesser and greater self-energies which will be defined in the following. The imaginary parts of the diagonal terms of 𝐺 < (𝐺 > ) provide the electron (hole) density. The off-diagonal terms are related to the spectral current between two successive space-points 𝐽 𝑐,𝑣 = 𝛽 𝑐,𝑣 G 𝑐,𝑣 < -𝛽 𝑐,𝑣 † G 𝑐,𝑣 > giving the total current 𝐽 = 𝐽 𝑣 +𝐽 𝑐 . Such a current is defined in position and in energy and can then be shown by a two-dimensional plot (Figs. 2a, b and c in the following).

The lesser and greater self-energies are defined as the sum of the self-energies relative to the influence of electron-photon interaction (optical), to the electron-phonon scattering (scattering) and to the boundaries condition with the semi-infinite reservoirs (reservoirs)

Σ 𝑐,𝑣 ≶ = Σ 𝑐,𝑣(𝑜𝑝𝑡𝑖𝑐𝑎𝑙) ≶ + Σ 𝑐,𝑣(𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑖𝑛𝑔) ≶ + Σ 𝑐,𝑣(𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟𝑠) ≶ . ( 5
)
Here we focus only on the optical component which is given by Σ 𝑐,𝑣(𝑜𝑝𝑡𝑖𝑐𝑎𝑙) >,< = 𝑀 𝑜𝑝 2 (𝑁 𝑜𝑝 + 1)𝐺 𝑣,𝑐 >,< and Σ 𝑐,𝑣(𝑜𝑝𝑡𝑖𝑐𝑎𝑙) <,> = 𝑀 𝑜𝑝 2 𝑁 𝑜𝑝 𝐺 𝑣,𝑐 <,> with 𝑀 𝑜𝑝 the optical matrix element and the number of photons 𝑁 𝑜𝑝 (given by the Bose distribution). Once we have the total Σ 𝑐,𝑣 ≶ , we can evaluate the retarded self-energy using Σ 𝑐,𝑣 𝑟 = (Σ 𝑐,𝑣 > -Σ 𝑐,𝑣 < )/2, which is needed to calculate the retarded Green function with Eq.(3). To assess the functioning of the cell, we will also rely on the electrons flux across the bandgap due to the processes of photon absorption and emission

𝐽 𝑎𝑏𝑠 = Σ 𝑐(𝑜𝑝𝑡𝑖𝑐𝑎𝑙) < 𝐺 𝑐 > = 𝑀 𝑜𝑝 2 𝑁 𝑜𝑝 𝐺 𝑣 < 𝐺 𝑐 > , ( 6
)
𝐽 𝑟𝑒𝑐 = Σ 𝑣(𝑜𝑝𝑡𝑖𝑐𝑎𝑙) < 𝐺 𝑣 > = 𝑀 𝑜𝑝 2 (𝑁 𝑜𝑝 + 1)𝐺 𝑐 < 𝐺 𝑣 > . ( 7
)
𝐽 𝑎𝑏𝑠 is proportional to the hole density in CB (𝐺 𝑐 > ) times the electron density is VB (𝐺 𝑣 < ), while 𝐽 𝑟𝑒𝑐 is proportional to the electron density in CB (𝐺 𝑐 < ) times the hole density in VB (𝐺 𝑣 > ). Finally, note that if electrons (holes) can only be extracted in the n(p)-contact, we have the total current 𝐽 = 𝐽 𝑎𝑏𝑠 -𝐽 𝑟𝑒𝑐 . This condition is respected in the devices investigated in the present work.

Since the diagonal elements of 𝐺 𝑐,𝑣 𝑟 give the LDOS and the diagonal elements of 𝐺 𝑐,𝑣 ≶ give the electron and hole densities, we can calculate the distribution functions of carriers in CB and VB. The LDOS also enables to determine the extraction energy 𝐸 𝑛 , since the state localized between the two barriers is visible on a two-dimensional plot of the LDOS versus position and energy (Fig. 3 in the following). All these quantities are obtained for a bias applied between the two reservoirs.

To obtain the open-circuit voltage, we interpolate the bias giving 𝐽 = 𝐽 𝑎𝑏𝑠 -𝐽 𝑟𝑒𝑐 = 0 (linear and exponential interpolations versus the bias for 𝐽 𝑎𝑏𝑠 and 𝐽 𝑟𝑒𝑐 , respectively).

III. RESULTS

Figure 2 represents the band diagrams and the current spectra obtained for three different contacts at a voltage of 0.7 V (μ 𝑝 =0 eV and μ 𝑛 =0.7 eV). This contact is either a simple tunnel barrier (Fig. 2a), thus nonselective, or a QD between two tunnel barriers (Fig. 2b andc), thus selective. In Figs. 2b andc the size of the QD varies, thus modifying the extraction energy 𝐸 𝑛 (0.94 eV and 1.02 eV respectively). In all three cases, we consider an InGaAs absorber, 12 nm thick, with an InP p-reservoir and an InGaP wetting layer.

The n-reservoir is also made of InP but is separated from the absorber by the contact. The two confining barriers of the n-contacts are in AlGaAsSb. As shown in Fig. 2, the absorption currents [START_REF] Galvani | Impact of Electron-Phonon Scattering on Optical Properties of CH3NH3PbI3 Hybrid Perovskite Material[END_REF] 𝐽 𝑎𝑏𝑠 are 21.7, 23.7 and 23.5 A cm -2 , for the cell without selectivity and for 𝐸 𝑛 =0.94 eV and 1.02 eV, respectively. As shown in Eq.( 6), 𝐽 𝑎𝑏𝑠 is proportional to the hole density in CB, 𝐺 𝑐 > , which is related to the LDOS and to the occupancy. The change of LDOS in the absorber with the different n-contacts (simple or double-barrier) is negligible. Another way to explain the variation of 𝐽 𝑎𝑏𝑠 is the extraction of carriers by the n-contact. In case of a reduced extraction, the electron density increases in absorber. This corresponds to a decrease of hole density in CB and thus to a decrease of 𝐽 𝑎𝑏𝑠 . We thus conclude that extraction is more efficient when a selective contact is considered. To explain this result, we represent in Fig. 3 the density-of-states in the cell with 𝐸 𝑛 =1.02 eV.

The third state of the absorber hybridizes with the state in the QD. [START_REF] Cavassilas | Hot-carrier solar cell NEGF-based simulations[END_REF] An electron on this state is thus delocalized in the contact and then in the reservoir. Moreover, thanks to the assistance of optical phonon-emission, this extraction is even more efficient. This phonon-assisted extraction can be seen on the current-spectra represented in Fig. 2, where in case of non-selective contact the current absorption (the positive one) is ballistic, while phonons contribute in the cases of selective contacts. Such a hybridization appears only when electrons are confined in absorber. In case of a bulk absorber, as considered theoretically [START_REF] Kirk | Fundamental limitations of hot-carrier solar cells[END_REF] and experimentally, [START_REF] Limpert | Single-nanowire, low-bandgap hot carrier solar cells with tunable open-circuit voltage[END_REF] a consequent reduction in current is observed with selective contact. Concerning the recombination current (the negative contribution on current spectra in Fig. 2), at a given bias, the amplitude is reduced with the selective contacts. This confirms that the isolation, at band-edge, is much more efficient with the double-barrier. Concerning the open-circuit voltage, 𝑉 𝑜𝑐_𝑛𝑢𝑚 , we obtain 0.695 V (non-selective), 0.715 V (𝐸 𝑛 = 0.94 eV) and 0.736 V (𝐸 𝑛 = 1.02 eV). We see that the non-selective contact cell offers the weakest 𝑉 𝑜𝑐_𝑛𝑢𝑚 and that 𝑉 𝑜𝑐_𝑛𝑢𝑚 increases with the extraction energy 𝐸 𝑛 . We then calculate the electronic distribution in the absorber and in the n-reservoir following the Kubo-Martin-Schwinger relation [START_REF] Michelini | Entropy production in photovoltaic-thermoelectric nanodevices from the non-equilibrium Green's function formalism[END_REF] . These distributions are plotted as a function of the energy on Fig. 4 for the three devices of Fig. 2. As expected, the distribution in the three n-reservoirs is exactly equal to the Fermi function with μ 𝑛 =0.7 eV and 𝑇=300 K. The distribution in the absorber, for the three cases, exhibits a ballistic component at large energy corresponding to the generated electrons before they emit phonons. These ballistic electrons are athermal and represent a negligible proportion of the total electronic density. The difference between the three devices concerns electrons having experienced scattering and is observable at lower energy (denoted thermal or pseudo-thermal). When considering a simple tunnel barrier (Fig. 4a), the distribution in the absorber is almost equal to the contact distribution, i.e. no hot-carrier effect is observed. With a selective contact (Fig. 4b andc), the distributions in the absorber are unusual and correspond to out-of-equilibrium electrons. We first see that they depict peaks. Observation of the distribution for each transverse wave-vector 𝑘 𝑡 shows that these peaks are related to the optical phonon, since they are spaced apart by the energy of the optical phonon. Independently of these peaks, we propose to fit (insets of Fig. 4) these two distributions by a Fermi function. We obtain μ 𝑛𝑎 =0.62 eV and 𝑇 𝑎 =430 K for 𝐸 𝑛 =0.94 eV and μ 𝑛𝑎 =0.6 eV and 𝑇 𝑎 =430 K for 𝐸 𝑛 =1.02 eV. Even though we cannot rigorously consider an equilibrium and thus a temperature, this result shows that such selective contacts isolate well enough the absorber, from the reservoirs, to allow the electrons in the absorber to maintain a hot pseudo-thermal distribution. This result also shows that a hot equilibrium can be obtained although optical phonons are at lattice temperature. Considering hot phonons should increase the electronic temperature. To confirm the link between 𝑉 𝑜𝑐 and this hot pseudo-equilibrium distribution, we modify equation (2) in order to consider the holes are at 𝑇 𝐿 . Indeed, in absorber we obtain holes at room temperature, for any considered cell, with the same Fermi level as in the left reservoir μ 𝑝 = 0 (not shown). This absence of hot holes was expected since we do not consider selective contact. We then obtain this new equation for 𝑉 𝑜𝑐_𝑎𝑛 (an for analytical):

𝑉 𝑜𝑐_𝑎𝑛 = (1 - 𝑇 𝐿 𝑇 𝑎 ) 𝐸 𝑛 + μ 𝑛𝑎 𝑇 𝐿 𝑇 𝑎 -μ 𝑝 . (8)
By applying this expression to the two cells with 𝐸 𝑛 =0.94 eV and 1.02 eV, and by considering the corresponding μ 𝑛𝑎 and 𝑇 𝑎 , we find respectively 𝑉 𝑜𝑐_𝑎𝑛 =0.717 V and 0.727 V (versus 𝑉 𝑜𝑐_𝑛𝑢𝑚 = 0.715 V and 0.736 V obtained by simulations). These values should be compared to 0.695 V, the numerical 𝑉 𝑜𝑐 obtained without hot-carriers. The agreement between the numerical and the analytical values confirms that the improvement of 𝑉 𝑜𝑐_𝑛𝑢𝑚 with the selectivity of contact is due to a hot-carriers effect. We also conducted calculations for a semi-selective contact considering a QW rather than a QD. In this case, with 𝐸 𝑛 =1.02 eV, we obtain numerically 𝐽 𝑎𝑏𝑠 =20.3 A cm -2 , 𝑉 𝑜𝑐_𝑛𝑢𝑚 =0.728 V, μ 𝑛𝑎 =0.62 eV and 𝑇 𝑎 =400 K, which gives 𝑉 𝑜𝑐_𝑎𝑛 =0.720 V. Even in the case with QW, the numerical and analytical values are in good agreement. Compared to the case with the QD contact having the same E 𝑛 , all the criteria show a less efficient solar cell. Particularly, the current spectra represented Fig. 5 shows that, likewise to the case with a simple barrier, the current is mainly ballistic. This means that the hybridization observed with the QD contact does not assist the extraction, explaining the smaller 𝐽 𝑎𝑏𝑠 (20.3 versus 23.5 A cm -2 ). However, while a selective contact seems necessary for optimal operation, with the semi-selectivity the effect of hot-carriers is significant. Since it is much easier to fabricate a semi-selective contact, this result is very encouraging for a future experimental demonstration of HCSC.

IV. DISCUSSION

In this section we propose a discussion based on a simple rate-model which is schematically represented Fig. 5. We consider three electronic states numbered 1, 2 and 3 of energies 𝐸 1 < 𝐸 2 < 𝐸 3 and with the distributions 𝑓 1 , 𝑓 2 and 𝑓 3 . By emitting or absorbing a phonon, electrons can change of state with the respective rates 𝜎 𝑒𝑚 and 𝜎 𝑎𝑏 given by

𝜎 𝑎𝑏 = 𝑀 2 𝑁 ( 9
)
𝜎 𝑒𝑚 = 𝑀 2 (1 + 𝑁) (10)
with 𝑀 the electron-phonon scattering matrix element and 𝑁 = 1 (𝑒𝑥𝑝(ℏ𝜔/𝑘 𝐵 𝑇 𝐿 ) -1) ⁄ the number of optical phonons of energy ℏ𝜔 which is given by the Bose distribution at 𝑇 𝐿 =300 K. Assuming 𝑛 > 𝑚, the corresponding electrons flux between the states 𝑛 and 𝑚 is given by 𝜎 𝑒𝑚 𝑓 𝑛 (1 -𝑓 𝑚 ) -𝜎 𝑎𝑏 𝑓 𝑚 (1 -𝑓 𝑛 ). We also consider a radiative generation flux 𝐺, corresponding to the laser excitation, injected on the state 3. This involves a recombination flux from each state 𝑛 given by 𝑅𝑓 𝑛 , with 𝐺 = 𝑅(𝑓 1 + 𝑓 2 + 𝑓 3 ) at 𝑉 𝑜𝑐 and with 𝑅 the interband recombination coupling (spontaneous emission) given by

𝑅 = 𝑀 𝑜𝑝 2 (1 -𝑓 𝑣 ) (11)
with 𝑀 𝑜𝑝 the interband optical matrix element and 𝑓 𝑣 the electronic distribution in the valence band. In stationary regime, the net flux on any state equals zero and we can calculate 𝑓 1 and 𝑓 3 versus 𝑓 2 and the other parameters

𝑓 1 = 𝜎 𝑒𝑚 𝑓 2 𝜎 𝑎𝑏 +𝑓 2 (𝜎 𝑒𝑚 -𝜎 𝑎𝑏 )+𝑅 (12) 𝑓 3 = 𝑅(𝑓 1 +𝑓 2 )+𝜎 𝑎𝑏 𝑓 2 𝜎 𝑒𝑚 -𝑓 2 (𝜎 𝑒𝑚 -𝜎 𝑎𝑏 ) . ( 13
)
In the following, we will compare the reservoir with doping but without illumination, and the absorber under illumination. We impose the same value of 𝑓 2 for the two cases. This will allow us to consider an isentropic transfer through a selective contact at 𝐸 2 . In dark, this value of 𝑓 2 is related to the doping, while under illumination and for a given 𝑅, it can be experimentally obtained by tuning the incident photon flux.

Fig. 6a shows result obtained for 𝑓 2 = 10 -4 , in the reservoir. In this case the resulting distribution 𝑓 𝑛 , which does not depend on 𝑀, exhibits the shape of a Boltzmann distribution (exponential versus energy). Considering that energy between the states equals the phonon energy, this Boltzmann distribution is at 300 K. This simple rate-model can then describe electrons at the equilibrium with the phonons.

We present in Fig. 6b the result in absorber under illumination, with 𝑅 = 0.2𝑀 2 . As suggested in the two equations ( 12) and ( 13), the distribution at high energy 𝐸 3 increases while it is reduced at low energy 𝐸 1 . But, we can no longer define a temperature because the distribution does not have an exponential shape. However, increasing the high energy distribution while reducing the low energy distribution is equivalent to heating the electrons. We thus obtain a pseudo hot equilibrium regime by simply generating electrons at 𝐸 3 , a higher energy than the average energy of the corresponding recombination (𝐸 1 𝑓 1 + 𝐸 2 𝑓 2 + 𝐸 3 𝑓 3 )/(𝑓 1 + 𝑓 2 + 𝑓 3 ). This result is in agreement with the numerical calculation presented in this work and shows that, even with cold phonons, it is possible to obtain a pseudo hot equilibrium.

Concerning the evolution with the different parameters, for a given 𝑅, we verified that hot-carrier effect increases with 𝐺. This is not surprising since experimentally a powerful illumination is needed to observe hot-carrier effect. We also verified that for a given 𝐺, this hot-carrier effect increases with 𝑅. This last result, considering Eq. 11, suggests that an empty VB (with a large hole density) and a strong optical coupling are both beneficial for the hot-carrier effect. We now consider that this pseudo hot distribution is an absorber connected to the cold distribution, defined as the reservoir. Fig. 7a shows when an isentropic extraction is considered, meaning with a selective contact whose energy 𝐸 𝑛 is located such that the hot and cold distributions are equal. In this case, at 𝑉 𝑜𝑐 , the hot distribution is not disturbed by the contact. On the other hand, Fig. 7b shows the case where a non-selective contact is considered. With such a contact the net electron flux is oriented from the reservoir into the absorber at low energy and in the inverse direction at high energy. This will thus reduce the distribution of the absorber at high energy while it will increase it at low energy. In other words, such a contact will cool the electrons in the absorber. These results are also in agreement with the numerical calculations conducted in the present work.

IV. CONCLUSION

This article shows that, under powerful laser illumination, an ultra-thin solar cell associated with energyselective contact exhibits a larger 𝑉 𝑜𝑐 without current degradation. In spite of the selective contact our quantum model shows that the current is not degraded thanks to a hybridization behavior. The improvement of 𝑉 𝑜𝑐 is related to a hot-carrier effect which can be explained by a simple rate model. Such an effect increases with the selectivity but still exists in realistic design such as contact made with QW. This result should be more visible by considering hot phonons and should not be reduced by the carriercarrier scattering. This behavior could be experimentally demonstrated by the 𝑉 𝑜𝑐 comparison of ultrathin solar cells with a simple barrier and with a double-barrier as contact. If the current is not degraded by the selectivity, as shown by our results, such a measurement would be a demonstration of a hotcarrier solar cell operation. However, it should be note that the hot carriers effect exists only if the incident light power is sufficient (10 4 suns here). This implies an important electron current density which can cause problems related to the practical extraction of the current. [START_REF] O'dwyer | Electronic and thermal transport in hot carrier solar cells with low-dimensional contacts[END_REF] Moreover, an interesting result shown in this article is that, assuming a sophisticated numerical model including realistic materials under illumination, we confirm the validity of the very general expression of 𝑉 𝑜𝑐 (Eq.(2)), in both QD and QW geometries.

In the future, a study concerning the design of the contact would be interesting to better understand the impact of hybridization and selectivity in cell under a concentrated solar spectrum. It would be also interesting to consider non-radiative recombination processes which should, as suggested in the present work, impact the hot-carrier effect.

ACKNOWLEDGEMENT

The authors thank ICEMAN (ANR-19-CE05-000) for financial support.

Figure 2

 2 Figure 2. a) Schematic representation of the considered cell. Band diagrams and two-dimensional plot of the current spectra 𝐽 (in arbitrary unit) of the ultrathin InGaAs solar cell (12 nm), b) with a non-selective contact, c) with a selective contact with 𝐸 𝑛 =0.94 eV (1.5 nm thick QD) and, d) with 𝐸 𝑛 =1.02 eV (1.2 nm thick QD). In all cases the reservoirs are in InP and contacts between the absorber and the n-reservoir are made with barrier of AlGaAsSb. For the selective contact, we assume a QD between the two barriers. The Fermi level in p-reservoir is 𝜇 𝑝 =0 eV, while in n-reservoir 𝜇 𝑛 =0.7 eV. For the three cells we give 𝐽 𝑎𝑏𝑠 and 𝐽 𝑟𝑒𝑐 calculated at bias equals to 0.7V. Note that, as suggested by the arrows in panel b), 𝐽 𝑎𝑏𝑠 corresponds to the positive component of 𝐽 (in yellow), while 𝐽 𝑟𝑒𝑐 corresponds to the negative one (in blue). We also give the open-circuit voltage which is the interpolated bias giving 𝐽 𝑎𝑏𝑠 = 𝐽 𝑟𝑒𝑐 .

Figure

  Figure Band diagram and two-dimensional plot of the local density of states (LDOS) (in arbitrary unit), in the absorber and contact, for one transverse mode 𝑘 𝑡 , calculated in the device presented Fig.2cwith 𝐸 𝑛 =1.02 eV. Between the two barriers, in the QD, the local density of states shows 2 states. The higher one is due to the hybridization with the third state in the absorber. The consequence of this hybridization is that electrons on this third state on the absorber is delocalized in the contact.

Figure 4 .

 4 Figure 4. Natural logarithm of the electronic distribution, versus energy, of cells presented in Fig. 2, in absorber (blue line) and in n-reservoir (yellow line). We also represent, in red, the fit of the distribution in the absorber obtained with a Fermi function. To obtain this fit we represent fit=ln(1/distribution-1) versus the energy. The corresponding curves are in inset.

Fig. 5 .

 5 Fig.5. Band diagrams and two-dimensional plot of the current spectra 𝐽 (in arbitrary unit) of the ultrathin InGaAs solar cell(12 nm), with a semi-selective contact (QW) with 𝐸 𝑛 = 1.02 eV (with a 1.2 nm thick InGaAs QW).

Fig. 6 .Fig. 7 .

 67 Fig.6. Schematic representation of the rate-model and of the resulting distributions a) at the thermodynamical equilibrium in the reservoir, and b) with a radiative generation on the top state 3 with R=0.5𝑀 in the absorber. For each state, the disk surface is proportional to the distribution. For both calculations f2 is imposed to 10 -4 , the number of phonons 𝑁 is unchanged (phonons stay cool) and the energies of the states are given by 𝐸 𝑛 = ℏ𝜔(𝑛 -1) with ℏ𝜔 = 35 𝑚𝑒𝑉 the phonon energy and 𝑛 the index of the state. Blue disks are chosen when distribution is cool while red disks represent a hot distribution.

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2W.

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