L Prat

S N'diaye

L Rigal

C Gourdon

Solid-liquid transport in a modified co-rotating twin-screw extruder-dynamic simulator and experimental validations

Keywords: Twin-screw extruder, Solid-liquid extraction, Raw plant substrate, Dynamic simulator

This work presents a dynamic transport model of a solid-liquid media through a twin-screw extruder (TSE). The application under consideration is the solid-liquid extraction of solute from raw plant substrate. Dynamic experiments are performed and compared with the simulated results for step functions on the solid feed rate and on the screw rotating speed. Despite some imperfections, results allow to validate the simulator.

Introduction

An important application domain of twin-screw extruders (TSE) is found in the agro-industry for starch conversion and protein texturisation [START_REF] Harper | Food extruders and their applications[END_REF][START_REF] Colonna | Transformations structurales de l'Amidon[END_REF]. These contactors allow the simultaneous combination of two unit operations: extrusion and cooking, thus performing both physical and chemical transformations of biopolymers in a single step. TSE can also be used as an extractor, to process essential oils [START_REF] Bouzid | Extraction des huiles essentielles par des techniques non conventionnelles[END_REF], lipids [START_REF] Isobe | A new twin-screw press design for oil extraction of dehulled sunflower seeds[END_REF], or wood pulping [START_REF] Anon | Procédé et installation de traitement en continu d'une matière cellulosique[END_REF][START_REF] Choudens | Les pâtes chimicothermomécaniques blanchies obtenues avec le procédé bi-vis[END_REF]. Its ability to carry out extraction and chemical reaction in a single continuous reactor makes TSE very valuable for the valorisation of agroresource wastes [START_REF]Technologie d'extrusion bi-vis et fractionnement de la matière végétale, 40 ans d'extrusion bi-vis chez CLEXTRAL[END_REF].

In the setting of solid-liquid extraction from raw plant substrate, twin-screw extruder is considered as a multifunction thermo-mechano-chemical reactor. This complex device (Fig. 1) is essentially constituted of solid and liquid feeding systems and of a variable speed motor that drags the two screws turning in a barrel. The temperature is regulated with induction belts and cooled by water circulation. Screw profiles are composed at will by the user with transport screw elements (direct pitch screw), with reversed screw elements (RSE) and with kneading elements (MAL) with different shapes and functions [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF]. Twin-screw extruder is mainly used for its capacity of transfer whatever the viscosity of products and the very large range of mechanical actions that it can exercise on the matter (shearing, bruising, compression, . . . ).

In order to facilitate the clarification of processes in the extruder, it is necessary to be able to develop some fast and sure methods to appraise parameters intervening in transport phenomena, in heat transfer and in mechanical effects within the device. Besides, in order to conduct the industrial implantation of these processes, the setting up of an efficient and robust supervision system will prove to be necessary. Indeed, the raw plant substrate heterogeneity and the fluctuation of solid feeding rate often leads to unsteady-states and can provoke the mechanical blockages of the system.

The objective of this work is to propose a dynamic simulator of the transport of a polyphasic medium raw plant-solvent in a co-rotative and intermeshing TSE. In the first step, a model will be proposed. This model integrates absorption phenomena of the solvent by the solid, as well as the expression of the solute in the final compression. In the second step, experiences will be presented in order to validate the dynamic simulator.

Presentation of the dynamic model

However, the current problem, in a certain way, is quit different from previous studies. If TSE hydrodynamics for food or polymer begin to be well known [START_REF] Colonna | Flow mixing and residence time distribution of maize starch within a twin-screw extruder with a longitudinally-plit barrel[END_REF][START_REF] Szydlowski | Modelling flow in an intermeshing corotating twin-screw extruder: flow in kneading discs[END_REF][START_REF] Tayeb | A basic model for a twin-screw extruder[END_REF], the two phase flow constituted by vegetable matter/liquid does not allow to adopt a mechanical approach for now.

Complexity of the phenomena in this process first lies in the complexity of the vegetable matter itself and second in the important coupling between chemical, mechanical and physical phenomena involved. Intrinsic physico-chemical properties classically used in chemical engineering are not easily accessible or are difficult to define for the vegetable matter. Moreover, local mass balances are complex and require drastically simplified hypothesis to be solved. Physical properties of the vegetable matter, like compressibility, are subject to thorough changes. Its internal structure is affected by the progress of the reactive extraction as NaOH breaks it, and by internal moisture occurrence. Chemically spreading, the vegetable matter is also uneasy to define. In fact, only general classes of compounds (cellulose, hemicelluloses, proteins, pectins, lipids, . . . ) are used. Each class contains biopolymers that can be very different in terms of molar weight, polymerisation degree as well as transport properties (diffusivity), solubility, etc.

Besides, thermo-mechano-chemical processes in twinscrew extruders exhibit use and possibility advantages which become disadvantages for their modelling. If a com- bination of thermal, mechanical and chemical effects in a same continuous reactor allows the development of a lot of different processes, it also leads to a high phenomena interdependence.

Model structure and establishment

The main theoretical difficulty for the establishment of the model lies in the coupling between the particle and intraparticle physico-chemistry of the vegetable matter on the one hand, and the mechanical and transport phenomena on the other hand (Fig. 2). Their links are direct via operation like crushing or compression, as well as, indirect via the residence time and consequently via the reaction or the adsorption process. In addition, an important retroaction of the screw element geometry on the behaviour within the previous elements prevails.

Furthermore, the filling ratio of the different elements evolves in space and in time [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF]. Otherwise, the study of a reactor completely closed to the observation presents many difficulties.

As it has been developed by many authors for polymer and food [START_REF] Janssen | Twin-Screw Extrusion[END_REF][START_REF] Altamore | An analysis of residence time distribution pattern in a twin-screw cooking extruder[END_REF][START_REF] Boissonat | Etude paramètrique et modélisation des écoulements de la matière par analyse de la distribution des temps de séjour dans trois cuiseurs extrudeurs bi-vis[END_REF], the matter transport has been studied via the residence time distribution (RTD) method [START_REF] Prat | Transport d'un milieu complexe polyphasique matériau végétal/solvant en extrudeur bi-vis, 6eme CFGP[END_REF]. Then a transport model has been proposed [START_REF] Prat | A one dimensional model for the prediction of extraction yields in a two phase modified twin-screw extruder[END_REF] for the twin-screw extruder used in raw plant substrate fractionation in steady-state. This work tackles the modelling of transport phenomena with a one-dimensional model. The example considers the extraction of hemicelluloses from poplar wood with an aqueous solution of sodium carbonate. A screw profile permitting this operation has been finalised [START_REF] Prat | Transport d'un milieu complexe polyphasique matériau végétal/solvant en extrudeur bi-vis, 6eme CFGP[END_REF][START_REF] N'diaye | Extraction of hemicelluloses from poplar, populus tremuloides, using an extruder-type twin-screw reactor: influence of the main factors[END_REF]. This profile includes transport screw elements, kneading elements (which assure the homogeneity of the matter), and reversed screw elements involved in the solid-liquid separation by matter compression over a filter. The adopted method consists in separating the extruder functional zones for which a specific model is proposed. These models are based on mass balances and take into account information acquired experimentally in order to integrate knowledge upon the geometry of the different screw elements and on the physico-chemical properties of the solid-liquid substrate. Two categories of screw elements are identified: elements that are always full of matter in steady-state and those that are not. Besides, two zones of the extruder can be differentiated: a zone where the two phases are present, and a zone where the only solid is present (after the filter).

Furthermore, the main process is considered as isothermal. This is mainly due to the presence of water all along the extruder and the low influence of temperature on the matter properties.

Partially filled elements

Elements of the screw profile that are partially filled are the transport screw to the exception of the one situated upstream of the reversed screw element. It will be considered, in these elements, that there are no phenomena of matter compression (i.e. there is no variation of the solid porosity). Besides, the mass transfer of the liquid phase towards the porous volume of the solid takes place in these elements that are not under mechanical constraints.

The mass balances for solid, liquid and internal liquid phases can be written in an dx length element as follows:

dm S dt = F S - Np(1 -α) dx m S (1) dm
L dt = F L - Np(1 -α) dx m L -A (2)
dm I dt = F I - Np(1 -α) dx m I + A (3)
where F I is the internal liquid flow entering in an element of volume, from the previous element, F L the liquid flow entering in an element of volume, from the previous element, F S the solid flow entering in an element of volume, from the previous element, m I the internal liquid mass (in the solid porosity) m L the mass of liquid, m S the mass of solid, N the screw rotating speed, p the pitch of the transport screw element and α is the factor of shape of the transport element. The term A is the flux of liquid transferred from the liquid phase toward the porosity of the solid. This global instantaneous flux can be written as:

A = K(m I max -m I ) (4)
where m I max is the maximal amount of liquid that can be contained by the solid. This term can be expressed with the porosity ε of the solid, the mass of solid m S and the ratio of the densities of the liquid and the solid is given by:

m I max = ε 1 -ε ρ L ρ S m S (5)
Finally, on the solid and liquid mass balances it is possible to add a term of mass inlet flow G per unit of time to the corresponding abscissa. In our case, this mass inlet flow appears at the beginning of the TSE (solid feeding zone in the first element and liquid feeding zone in the second element).

Filled elements

The filled elements of the screw profile are the kneading elements and the RSE. Concerning the transport screw situated upstream of the RSE, previous experiments show that it is filled to the half of its length [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF][START_REF] Prat | Transport d'un milieu complexe polyphasique matériau végétal/solvant en extrudeur bi-vis, 6eme CFGP[END_REF]. Indeed, with the solid-liquid system and the processing conditions we used, the pressure drop in the RSE does not seem to have a major influence on the behaviour of the matter in this element. Kneading elements contain the two phases, liquid and solid. The other filled elements are situated after the liquid outlet and then, only the liquid phase is the one contained in the porosity of the solid. The compression of the solid takes place in these elements.

Mass balances for solid and liquid can be expressed in an dx length element by:

dm S dt = F S - v S m S dx (6) dm L dt = F L - v L m L dx ( 7
)
otherwise it is possible to add the two following constraints:

m L ρ L + m S ρ S = V (8) v S = βv L (9)
as the element is full, it becomes:

d dt m S ρ S + m L ρ L = 0 = F S ρ S + F L ρ L - m S dx βv L ρ S - m L dx v L ρ L ( 10
)
and then

F S ρ S + F L ρ L = v L m S dx β ρ S + m L dx 1 ρ L (11)
so, by using Eqs. ( 10) and [START_REF] Tayeb | A basic model for a twin-screw extruder[END_REF] in Eqs. ( 6) and ( 7):

dm S dt = F S -β(ρ L F S + ρ S F L ) m S m L + βm S ( 12
) dm L dt = F L -(ρ L F S + ρ S F L ) m L m L + βm S (13)

Numeric resolution

Equations are discretised at the second-order in space and the first-order in time. The resolution of the system thus formed is solved with a Runge-Kutta type algorithm.

Simulations and experimental validations

Description of the process and the equipment

The application under consideration is a solid-liquid extraction process of hemicelluloses from poplar wood (populus tremuloïde) via alkaline solubilisation. The used twin-screw extruder is a CLEXTRAL BC45 that includes two co-rotative and intermeshing screws.

Five different kinds of elements are used in the screw profile: T2F (trapezoidal two-flight screw), C2F (two-flight screw), MAL2 (bi-lobe kneading element) always positioned with a π/2 shift (i.e. in a neutral position for transport compared with conveying screw elements), MAL0 (right-handed mono-lobe kneading element) always positioned with a π/13 shift (i.e. in a positive position for transport) and C2FC (reversed two-flight screw element). The reverse screw element has peripheral slots grooved in the screw flight for leakage flow. Finally, the pitch of the transport screw elements, varies from 66 to 25 mm. The exact screw profile used is described on Fig. 1.

The ground wood (at an average size of 15 mm × 2 mm × 2 mm) is introduced in head of device by a feeding screw calibrated in mass, at the same time as the liquid phase containing 5% mass in sodium carbonate. A balance permits the registration of the mass in solid outlet (at the die).

Model parameters

The considered model requires the knowledge of physico-chemical parameters of the solid and liquid phases (density, porosity), as well as the geometric parameters of the different elements (volume, surface). In the studied case, ρ L = 1000 kg m -3 , ρ S = 1220 kg m -3 , ε = 0.78, and K = 0.01 s -1 . These parameters have been determined independently of the extruder [START_REF] Prat | Solid-liquid reactive extraction with raw plant substrate[END_REF].

Finally, the model has an adjustable parameter by type of element of screw:

• The parameter α is for transport screw elements. This parameter characterises the slip velocity of matter in relation to the longitudinal screw velocity. • The parameter β is for the kneading elements. This parameter is the ratio of the solid velocity by the liquid velocity and allows to calculate the mass retention in the screw elements. β represents in a global point of view rubbings in kneading elements (i.e. the viscous rubbings and rubbings between matter and barrel).

The values of these adjustable parameter are α = 0.3; β MAL0 = 1.42 (for the MAL0 kneading element) and β MAL2 = 2.05 (for the MAL2 kneading element). These values have been obtained by parametric identification in steady-state and their physical meaning have been discussed in [START_REF] Prat | A one dimensional model for the prediction of extraction yields in a two phase modified twin-screw extruder[END_REF]. In the present paper, the objective is to use these results without changes to predict the dynamics of the system and discuss their accuracy and relevance in that case.

Experiments

Once the steady-state is reached, variations are done on the solid feed rate and on the screw rotation velocity.

Step functions answers of the system are then measured on the solid instantaneous outlet.

The initial steady-state is obtained with the following values for the operating parameters: screw rotating speed, N = 2.98 turns s -1 ; liquid feed rate, Q L = 5.3 g s -1 ; solid feed rate, Q S = 1.09 g s -1 . Table 1 describes the different variations achieved on Q S and N.

There are three step functions on the solid feed rate and three on the rotating speed screw. The solid phase outlet rate is measured every seconds and is reported in Fig. 3.

Solid feed rate step function answers

Fig. 4 describes the answer of the system to a positive and a negative step function on the solid feed rate. Between the two landings in steady-state, the curve presents two dynamic phases separated by a short steady-state.

The first is owed to the change of retention in the kneading element the nearest to the solid outlet, and the second to the kneading element the most distant. Indeed, the reversed screw element and the half transport screw upstream of the RSE remain full because these two elements only contain the solid phase (they are situated after the filter).

Besides, a solid flux change has a direct influence on the liquid flux because of the variations of the internal liquid mass. Indeed, for a liquid flux of 5 g s -1 , when the solid flux goes from 1 to 2 g s -1 , the internal liquid quantity grows from 1.5 to 3 g s -1 (for a solid weight percentage in the outlet solid ω of 40%). In steady-state

m I = 1 -ω ω m S

Screw rotating speed step function answers

Fig. 5 describes the answer of the system to a positive followed by a negative step function on the screw rotating speed. The curve presents two peaks separated by a steady-state. The model prediction is not satisfactory. Indeed, according to the model, the variations of rotation speed have only a direct influence on the amount of matter in the transport screws. However, it seems that the rate of compaction in the reversed screw element also depends on this parameter [START_REF] Prat | A one dimensional model for the prediction of extraction yields in a two phase modified twin-screw extruder[END_REF]. The model must then be improved in order to better represent the behaviour of this element and then to take into account the dynamic influence on the physical structure on the matter. Of course, this study is far from being easy and will need further improvement on the raw plant knowledge.

Conclusion

The proposed model represents the experimental dynamic results in a correct way in the studied cases. Besides, values of the parameters β 1 and β 2 obtained in steady-state by RTD methods are preserved. The coefficient K, characterising the apparent global kinetic of transfer of the liquid phase toward the solid did not appear in the steady-state model. Indeed, the time of contact of the two phases in the device is sufficient for the solid to be saturated in liquid at the filter abscissa. Therefore, the influence of this parameter can only be studied in dynamic experiments.

The model must be improved on the reversed screw element to take into account the effect of the rotating speed on the mechanical phenomena. But as far as the considered operation is solid-liquid extraction, the extruder part situated after the filter has less importance.

The proposed model, in spite of its imperfections seems to be adapted to raw plant substrate fractionation in the twin-screw extruder. It can be used to characterise transport phenomena of solid-liquid medium in few experiments and then to avoid fastidious residence time distribution experimental phases. This parameters identification step will, of course, need further study on the mathematical structure as the model sensitivity to the adjustable parameters.

Fig. 1 .

 1 Fig. 1. Twin-screw extruder description.

Fig. 2 .

 2 Fig. 2. Phenomena and coupling analysis.

Fig. 4 .

 4 Fig. 4. System answer to step functions on the solid feed rate. Model (grey line) and experimental (black line) answers.

Fig. 5 .

 5 Fig. 5. System answer to step functions on the screw rotating speed. Model (grey line) and experimental (black line) answers.

Table 1

 1

	Achieved experiments
	Time (s)	Q S (g s -1 )	N (turns s -1 )
	0	1.09	2.98
	290	1.9
	770	0.9
	1380		3.66
	1700		2.98
	1980		3.66
	2160	1.9

Acknowledgements

The authors thanks C. Doumeng (CRITT Agroressources-CATAR) for her competence and technical help.


Julien Legendre

Pierre-Olivier Chapuis

GaAs-Based Near-Field Thermophotonic Devices: Approaching The Idealized Case With One-Dimensional PN Junctions

Keywords:

A C T Thermophotonics (TPX) is a technology close to thermophotovoltaics (TPV), where a heated lightemitting diode (LED) is used as the active thermal emitter of the system. It allows to tune the heat flux, by means of electroluminescence, to a spectral range matching better the gap of a photovoltaic cell. The concept is extended to near-field thermophotonics (NF-TPX), where enhanced energy conversion is due to both electric control and wave tunneling. We perform a thorough numerical analysis of a GaAsbased NF-TPX device, by coupling a near-field radiative heat transfer solver based on fluctuational electrodynamics with an algorithm based on a simplified version of the drift-diffusion equations in 1D. This allows for the investigation of the emission and absorption profiles in the LED and the photovoltaic (PV) cell, and for the scrutiny of the impact of key parameters. We also demonstrate that the performance obtained with this algorithm can approach idealized cases for improved devices. For the considered simplified architecture and 300 K temperature difference, we find a power density output of 1 W.cm -2 , underlining the potential for waste heat harvesting close to ambient temperature.

Introduction

The necessity of a change in our relation with energy needs no longer to be proven. This change has to happen on several important topics: the generation of clean energy, the reduction of consumption of machines and buildings, and the recuperation of energy losses. On this last subject, recent solid-state heat engines avoid notably (potentially polluting) refrigerants and moving parts (thus vibrations). Thanks to these benefits, solid-state heat engines can be interesting candidates for both terrestrial and spatial applications. Several of these engines have seen good recent development in the last 20 years thanks to dynamic research [START_REF] Green | Energy conversion approaches and materials for high-efficiency photovoltaics[END_REF][START_REF] Melnick | From thermoelectricity to phonoelectricity[END_REF]: we can cite thermoelectric devices [START_REF] Chen | Manipulation of Phonon Transport in Thermoelectrics[END_REF][START_REF] Mao | Advances in thermoelectrics[END_REF], thermophotovoltaic (TPV) devices [START_REF] Daneshvar | Thermophotovoltaics: Fundamentals, challenges and prospects[END_REF], thermionic devices [START_REF] Schwede | Photon-enhanced thermionic emission for solar concentrator systems[END_REF][START_REF] Campbell | Progress Toward High Power Output in Thermionic Energy Converters[END_REF] or hot-carrier cells [START_REF] König | Hot carrier solar cells: Principles, materials and design[END_REF][START_REF] Li | Slow Hot-Carrier Cooling in Halide Perovskites: Prospects for Hot-Carrier Solar Cells[END_REF]. In the TPV field, if a large fraction of the research focuses on the development of improved emitters [START_REF] Sakakibara | Practical emitters for thermophotovoltaics: a review[END_REF] or cell rear-face reflectors [START_REF] Fan | Near-perfect photon utilization in an air-bridge thermophotovoltaic cell[END_REF], advanced concepts are also under development [START_REF] Tervo | Near-field radiative thermoelectric energy converters: a review[END_REF], including hybridization with thermionic devices [START_REF] Datas | Hybrid thermionic-photovoltaic converter[END_REF][START_REF] Zeneli | Performance evaluation and optimization of the cooling system of a hybrid thermionic-photovoltaic converter[END_REF], near-field (NF) enhancement [START_REF] Whale | Modeling and performance of microscale thermophotovoltaic energy conversion devices[END_REF][START_REF] Park | Performance analysis of near-field thermophotovoltaic devices considering absorption distribution[END_REF][START_REF] Francoeur | Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators[END_REF] and thermophotonics (TPX) [START_REF] Harder | Thermophotonics[END_REF][START_REF] Zhao | Self-sustaining thermophotonic circuits[END_REF]. In this work, we aim to study the combination of these last two concepts through the numerical analysis of a near-field thermophotonic (NF-TPX) device [START_REF] Zhao | Near-Field Thermophotonic Systems for Low-Grade Waste-Heat Recovery[END_REF]. In a near-field thermophotovoltaic (NF-TPV) device, by decreasing the gap distance between the emitter and the photovoltaic (PV) cell to the order of hundreds of nanometers or below, evanescent waves start to participate to the heat transfer along with propagative modes [START_REF] Zhang | Nano/Microscale heat transfer[END_REF]. Heat transferred to the PV cell can then be increased by a factor 10 to even 100 for small gap distances [START_REF] Francoeur | Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators[END_REF], exceeding therefore the blackbody limit and increasing by a similar order of magnitude the electrical power produced by the PV cell. However, Figure 1: Schematic of a GaAs near-field thermophotonic device. The LED is maintained at 600 K, while the PV cell is kept at 300 K. In the nanogap, propagative and evanescent modes participate to the radiative heat transfer. Absorption of the radiation by the photovoltaic cell allows for the production of electrical power, a fraction of which is redirected towards the light-emitting diode for enhancement of the radiative heat transfer by electroluminescence. Ideal mirrors (in grey) allow for the confinement of photons inside the device.

the emitter cannot be controlled dynamically. The practical capabilities of such device have already been demonstrated [START_REF] Fiorino | Nanogap near-field thermophotovoltaics[END_REF][START_REF] Inoue | One-Chip Near-Field Thermophotovoltaic Device Integrating a Thin-Film Thermal Emitter and Photovoltaic Cell[END_REF][START_REF] Bhatt | Integrated near-field thermo-photovoltaics for heat recycling[END_REF][START_REF] Lucchesi | Near-Field Thermophotovoltaic Conversion with High Electrical Power Density and Cell Efficiency above 14%[END_REF][START_REF] Mittapally | Near-field thermophotovoltaics for efficient heat to electricity conversion at high power density[END_REF][START_REF] Inoue | Integrated Near-Field Thermophotovoltaic Device Overcoming Blackbody Limit[END_REF]. Thermophotonic devices rely on the recent development of light-emitting diodes (LED). Thanks to the electroluminescent effect, the emission spectrum of the LED can be tuned by the application of a voltage, and can also exceed the blackbody limit. If most of this development was directed towards lighting applications, recent research has pointed out the interest of LEDs for thermodynamic applications as an electroluminescent refrigerator/heat pump [START_REF] Oksanen | Thermophotonic heat pump-a theoretical model and numerical simulations[END_REF][START_REF] Sadi | Electroluminescent Cooling in III-V Intracavity Diodes: Efficiency Bottlenecks[END_REF][START_REF] Sadi | Thermophotonic cooling with light-emitting diodes[END_REF] or as a key part of a TPX device. In the latter, the LED is placed in front of a PV cell and is heated above ambient temperature (in a heat engine configuration). Thanks to the electroluminescence effect, the radiative power transferred to the PV cell can be increased by order of magnitudes compared to the TPV case [START_REF] Harder | Thermophotonics[END_REF]. However, the LED is consuming electrical power: a TPX device can therefore produce electrical power only when made of high-efficiency LED and PV cell, this last argument being the reason of the recent interest in the device. Indeed, with the maturation of the LED technology, the achievable efficiency has increased largely in the last decades. Among the well-known semiconductors regularly used for PV and TPV applications, gallium arsenide (GaAs) junctions has reached extremely high internal and external quantum efficiencies (above 99% [START_REF] Bender | Development of high quantum efficiency GaAs/GaInP double heterostructures for laser cooling[END_REF]), making it an ideal candidate for TPX application as underlined by the Aalto group [START_REF] Sadi | Thermophotonic cooling with light-emitting diodes[END_REF], even with a larger bandgap energy than what is usually used for TPV (i.e. closer to visible than to midinfrared). By combining near-field effects with electroluminescent enhancement in a NF-TPX device, the radiative heat transfer increase is coupled and permits a dramatic increase of the electrical power output. Due to the limited temperatures current LEDs can withstand (600 to 700 K, see the review on high-temperature pn junctions in [START_REF] Vaillon | Solar Cells Operating under Thermal Stress[END_REF]), this kind of device is ideal for low-grade heat recovery application. This could allow for a photonic device to be performant in a range where standard TPV is barely used [START_REF] Daneshvar | Thermophotovoltaics: Fundamentals, challenges and prospects[END_REF][START_REF] Sakakibara | Practical emitters for thermophotovoltaics: a review[END_REF][START_REF] Fan | Near-perfect photon utilization in an air-bridge thermophotovoltaic cell[END_REF] because of its poor efficiency [START_REF] Green | Energy conversion approaches and materials for high-efficiency photovoltaics[END_REF], being then competitive against thermoelectrics, a technology commonly used in this range of temperature [START_REF] Mao | Advances in thermoelectrics[END_REF]. In the first article on NF-TPX written by Zhao et al. [START_REF] Zhao | Near-Field Thermophotonic Systems for Low-Grade Waste-Heat Recovery[END_REF], a 1D near-field radiative heat transfer (NF-RHT) model was coupled with a 0D detailed balance (DB) model for the computation of the currents. We propose here to analyze the performance of a GaAs NF-TPX device, by coupling a similar NF-RHT code with an improved charge carrier behaviour model relying on the solution of the simplified drift-diffusion (SDD) equations in 1D, often used in standard PV studies [START_REF] Nelson | The Physics of Solar Cells[END_REF][START_REF] Kitai | Principles of Solar Cells, LEDs and Diodes: The role of the PN junction[END_REF].

Modelling

The NF-TPX device considered is shown in Figure 1. The PV cell is a GaAs pn junction, while the LED is made of a GaAs-based alloy (this matter is discussed in the next paragraph). The p-and n-regions are respectively the front and back layers of the two junctions. The LED is heated at 600 K, which allows for large radiative heat transfer while keeping the quantum efficiency high [START_REF] Maros | High temperature characterization of GaAs single junction solar cells[END_REF][START_REF] Sun | Thermal stability of GaAs solar cells for high temperature applications[END_REF]. The PV cell is kept at ambient temperature (300 K). On the back surface of the junctions, two perfect mirrors (i.e., with a dielectric function with an infinitely large real part and a null imaginary part) allows for the confinement of the photons inside the device. The main device parameters are shown in Table 1. On the right of Figure 1 are provided the main electrical parameters, where is the voltage and the current density. represents the electrical power density produced by the device.

The modelling process can be divided into two main steps: a) calculation of the near-field radiative heat transfer (including both below and above-bandgap photons, see Section 2.1) b) calculation of the I-V characteristic through the modelling of the charge carrier behaviour (see Section 2.

2)

The different modelling parameters used in this work can be found in Table 2. By taking into account the variation of GaAs bandgap with temperature [START_REF] Gonzalez-Cuevas | Calculations of the temperature and alloy composition effects on the optical properties of Alx Ga1-x Asy Sb1-y and Gax In1-x Asy Sb1-y in the spectral range 0.5-6 eV[END_REF], the bandgap of a GaAs LED goes down to 1.28 eV, which has a significant impact on performances due to bandgap mismatch with the PV cell ( 300 K = 1.42 eV). Instead of pure GaAs, the LED is therefore made of a GaAs-based alloy, which displays a bandgap of 1.42 eV at 600 K. In addition, optical properties of such alloys are very close to those of GaAs at 300 K (precisely because the bandgaps are matched). This allows us to model the performance of a GaAs-based LED using well-known GaAs optical properties at 300 K and electrical properties at 600 K. In this paper, we indistinctly refer to the device as 'GaAs NF-TPX' or 'GaAs-based NF-TPX', the LED being always made of GaAs-based alloys to keep its bandgap matched with that of the PV cell.

Near-field radiative heat transfer

The near-field radiative heat transfer between the LED and the PV cell is simulated using the Fluctuational Electrodynamics theory developed by Rytov [START_REF] Rytov | Principles of Statistical Radiophysics[END_REF], which allows for an accurate description of both propagative, frustrated and surface modes, the two latter being evanescent modes that are not described by the macroscopic framework. The random movement of charges at any point, characterized by the Fluctuation-Dissipation Theorem, can be linked to the electromagnetic wave perceived at any other point through Green tensors. Then, the radiated power is simply obtained as the mean value of the Poynting vector. By using Francoeur's formalism [START_REF] Francoeur | Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method[END_REF], the spectral photon flux density , which is related to the spectral heat flux density as = ℏ , emitted at a layer s and received at a position in layer l can be expressed as , ( ) = 0 -0  ( ) = , ( ) -, ( ). [START_REF] Green | Energy conversion approaches and materials for high-efficiency photovoltaics[END_REF] In this expression, 0 is the modified Bose-Einstein distribution, which is tuned by means of electroluminescence through the value of the electrochemical potential (defined as the difference between the electron and hole quasi Fermi levels = -):

0 ( , , ) = exp ℏ ∕ -1 -1 , ℏ < exp (ℏ -) ∕ -1 -1 . ℏ ≥ (2)
E is the gap energy, ℏ is the reduced Planck constant, the radiation angular frequency, the Boltzmann constant and T the temperature. In this work, we consider that the electrochemical potential is constant and equal to , a common approximation whose precision has been studied in [START_REF] Callahan | Coupled Charge and Radiation Transport Processes in Thermophotovoltaic and Thermoradiative Cells[END_REF]. By doing this approximation, we assume that the Bose-Einstein distribution is similar in all points of the LED (resp. PV cell), therefore that the net photon flux density between two points of the same device is null. The other factor,  ( ), corresponds to an equivalent transmission coefficient between the layer s and any point of the layer l, and can be obtained from the transmission coefficient  ( , ) between two points positioned respectively in layers s and l

 ( ) = ∫  ( ′ , ) ′ (3a)  ( , ) = 2 ℜ ′′ 1 ∫ +∞ 0 , ( ) (3b)
, ( ) =

⎡ ⎢ ⎢ ⎢ ⎣ ( , , , ) * ( , , , ) + ( , , , ) * ( , , , ) - ( , , , ) * ( , , , ) ⎤ ⎥ ⎥ ⎥ ⎦ . ( 3c
)
The terms g correspond to the different Weyl components of the Green tensors, is the parallel component of the wavevector k, is the vacuum wavenumber corresponding to the given frequency and ′′ is the imaginary part of the dielectric function . In this work, we assume that the LED and PV cell temperatures are kept constant; thus, below-bandgap photons do not have any impact on the electrical performance, only on the device efficiency. In the following, the photon flux density used in the electrical models is therefore defined as the integral of its spectral quantity over above-bandgap photons:

= ℏ ≥ , = ∫ +∞ ∕ℏ ,
.

Charge carrier behaviour

Once the spectral photon flux density is obtained, the charge carrier behaviour can be modeled. In order to do so, we use two different methods:

a) The detailed balance (DB) method (see Section 2.2.1) allows to easily obtain the current thanks to a balance between photons and charge carriers. This method has been used in the literature for the computation of the currents in a TPX device [START_REF] Harder | Thermophotonics[END_REF][START_REF] Zhao | Near-Field Thermophotonic Systems for Low-Grade Waste-Heat Recovery[END_REF][START_REF] Sadi | Thermophotonic cooling with light-emitting diodes[END_REF]. b) The simplified drift-diffusion (SDD) method, in which the drift-diffusion equations are solved using standard approximations, is routinely used in photovoltaics [START_REF] Nelson | The Physics of Solar Cells[END_REF][START_REF] Kitai | Principles of Solar Cells, LEDs and Diodes: The role of the PN junction[END_REF] but has not been used for TPX devices until now.

The use of these two methods allows for a complete description of the device performance. By using the DB method, we can estimate the ideal capabilities of such a system; then, thanks to the SDD method, the power output of a specific device can be obtained and its efficiency estimated in comparison to the ideal case.

Detailed balance (0D)

The photon flux density is known at any point of the system. In order to obtain a direct relation with the current densities, several assumptions have to be made [START_REF] Harder | Thermophotonics[END_REF]: and ̃ = (1∕IQE -1) ( = 0).

Since = ( ̃ -̃ ), we obtain = - - 1 IQE -1 - ( = 0) = ( -) -( + ) - 1 IQE -1 0 -0 ( = 0)  ( -) - ( + ) . (4)
This expression can be simplified if IQE = 1:

= ( -) -( + ) . ( 5
)
These two equations are developed in [START_REF] Harder | Thermophotonics[END_REF][START_REF] Sadi | Thermophotonic cooling with light-emitting diodes[END_REF]. In [START_REF] Zhao | Near-Field Thermophotonic Systems for Low-Grade Waste-Heat Recovery[END_REF], a more thorough model has been developed. However, in the case of this work, we aim to obtain a simple and ideal (or quasi-ideal if IQE < 1) model, therefore refining the DB model is not of interest for us. Note that when using the DB method, we consider that both devices (labeled respectively s and l) are homogeneous semi-infinite media for the radiative computation, i.e. that  ( + ) ≈ 0 (thus ( + ) ≈ 0). This allows to obtain ideal or quasi-ideal performances which are independent of the LED and PV cell geometries considered. In addition, when considering quasi-ideal cases, the IQE chosen is identical for the LED and the PV cell.

Drift-Diffusion (1D)

For a thorough modelling of a TPX system, Poisson, continuity and drift-diffusion equations should be solved simultaneously in order to obtain the local charge carriers distribution and , electric field inside the junction and electron and hole currents and . In 1D, these equations are expressed as

( ) = - ( ) -( ) + ( ) - ( ) (6)
( ) = ( ( ) -( )) (7a)
( ) = -( ( ) -( )) (7b)
( ) = ⋅ ( ) ( ) + ( ) (8a)
( ) = ⋅ ( ) ( ) - ( ). (8b)
These equations being coupled, solving the complete problem would need to implement an iterative process [START_REF] Blandre | High-injection effects in near-field thermophotovoltaic devices[END_REF][START_REF] Gummel | A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations[END_REF]. However, standard approximations are regularly used to simplify the problem [START_REF] Nelson | The Physics of Solar Cells[END_REF][START_REF] Kitai | Principles of Solar Cells, LEDs and Diodes: The role of the PN junction[END_REF]: a) the depletion approximation, in which the electric field is analytically defined and becomes independent of the illumination. b) the low-injection approximation, in which we assume that the illumination is low enough to have constant majority carrier densities (equal to the doping level).

By using these approximations, the problem can be decoupled, obtaining one differential equation for the minority carrier density to be solved in the quasi-neutral regions on the p-side and on the n-side. This differential equation is expressed for electrons in the p-region as

2 Δ 2 ( ) - Δ ( ) - , + ( ) - ( ) = 0. (9)
In this equation, Δ represents the variation of the electron density in the p-region compared to the equilibrium value 0 . corresponds to the position in the p-region, and the differential equation should be solved between the limit of the depletion region in the p-region , and the boundary of the region . The boundary conditions are

Δ ( , ) = 0 exp -1 (10a) Δ ( ) = -Δ ( ). (10b)
Note that the differential equation and the boundary conditions are identical for holes in the n-region.

corresponds to the surface recombination coefficient for electrons at the front interface (see Table 2). We consider equivalent surface recombinations at the front and back interfaces of the junctions:

= = 5 × 10 2 m.s -1 .
, is the diffusion coefficient, obtained from the mobility , through Einstein's equation:

, = , ⋅ ∕ . ( 11
)
For the SDD method, as for the DB method, the generation of e-h pairs is computed from the results of the nearfield radiative heat transfer model. Here, the two layers (pand n-regions) of each device are considered for the radiative and charge transport computations, as shown in Figure 1. In order to have the localized photon flux density, it is therefore necessary to sum over the two emission layers s, which can be those of the LED or of the PV cell depending on the device considered. In addition, the volumetric generation rate is needed instead of the global one, and can be written as

= - ( ). (12)
Similarly, we write the radiative recombination rate as

= - ( ). (13)
Note that in the existing literature for TPV (including in the near-field), radiative recombinations are modeled either by using a radiative recombination coefficient B [START_REF] Francoeur | Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators[END_REF][START_REF] Blandre | High-injection effects in near-field thermophotovoltaic devices[END_REF] or by using an iterative process that allows for a precise modelling of photon recycling and transfer between the two bodies [START_REF] Callahan | Coupled Charge and Radiation Transport Processes in Thermophotovoltaic and Thermoradiative Cells[END_REF]. Of course, our modelling should tend towards precision (thus the second solution). However, in this work, we use an intermediary between these two models as shown in Eq. 13, which has several advantages:

1. compared to the use of a coefficient , using the local emission rate for computing the radiative recombinations allows to account for the influence of the geometry and of near-field effects. 2. compared to the precise modelling, this method does not need any iteration, simplifying the calculations. 3. the expressions of the radiative generation and recombination rates are similarly expressed for the DB and the SDD methods, allowing for an easier comparison.

The carrier lifetime related to non-radiative recombinations is computed using classic and coefficients (see Table 2), taking into account Shockley-Read-Hall (SRH) recombinations and Auger recombinations:

- = + ⋅ 2 -1 (14)
Once the charge carrier densities are known, the current density can then be computed as:

= + + ( 15
)
where and are the minority carrier current densities at the depletion region boundary (whose expression can be obtained from Eq. 8 by setting = 0) and is the current related to the depletion region, given by (neglecting nonradiative recombinations due to the depletion approximation):

= ∫ ( ) - ( ) (16)
The doping levels selected in Table 1 have been chosen so that these approximations are valid: by taking as a reference the work from Blandre et al. [START_REF] Blandre | High-injection effects in near-field thermophotovoltaic devices[END_REF], a doping level of 10 17 cm -3 for both the p-and n-regions would be enough to consider that the system is in the low-injection regime. In order to ensure that the approximation is correct, we use larger doping levels: a donor doping level of 10 18 cm -3 and an acceptor doping level of 10 20 cm -3 . Since non-radiative recombinations are neglected in the depletion region (see Eq. [START_REF] Park | Performance analysis of near-field thermophotovoltaic devices considering absorption distribution[END_REF], is only approached: therefore, if it becomes a predominant part of the current, this could increase the error made by using the SDD model and require the use of a more precise model. For the geometries considered in this paper (see Table 1), only accounts for 1 to 10% of the total current density, because the doping levels are high enough to keep the depletion region thin (of the order of tens of nanometers). We thus expect the SDD model to be precise enough for these different cases.

Analysis of the reference case

Absorption and Emission

Before looking at the performance of a NF-TPX device, it is interesting to study the net absorption and emission profiles at a 10 nm gap distance, respectively for the LED and the PV cell. The profiles obtained with the reference case (see Table 1) are shown in Figure 2.

Looking first at above-bandgap quantities, which are the most important for power generation, we observe as expected an extremely large increase in the absorption and emission rates, with the mean increase factor from the farfield (FF) case to the near-field (NF) case with electroluminescent enhancement (for LED and PV cell voltages of 1 V) reaching 10 9 . Most of this increase is due to electroluminescence: going to the near-field only accounts for a mean increase rate of the order of 10. The dramatic effect of electroluminescence on the radiative heat transfer however comes with an increased electrical power consumption, and does not necessarily cause an increase in the net power density produced. This will be discussed in Section 3.3. In the insets of Figures 2a and2b are shown the total emission and absorption rates. Their variations are quite different compared to their above-bandgap equivalent. First, the rates abruptly change at the p-n boundary: because the doping levels and the majority carriers are different, the dielectric functions obtained with the Drude model (see Table 2) are different for the p and n layers, causing the photons to be reflected back and causing the emission (or absorption) discontinuity. The rates of increase between the three cases studied are also quite different. Near-field effects allow for a much larger increase, up to 10 4 at the front surface. Meanwhile, the impact of electroluminescence is nearly negligible, since the orange and yellow curves are almost superimposed. This can be better seen using Figures 2c and2d, where the spectral emission and absorption rates are shown. Two regions can be distinguished: a) a first one in the infrared (below 0.3 eV, hence above 4 µm), corresponding to the usual photon energy range of radiation for bodies close to room temperature b) a second one in the visible (between 1.42 and 2 eV, hence between 600 and 900 nm), corresponding to the electrically-induced emission/absorption (i.e., electroluminescence).

As the electroluminescent peak is low compared to the thermal peak (see Figure 2e), we cannot see the influence of the voltage on the total rates. One can notice the similarity between the LED emission and the PV cell absorption profiles. These two profiles are indeed symmetrical, because of the device symmetry (in terms of geometry, of optical properties and of doping levels).

Parametric Study

We now investigate the global influence of two key parameters, the LED temperature and the gap distance, on the ideal performance achievable by a GaAs NF-TPX device. In order to do so, we perform a parametric study using the DB method around the reference case (see Table 1). Results are reported in Figure 3. The LED temperature impact on the device is computed for two different IQEs of 1 and 0.9 (frequency and temperature independent). In the ideal scenario (IQE = 1), it is shown to have only limited effect on the power output, with an increase rate of a factor 2 between 550 K and 650 K. However, this increase is much

PV cell

Far-field Near-field Near-field, U=1V

10 -8 10 -7 10 -6 10 -5 10 28 larger for lower IQE, reaching a factor 16 in the same range of temperature for IQE = 0.9. This shows the importance of using devices with nearly ideal IQE for low temperature application, as stated in the introduction.

In Figure 3b is shown the influence of the gap distance on the above-bandgap radiative heat flux and on the different electrical powers involved, for an IQE of 0.9. As shown in [START_REF] Bernardi | Impacts of propagating, frustrated and surface modes on radiative, electrical and thermal losses in nanoscale-gap thermophotovoltaic power generators[END_REF], it is interesting to study the variation of amplitude of the different modes composing the heat flux, i.e. propagative modes, frustrated modes (which propagate in GaAs but are evanescent in vacuum) and surface modes (which are evanescent on each side of the interface). Three different regions can be observed: a) above a few microns: it is the FF regime, with constant values for the different powers.

b) between hundreds of nanometers and a few microns: propagative modes are still dominant in the radiative heat flux, but cavity effects cause oscillations of the powers around the far-field value. c) below hundreds of nanometers: frustrated evanescent modes become dominant, causing a large increase of the powers at first, before a slower increase below tens of nanometers. At 1 nm, surface modes are still in minority, but have exceeded propagative modes.

Looking more closely at the power levels consumed by the LED and produced by the PV cell, we can note their proximity, caused by an IQE already far from the ideal case and causing the net power output to be quite small compared to them. By computing the ratio between the radiative heat flux exchanged and the electrical power produced or consumed, we can obtain an estimation of the conversion efficiencies at maximum power. We first focus on above-bandgap photons. Above-bandgap efficiencies are nearly independent of the gap distance, reached at the same voltages for any of the gap Figure 3: Influence of (a) the LED temperature and (b) the gap distance on the power produced by the device. For both cases, the parametric analysis is performed around the reference case. In (b), an IQE of 0.9 is considered.

distance considered ( = 1.00V and = 1.16V), and equal to

, ℎ -= ℏ ≥ = 78%, ( 17a
) , -ℎ = ℏ ≥ = 132%, (17b)
showing thus the quality of the conversions of each device. Note that the efficiency is greater than 1 for the LED because it is heated: it will still have a positive net emission even with no electrical power consumed. Since most of the power produced by the PV cell is sent back to the LED, the net conversion efficiency is much lower:

= ℏ ≥ = 2%. ( 18
)
This net conversion efficiency should not be confused with the efficiency of the device, which would be expressed as the ratio between the device power output and the heat supplied to the LED in order to keep its temperature constant. We obtain

Σ = ℏ ≥ - = 9%. (19)
equivalent to a scaled efficiency (i.e. the efficiency divided by the related Carnot efficiency) of 18%, close to what is found for NF-TPV (e.g., 23% in [START_REF] Francoeur | Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators[END_REF]). These results are obtained for above-bandgap photons, providing an upper bound for the overall efficiency. A complete evaluation of the efficiency is performed at = 10 nm. We model GaAs dielectric function below the bandgap with a decoupled Drude-Lorentz model. Most of the values needed in this model are given or can be computed thanks to [START_REF] Adachi | GaAs and Related Materials[END_REF]; for the variation of the Drude damping coefficient, we use the expression given in [START_REF] Losego | Conductive oxide thin films: Model systems for understanding and controlling surface plasmon resonance[END_REF]. However, since the Drude model depends on the doping levels, we have to consider a specific geometry. By taking the reference case, the junctions are thick enough so that the hypothesis of semi-infinite junctions remains true. The power densities are thus similar to those computed above. The results obtained are provided in Table 3: as shown in the previous section, below-bandgap photons still dominate the photon flux for an IQE of 0.9, worsening therefore the different efficiencies and increasing the heating (resp. cooling) power required to keep the LED (resp. the PV cell) at constant temperature. While only 10 4 W.m -2 are needed when considering above-bandgap photons, this rises up to 10 6 W.m -2 by considering all photons. This underlines the thermal management issue of such devices. 2.9 ×10 3 9.9×10 -2 1.0

I-V and P-V characteristics

We now look at the characteristics obtained for a NF-TPX device, starting with the results obtained with the DB method for an IQE of 0.9 (Figure 4a). Since both the LED and the PV cell voltages vary, the characteristic of a NF-TPX device is a 3D plot. The region in which the net electrical power output is positive is strictly limited by two inequalities: Even if the characteristic is limited by these two inequalities, other phenomena determine where power can be extracted from the device. For a given LED voltage, if the PV cell voltage is too large, electroluminescence from the PV cell is important enough to counterbalance the difference of temperature with the LED and change the direction of the net radiative heat flux. In addition, when decreasing the IQE, the range of acceptable PV cell voltage for a given LED voltage becomes narrower, resulting in the decrease of the power output. If the maximum power point (MPP) of an ideal device is close to the point ( ∕ , ∕ ), the real MPP is located at lower voltages. A quick estimation of its location can be found at the intersection of two straight lines that qualitatively envelop the characteristic of the device with a good precision and correspond to the two limiting factors stated above:

a) = + ∕ - + ∕ ⋅ ln ∕
: balance between thermally-and electrically-induced emission for the LED and the PV cell ( corresponds to the Carnot efficiency). This expression is derived from = 0, which can be reduced to Δ 0 = 0 by assuming a constant transmission coefficient. As a first approximation, the last term of the equation can be neglected. b) = ∕IQE: reduction of the PV cell voltage range available with the IQE.

In Figure 4b, we show the I-V curve of the LED and the PV cell. The voltage given on the horizontal axis is, respectively, the LED voltage and the PV cell voltage. For the LED (resp. the PV cell) characteristic, the PV cell (resp. the LED) voltage is kept either at its optimum value (i.e. = 1.16 V and = 1 V) or at 0. For the PV cell, increasing the LED voltage (from the dashed blue line

Figure 4: (a) Full V-V characteristic of the device, obtained with the 0D model and for an IQE of 0.9. Precise expression of the upper limit is given in the text. (b) I-V characteristics of the LED and the PV cell around the maximum power point from (a). (c) Simplified P-V characteristics of the device, where the power is computed from the full characteristics as ( ) = max ( , ), 0 ≤ < ∕ . See Section 4 for an explanation on the optimization.

to the full blue line) raises significantly the short-circuit current by means of electroluminescence. For the LED, we can notice that the variation of the characteristic with the PV cell voltage (from the dashed orange line to the full orange line) is slower; however, for low LED voltage, electroluminescence from the PV cell is large enough to reverse the direction of the net heat flux, thus changing the sign of the short-circuit current and allowing for a power production from the LED. In this case (low LED voltage, high PV cell voltage), the LED works as a PV cell while the PV cell works as a LED. In fact, this regime where the high-temperature device is also the high-voltage device corresponds to the heat-pump/refrigerator configuration of the NF-TPX device. In order to ease the understanding, we use simplified characteristics in the following, as shown in Figure 4c. In this case, we plot the variation of the power output only with respect to the LED voltage. For each LED voltage, we take the maximum power output reached for the PV cell voltages considered: ( ) = max ( , ), 0 ≤ < ∕ . This way of presenting the performance has several advantages: a) the comparison of the characteristic of different devices is made easier, b) the maximum power produced by the TPX device, along with the power produced by the related TPV device (at = 0), is made more readable.

In black is traced the characteristic obtained with the DB method, for an IQE of 0.9 and 1, with respective maximum output powers of =0.9 = 1.3 × 10 3 W.m -2 and =1 = 9.7 × 10 5 W.m -2 . Two main domains can be observed, with a fairly linear evolution first, and a sudden decrease in the electrical power output due to the exponential variations caused by the Bose-Einstein distributions. It is also interesting to note that for an IQE of 0.9, the characteristic is nearly identical to the one obtained for IQE = 1 up to a LED voltage of 0.9 V, meaning that the reduction of the power output seems to be mainly affected by the reduction of the achievable voltage range. In color are then plotted the results obtained with the SDD method, in the FF (blue) and in the NF (orange). The maximum performance of the device is given respectively as = 5.8 × 10 -6 W.m -2 and = 3.7 × 10 -4 W.m -2 . Even in the NF, the performance of the device considered is mediocre. Still, using TPX instead of TPV (i.e., with = 0 V) shows a quite noteworthy improvement ( -∕ -≃ 4). The different maximum power outputs and their respective LED and PV cell voltages are summarized in Table 4. In the next section, we will thus focus on the understanding of the difference in the results obtained between the DB and the SDD methods, before trying to improve the device considered.

Towards improved devices

Surface recombinations

A first means to increase the performance of the device is to decrease surface recombinations. By nature, these are non-idealities caused by the device (i.e., not necessarily directly related to a given material) that limit the collection of charges. In Figure 4c, we show the P-V characteristic obtained when surface recombinations are removed (in yellow). If this allows for moderate improvement, with the maximum power output increased by a factor 4 compared to the reference case and reaching =0 = 1.5 × 10 -3 W.m -2 , reducing the surface recombinations is still important to approach quasi-ideal and ideal cases as will be discussed in Section 4.3.

Carrier diffusion length

Two other parameters of interest could be studied in order to search for improved performance: the carriers diffusion coefficient D (Eq. 11) and the non-radiative lifetime (Eq. 14). Unlike surface recombinations coefficients, these two parameters are physically only related to the material and thus cannot be changed easily. However, they provide good information about the charge carrier movement capabilities inside the device, through the diffusion length = √ . By varying , we modify the distance carriers will be able to travel, a process which is thus equivalent to changing the geometry for electrons and holes, but not for photons. By doing this theoretical analysis, we can easily observe how close we could approach the ideal case; this provides information about what could be obtained when optimizing the geometry. For the sake of simplicity, this study is performed separately for the LED and the PV cell, with the other device kept as a passive emitter or receiver. We choose to change the diffusion length around the value ,0 obtained using the physical parameters from Table 2. Since both D and change when the diffusion length varies, we force them to vary at the same rate: = √ ( ⋅ 0 )( ⋅ 0 ) = ,0 . The results, shown in Figure 5, reveal that the I-V characteristic of the two devices tends towards the ideal case when the diffusion lengths considered reach a hundred times the physical value. By a really rough approximation, we could state that increasing the diffusion length by a factor is similar to decreasing the geometry by the same factor. While it is probably not possible to decrease all the dimensions by a factor 100, we can however note that the dark current is already quite close to the ideal value for = 10.

Improving the geometry

We now try to optimize the geometry in order to see if we could approach the ideal case. This optimization is performed using the fmincon solver of MATLAB, an algorithm searching for local minima; if this solution would not be relevant for finding the optimal geometry, since the result of the process can vary with the starting point, it however returns geometries that are improved in comparison to the reference case. We set the thickness for each region in the range [10 nm;10 µm], while eliminating surface recombinations. The result of this early optimization process is given in Figure 4c (green), while the improved geometry parameters are given in Table 1. The maximum power output is much larger than those obtained for all the other cases, reaching a value of , =0 = 1.1 × 10 4 W.m -2 . It even exceeds what was obtained with the DB method for a quasi-ideal NF-TPX device with an IQE of 0.9, and the performance reached by a FF-TPV device with an emitter at more than 1400 K [START_REF] Fan | Near-perfect photon utilization in an air-bridge thermophotovoltaic cell[END_REF]. It is also close to what is found for NF-TPV for 700 K temperature difference [START_REF] Vaillon | Micron-sized liquid nitrogencooled indium antimonide photovoltaic cell for near-field thermophotovoltaics[END_REF], and for thermoelectric devices for 300 K temperature difference [START_REF] El Oualid | High Power Density Thermoelectric Generators with Skutterudites[END_REF]. This result is invigorating for the development of NF-TPX device: it shows that even when getting rid of idealizing approximations, we can reach really high electrical power densities by improving both the contacts at the boundaries of the LED and the PV cell and their geometries for efficient charge carrier management. The device performance could be even further improved by optimizing the doping level of the different layers; however, the use of the SDD method limits the range of doping levels for which the result obtained is accurate. Solving instead the full set of equations (see Eq. 6,7 and 8) would permit a correct optimization of the doping levels. It is important to notice that the improvement of the geometry alone would not allow for such a large increase of the power output: when optimizing the geometry while keeping surface recombinations on the front surface (in purple in Figure 4c, see Table 1 for the geometry obtained), we reach = 3.0 × 10 -3 W.m -2 , in the same order of magnitude as what was found using the reference case without surface recombinations. The optimized geometry exhibits p-regions that are only 10 nm thin (i.e., the lower bound of the thickness range tested); if reaching such thickness seems difficult to achieve, it highlights nonetheless the necessity to reduce the thickness of the front layer in order to achieve high performance.

In Figure 6, we show the variation of electric field, electrostatic potential and quasi Fermi levels inside the LED and the PV cell, at the MPP. We notice that the quasi-Fermi levels (in color in Figures 6e and6f) are nearly constant: the assumption that the electrochemical potential is nearly constant and equal to seems therefore valid for this specific case.

Conclusion and Outlooks

We have developed a model coupling near-field radiative heat transfer and charge carrier transport using the simplified drift-diffusion method, allowing for an accurate representation of the phenomena occurring in a near-field thermophotonic device. The difference between the results obtained with the Detailed Balance (in 0D) and the simplified driftdiffusion (in 1D) methods underlines the necessity to use the latter for a precise estimation of a device performance. By carrying an extensive analysis of a GaAs-based nearfield thermophotonic device, we have also identified that the diffusion length and the surface recombinations have strong impacts on the performance. Through the improvement of the contacts and of the geometry, we are capable of reaching power densities as high as = 1.1 × 10 4 W.m -2 , i.e. close to 35 W.m -2 .K -1 . This is a promising result for near-field thermophotonic devices, especially since such performance was obtained in spite of the high GaAs bandgap energy compared to thermal energy at 600 K. In addition, the doping concentrations are fixed in this study, thus not optimized. Of course, the numerical study developed above has still room for improvement. In future work, the model itself should be improved, by getting rid of the approximation = in the modified Bose-Einstein distribution (see Eq. 2) and by solving fully the Poisson, drift-diffusion and continuity equations (see Eq. 6,7,8) instead of using the simplified drift-diffusion method. These developments have already been addressed for near-field thermophotovoltaic applications respectively in [START_REF] Callahan | Coupled Charge and Radiation Transport Processes in Thermophotovoltaic and Thermoradiative Cells[END_REF] and [START_REF] Blandre | High-injection effects in near-field thermophotovoltaic devices[END_REF], and could be applied to near-field thermophotonics, allowing for an accurate description of the photon recycling and of the charge carrier behaviour in general. Apart from the algorithm itself, the device considered could also be modified to approach more realistic geometries, by considering the use of pin junctions [START_REF] Blandre | Thermal radiation at the nanoscale: near-field and interference effects in few-layer structures and on the electrical performances[END_REF] for instance.

  a) absorption -each above-bandgap photon absorbed generates a unique electron-hole (e-h) pair: ̃ = = ( -) -( + ) (defining ̃ = ∫ and ± as the limits of the layer considered). b) radiative recombination -photon recycling is neglected, thus the local radiative recombination corresponds to the photon flux density emitted towards the other device: ̃ = = ( -) -( + ). c) non-radiative recombination -the level of non-radiative recombinations and of thermal generation of e-h pairs is set by the Internal Quantum Efficiency (IQE): ̃ = (1∕IQE -1)

Figure 2 :

 2 Photon energy (eV)

  hypothesis considered, we always have | | ≥ | | (they are equal if IQE = 1), having as a consequence that must be greater than in order to produce electrical power. b) < ∕ : if the voltage reaches this value, the modified Bose-Einstein distribution diverges.

Figure 5 :

 5 Figure 5: Impact of the diffusion lengthon the dark current and the short-circuit current of (a) an LED at 600 K, facing a GaAs absorber at 300 K and (b) a PV cell at 300 K facing a GaAs emitter at 600 K. Since the value of the diffusion length changes in the p-region and in the n-region, the variation of the dark current is shown as a function of the factor defined as = ,0 . Similar tendencies are found for the short-circuit current.

Figure 6 :

 6 Figure 6: (a,b) Electric field, (c,d) Electrostatic potential and (e,f) Quasi-Fermi levels of the LED and the PV cell composing the device at the maximum power point reached without surface recombinations, with the optimized geometry at 10 nm. The depletion region is represented in grey.

Table 1

 1 Device parameters. The first value given is related to the LED, the second one to the PV cell. Opt. 1 refers to the optimized values obtained with surface recombinations, Opt. 2 to those obtained without surface recombinations (see Section 4.3).

	Parameter	Unit Symbol Ref. Opt. 1 Opt. 2
	Temperature	K		600
				300
	P-region doping	cm -3		10 20
				10 20
	N-region doping	cm -3		10 18
				10 18
	Gap distance	nm	d	10
	Mirror thickness	nm		10
	P-region thickness nm		400 10	10
				400 10	10
	N-region thickness nm		10 4	10 4	500
				10 4	10 4	10 3

Table 2

 2 Modelling parameters.

	Parameter	Unit	Symbol Value Reference
	Bandgap energy	eV		1.42	[38]
	Dielectric function	-
	Interband				[37]
	Drude-Lorentz				[39, 40]
	Static rel. permittivity -		12.9	[41]
	Electron rel. eff. mass -	*	0.063	[41]
	Hole rel. eff. mass Electron mobility	-cm 2 .V -1 .s -1	* ℎ	0.53	[41] [42]
	Hole mobility	cm 2 .V -1 .s -1	ℎ		[42]
	SRH recomb. coeff.	s -1		3×10 5	[29]
	Auger recomb. coeff. cm 6 .s -1		10 -30	[29]
	Surf. recomb. coeff.	m.s -1		5×10 2	[43]

Table 3

 3 Device efficiency for a 10 nm gap distance.

		(W.m -2 )	(%)	(%)	(%)	Σ (%)
	Above	5.9 ×10 4 78		1.3 ×10 2 2.2	9.1
	All photons 1.3 ×10 6 3.5

Table 4

 4 Maximum power point reached for the different cases studied in Figure4c.

	Method Case	P (W.m -2 ) U LED (V) U PV (V)
	SDD	Far-field	5.8×10 -6	0.00	0.58
		Ref. case	3.7×10 -4	0.13	0.69
		W/o surf. rec.	1.5×10 -3	0.22	0.75
		Optimized	3.0×10 -3	0.27	0.77
		Opt. w/o surf. rec. 1.1×10 4	1.13	1.25
	DB	IQE = 0.9	1.3×10 3	1.00	1.16
		IQE = 1	9.7×10 5	1.38	1.4

Acknowledgements

We thank E. Blandre for the help given, P. Kivisaari, T. Sadi, J. Oksanen from Aalto University for constructive discussion, and E.J. Tervo for clarification.

The project TPX-Power has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 951976.