Pablo Caño
email: pablo.cano@ies.upm.es

Manuel Hinojosa

Iván García

David Fuertes

Marrón Carmen

María Ruiz-Herrero

Richard Beanland

Andrew Johnson

Ignacio Rey-Stolle
email: ignacio.reystolle@upm.es

David Fuertes Marrón

Carmen M Ruiz

Solar Energy GaAsP/SiGe tandem solar cells on porous Si substrates

Keywords: III-V on silicon, GaAsP/SiGe, porous silicon, reverse buffer layers, tandem on silicon III-V on silicon, GaAsP/SiGe, porous silicon, reverse buffer layers, tandem on silicon 00-01, 99-00

III-V compound semiconductors and SiGe alloys can be combined to develop multijunction solar cells on Silicon substrates with optimum bandgap combinations. Current implementations of such devices have reached efficiencies over 20%, using thick -and thus costly-buffer layers which induce the appearance of cracks in large area samples. As a strategy to mitigate these two issues (thick buffers and cracking), a GaAsP/SiGe tandem solar cell has been developed employing group IV reverse graded buffer layers grown on Ge/Si virtual substrates with a subsurface Silicon porous layer. Reverse buffer layers facilitate a reduction in the threading dislocation density with limited thicknesses but can also induce cracks. To minimise this, a porous silicon layer has been incorporated close to the Ge/Si interface so that the ductility of this layer suppresses crack propagation. In terms of solar cell performance, this porous layer reduces the problem of cracks, not totally supressing them though. Accordingly, the low shunt resistance observed in previous designs has been increased thus improving solar cell efficiency, which is still notably behind designs using thicker forward graded buffer layers. The first results of this new architecture are presented here.

Introduction

III-V multijunction solar cells are very efficient, repeatedly breaking conversion records in recent years [START_REF] Geisz | Six-Junction III-V Solar Cells with 47.1% Conversion Efficiency Under 143 Suns Concentration[END_REF]), but still expensive for terrestrial applications.

Conversely, silicon solar cells whose technology dominates the terrestrial market [START_REF] Battaglia | High-Efficiency Crystalline Silicon Solar Cells: Status and Perspectives[END_REF]) are approaching a standstill having essentially reached their practical efficiency limit [START_REF] Yoshikawa | Exceeding Conversion Efficiency of 26% by Heterojunction Interdigitated Back Contact Solar Cell with Thin Film Si Technology[END_REF]. Therefore, the integration of III-V semiconductors on silicon substrates has been the target of numerous research lines from the 1980s [START_REF] Suzuki | Growth Mechanism of GaP on Si Substrate by MOVPE[END_REF], Hayashi et al. 1994a) based on the premise of high performance III-V semiconductor multijunction solar cells combined with the low-cost advantages of large area silicon substrates (Hayashi et al. 1994b[START_REF] Kurtz | A Comparison of Theoretical Efficiencies of Multi-Junction Concentrator Solar Cells[END_REF][START_REF] Supplie | Metalorganic Vapor Phase Epitaxy of III-V-on-Silicon: Experiment and Theory[END_REF][START_REF] Essig | Raising the One-Sun Conversion Efficiency of III-V/Si Solar Cells to 32.8% for Two Junctions and 35.9% for Three Junctions[END_REF]. Other alternatives for Si-based tandem solar cells have been proposed as perovskites on Si [START_REF] Leijtens | Opportunities and Challenges for Tandem Solar Cells using Metal Halide Perovskite Semiconductors[END_REF], or thin films on Si [START_REF] Ramanujam | Copper Indium Gallium Selenide Based Solar Cells -a Review[END_REF]Singh 2017, Valentini et al. 2019), to name the most relevant. Recently, perovskites on Si have achieved higher efficiencies than III-V/Si architectures exceeding 29% [START_REF] Al-Ashouri | Monolithic Perovskite/Silicon Tandem Solar Cell with >29% Efficiency by Enhance Hole Extraction[END_REF]) as compared to 25.9% for the best device reported with III-V/Si (Feifel et al. 2021). However, the limited reliability and durability of perovskites [START_REF] Wu | Ultrahigh Durability Perovskite Solar Cells[END_REF]) in contrast to III-Vs [START_REF] Nuñez | Estimation of Activation Energy and Reliability Figures of Space Lattice-Matched GaInP/Ga(in)as/Ge Triple Junction Solar Cells from Temperature Accelerated Life Tests[END_REF]) is a big hurdle for their industrial deployment, which makes the research on III-V/Si design a solid alternative.

Among the wealth of approaches to combine III-V materials and silicon for PV applications, monolithic structures are of particular interest for their straightforward integration in current manufacturing lines and lower production cost than other approaches. In this monolithic integration III-V and group-IV compounds are grown heteroepitaxially on a single substrate to produce a multijunction solar cell, which is then processed as a monolithic device, i.e. forming metallic contacts both at the front and rear and depositing an antireflection coating. However, the direct growth of III-V and group-IV semiconductors on silicon must tackle key difficulties such as the large differences in lattice parameter, thermal expansion coefficients or polar on non-polar growth.

Therefore, smart engineering of the buffer layer between the silicon and the III-V layers is essential in order to accommodate these dissimilar parameters. III-V/Si architectures have been demonstrated for optoelectronics with lasers [START_REF] Groenert | Monolithic Integration of Room-Temperature Cw GaAs/AlGaAs Lasers on Si Substrates Via Relaxed Graded GeSi Buffer Layers[END_REF], bipolar transistors [START_REF] Lew | High Gain AlGaAsGaAs Heterojunction Bipolar Transistor Fabricated on SiGeSi Substrate[END_REF], photodetectors [START_REF] Luan | High-Quality Ge Epilayers on Si with Low Threading-Dislocation Densities[END_REF] or light emitting diodes [START_REF] Yang | Monolithic Integration of III-V Optical Interconnects on Si using SiGe Virtual Substrates[END_REF]. As for monolithic III-V/Si photovoltaic devices, GaAsP/Si dual junctions using GaP buffer layers (Hayashi et al. 1994a[START_REF] Lepkowski | Loss Analysis and Design Strategies Enabling >23% GaAsP/Si Tandem Solar Cells[END_REF], Fan et al. 2020a, Fan et al. 2020b[START_REF] Fan | 20%-Efficient Epitaxial GaAsP/Si Tandem Solar Cells[END_REF][START_REF] Caño | Growth of GaP Layers on Si Substrates in a Standard MOVPE Reactor for Multijunction Solar Cells[END_REF][START_REF] Saenz | Nucleation of High-Quality GaP on Si through V-Groove Si Substrates[END_REF], GaInP/GaAs tandem cells on Ge/Si virtual substrates [START_REF] Ginige | Characterization of Ge-on-Si Virtual Substrates and Single Junction GaAs Solar Cells[END_REF][START_REF] Wang | Fabrication and Characterization of Single Junction GaAs Solar Cells on Si with as-Doped Ge Buffer[END_REF][START_REF] Kim | InGaP Solar Cell on Geon-Si Virtual Substrate for Novle Solar Power Conversion[END_REF][START_REF] Bioud | Uprooting Defects to Enable High-Performance III-V Optoelectronic Devices on Silicon[END_REF], García et al. 2021) and GaAsP/SiGe dual junctions grown on inactive Si substrates have been developed [START_REF] Schmieder | Analysis of Tandem III-V/SiGe Devices Grown on Si[END_REF][START_REF] Pitera | Monolithically Integrated Thin Film III-V/Si Solar Panel on Wafer for Active Power Management[END_REF]). In the latter design, an extra degree of freedom for bottom cell bandgap tunability is provided by changing the composition of the SiGe alloy. A further step ahead is the addition of minute amounts of tin to form SiGeSn alloys, which provide both bandgap and lattice constant adjustability, at the price of having to sort out big difficulties in material growth and quality (Roucka et al. 2016, Soref and[START_REF] Soref | Predicted Band Gap of the New Semiconductor SiGeSn[END_REF]. Anyhow, Si1-xGex alloys allow a reasonable approach to the optimum bandgap combination for a dual-junction solar cell [START_REF] Connolly | Designing III-V Multijunction Solar Cells on Silicon[END_REF].

Actually, the ~0.95 eV-subcell for an ideal current matching in 3-and 4-junction solar cell designs can be achieved with Si1-xGex, with x~70-80%. [START_REF] Friedman | Analysis of the GaInP/GaAs/1-eV/Ge Cell and Related Structures for Terrestrial Concentrator Application[END_REF].

Moreover, SiGe alloys have reached a significant level of maturity in terms of material growth and quality, since they are used in the electronic industry for the fabrication of bipolar transistors, MOS transistors, CMOS and BiCMOS technologies [START_REF] Haddara | Silicon-Germanium: Properties Groth and Applications[END_REF]. In fact, GaAsP/SiGe tandem cells on Si have reached efficiencies over 20% [START_REF] Wang | Current Matched Three-Terminal Dual Junction GaAsP/SiGe Tandem Solar Cell on Si[END_REF][START_REF] Pitera | Monolithically Integrated Thin Film III-V/Si Solar Panel on Wafer for Active Power Management[END_REF][START_REF] Conrad | Improved GaAsP/SiGe Tandem on Silicon Outdoors and Under Concentration[END_REF]. In these structures, a Si1-xGex buffer is grown on the silicon substrate as a graded layer, increasing Ge content until the desired Ge composition is reached [START_REF] Fitzgerald | Relaxed GeXSi1-X Structures for III-V Integration with Si and High Mobility Two-Dimensional Electron Gases in Si[END_REF], Wang et al. 2016[START_REF] Faucher | Single-Junction GaAsP Solar Cells Grown on SiGe Graded Buffers on Si[END_REF], Conrad et al. 2016). The thickness of such buffer layers is typically between 5 and 15 µm in order to reduce the threading dislocation density and achieve the target bandgap (defined by the lattice parameter) while retaining sufficient quality in the surface morphology [START_REF] Schmieder | Analysis of Tandem III-V/SiGe Devices Grown on Si[END_REF][START_REF] Groenert | Monolithic Integration of Room-Temperature Cw GaAs/AlGaAs Lasers on Si Substrates Via Relaxed Graded GeSi Buffer Layers[END_REF][START_REF] Faucher | Single-Junction GaAsP Solar Cells Grown on SiGe Graded Buffers on Si[END_REF], Wang et al. 2016). Accordingly, following the terminology coined by Shah and coworkers [START_REF] Shah | Reverse Graded Relaxed Buffers for High Ge Content SiGe Virtual Substrates[END_REF][START_REF] Capellini | Strain Relaxation in High Ge Content SiGe Layers Deposited on Si[END_REF]), we will refer to this kind of graded buffer layer as a forward-graded buffer.

Here, an innovative architecture for the integration of III-V compounds and SiGe alloys on silicon substrates is presented, using so-called reverse-graded buffers [START_REF] Shah | Reverse Graded Relaxed Buffers for High Ge Content SiGe Virtual Substrates[END_REF][START_REF] Capellini | Strain Relaxation in High Ge Content SiGe Layers Deposited on Si[END_REF]. This III-V/SiGe/Si approach is aggressive in terms of growth; a germanium layer is directly grown by CVD on the silicon substrate, despite the large lattice mismatch, utilizing a growth scheme that has demonstrated dislocation densities as low as 5x10 6 cm -2 [START_REF] Lee | Reduction of Threading Dislocation Density in Ge/Si using a Heavily as-Doped Ge Seed Layer[END_REF]. Then, a Si1-xGex graded buffer layer is grown, decreasing from pure Ge to the desired final target (Ge ~75%). Because of the almost complete relaxation of the initial Ge layer and the reduction in lattice constant with increased silicon content, reverse-graded buffers are subject to tensile strain. The tensile strain facilitates the movement of glissile threading dislocations, aiding their annihilation. Reverse-graded buffers thus offer promise for both lower TDD and smoother surfaces [START_REF] Xie | Semiconductor Surface Roughness: Dependence on Sign and Magnitude of Bulk Strain[END_REF][START_REF] Capellini | Strain Relaxation in High Ge Content SiGe Layers Deposited on Si[END_REF]. Furthermore, for Ge contents over 50%, reverse-graded buffers are thinner due to the smaller change in composition that is needed, which decreases costs [START_REF] Shah | Reverse Graded Relaxed Buffers for High Ge Content SiGe Virtual Substrates[END_REF][START_REF] Ward | Design Rules for Dislocation Filters[END_REF]. On the other hand, the high tensile strain and the larger differences in thermal expansion coefficients can result in the appearance of cracks [START_REF] Shah | Reverse Graded SiGe/Ge/Si Buffers for High-Composition Virtual Substrates[END_REF]) that cause a degradation in the device performance, as we have shown in a previous work [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF]. In fact, the differences in thermal expansion coefficients in the buffer layer itself is lager in reverse buffers than in forward buffers, due to the abrupt change between Si and Ge.

As way to sort out some of the aforementioned problems, in this new architecture, a porous silicon layer on the substrate subsurface has been implemented. Porous silicon has been previously used in optoelectronic [START_REF] Zheng | Highly Sensitive Photodetector using Porous Silicon[END_REF][START_REF] Tsai | Rapid-Thermal-Oxidised Porous Si Photodetectors[END_REF]) and PV applications [START_REF] Smestad | Photovoltaic Response in Electrochemically Prepared Photoluminescent Porous Silicon[END_REF][START_REF] Jiménez-Cruz | Porous Silicon Infiltration with Advanced Materials for their use in Third Generation of Solar Cells[END_REF] as window layers, passivation or anti-reflection coatings [START_REF] Menna | Porous Silicon in Solar Cells: A Review and a Description of its Application as an AR Coating[END_REF][START_REF] Prasad | Porous Silicon Oxide Anti-Reflection Coating for Solar Cells[END_REF]. Here, the porous silicon layer is used to absorb strain energy due to its lower stiffness compared to bulk silicon [START_REF] Menna | Porous Silicon in Solar Cells: A Review and a Description of its Application as an AR Coating[END_REF][START_REF] Barla | Determination of Lattice Parameter and Elastic Properties of Porous Silicon by X-Ray Diffraction[END_REF], since it has lower Young's modulus [START_REF] Karbassian | Porosity -Process[END_REF]. This way, we think that the Si/porous-Si structure should behave similarly as a partially compliant substrate [START_REF] Ayers | Compliant Substrates for Heteroepitaxial Semiconductor Devices: Theory, Experiment, and Current Direction[END_REF].

In order to understand the potential of reverse buffer layers in the context of III-V/Si multijunction solar cell development, we herein present the results of GaAsP/SiGe tandem solar cells grown on silicon substrates with porous layers. In this paper, we report the characterization of these structures and solar cells, highlighting the issues that arise and comparing them with tandem GaAsP/SiGe devices grown on standard silicon wafers from our previous works [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF].

Experimental

Fig. 1 shows the structure of the solar cells in this study with doping levels and thicknesses. The samples were grown following a combined approach using both chemical vapor deposition (CVD) and molecular beam epitaxy (MBE). Firstly, porous silicon was created by means of electrochemical etching on 6-inch (100) 6º off towards

[110] silicon wafers with a resistivity of 0.01 𝛺cm. Then, all group-IV layers were grown in an ASM Epsilon LPCVD reactor at a temperature of ~650 °C using standard precursors (SiH4 and GeH4 for the alloys and boron and phosphorus for the p-type and n-type doping respectively). An initial 3.5 μm germanium layer was grown directly on the silicon wafer followed by a ~1.5 µm SiGe reverse-graded buffer, changing composition gradually from pure germanium to 76% Ge. Afterwards, the Si0.24Ge0.76 bottom cell with Eg ~1 eV was grown. Before the transfer to a Veeco Gen2000 MBE reactor, the Si0.24Ge0.76 bottom cell was capped with a 5 nm Ge layer in order to prevent oxidation of the Silicon in the SiGe, which is difficult to remove thermally in the MBE chamber. In the second growth III-V layers were grown lattice matched to the Si0.24Ge0.76 bottom cell, forming the tunnel junction and the GaAs0.75P0.25 top cell (Eg ~1.65 eV) [START_REF] Pitera | Monolithically Integrated Thin Film III-V/Si Solar Panel on Wafer for Active Power Management[END_REF][START_REF] Wang | Current Matched GaAsP/SiGe Tandem Device on Si Over 20% Efficiency Under Indoor Measurement[END_REF][START_REF] Diaz | Tandem GaAsP/SiGe on Si Solar Cells[END_REF]) at ~650 °C with a growth rate of ~1 µm/h. In our previous work on conventional Si substrates, the growth was carried out under similar conditions. High-resolution X-ray diffraction reciprocal space maps (RSMs) of the structures were obtained using an X'Pert Pro MRD tool.

An important remark about this structure should be made at this point. As shown in Fig. 1, the SiGe bottom cell is just 1 µm thick. This value is far from the ~5-6 µm needed for a current matched device, so our solar cells will be severely bottom cell limited. This design decision was made to minimize the risk of epilayer peel-off during the processing of the solar cells, which can be a consequence of the inclusion of porous silicon. In our experience, the handling of large area (6 inch) wafers with thick epilayers is not easy.

Oftentimes, minor shocks during the manual handling of the wafers cause the epilayers to peel off catastrophically, ending up shattered into a thousand pieces. To minimize this risk, we limited the thickness of the bottom cell to one micron. Therefore, the goal of this study is an initial assessment the joint use of porous silicon layers and reverse graded buffers to implement functional GaAsP/SiGe tandem solar cells rather than the achievement of a high efficiency which is out of reach with a severe current mismatch.

The epi-wafers were processed into ~0.10 cm 2 (3.2 mm × 3.1 mm) solar cells with conventional photolithography techniques. Since group-IV and III-V semiconductors have different processing requirements, both front and rear contacts were specifically designed to reduce thermal budget. In this way, the high temperatures of conventional silicon metallization were avoided, minimizing the risk of crack propagation and guaranteeing material compatibility. The front contact was formed using electroplated gold (~600 nm thick) without any alloying. The rear contact was deposited with Electron Beam Physical Vapor Deposition (EBPVD) and consisted of a stack of Pd(50nm) / Ti(50nm) / Pd(50nm) / Al(1000nm) alloyed at 170ºC for 600 s. No antireflection coating (ARC) was deposited on the cells. The devices were isolated though mesa etching using a NH4OH-H2O2-H2O (2:1:10) solution for arsenides. This solution is also suitable for etching SiGe. In the case of phosphides, HCl-H2O (1:1) was used to remove them.

After device fabrication, External Quantum Efficiency (EQE) was measured using a custom-made tool based on a 1000W Xe lamp, a triple-grating monochromator (JOBIN-YVON) and a lock-in amplifier. I-V curves were measured using the four-probe method with a Keithley 2602 source-meter instrument and a home-made AAA solar simulator based on a 1000-W Xe-lamp and an ORIEL 68820 stabilized power supply.

Capacitance-voltage (C-V) measurements were carried out with an HP 4284 LCR meter.

Curves were collected with a 5 mV signal at 30 kHz, which is considered to be highfrequency [START_REF] Recart | Application of Junction Capacitance Measurements to the Characterization of Solar Cells[END_REF]. Spectral Photovoltage (SPV) measurements were performed in two-probe contact mode at room temperature using a 200W QTH-lamp filtered through a 1/8 monochromator (Oriel) as probe beam and a lock-in amplifier (Stanford).

Results and Discussion

Material Characterization

A cross-section of the whole structure can be observed in the Transmission Electron Microscopy (TEM) image in Fig. 2, where the most relevant layers in the structure have been identified. The thicknesses of the layers agree well with the nominal values in Fig. 1. As can be seen, the porous silicon layer is clearly delineated between the underlying substrate and overlying CVD-grown Si. The other group-IV layers have sharp and flat interfaces. Nonetheless, the interface between the SiGe subcell and the GaAsP top cell shows some unevenness, which is even more noticeable at the GaAsP/GaAs interface, which is especially susceptible to defects [START_REF] Sharma | Controlling Epitaxial GaAsXP1-X/Si1-YGeY Heterovalent Interfaces[END_REF]). This can be observed in figure 3, which shows TEM images of key parts of the structure at higher magnification. were grown on standard silicon wafers, a TDD of ~8x10 5 cm -2 was measured [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF]). This level is comparable to TDDs reported in other studies of forward buffers in Si1-xGex with x ~80% [START_REF] Ringel | Single-Junction InGaP/GaAs Solar Cells Grown on Si Substrates with SiGe Buffer Layers[END_REF][START_REF] Groenert | Monolithic Integration of Room-Temperature Cw GaAs/AlGaAs Lasers on Si Substrates Via Relaxed Graded GeSi Buffer Layers[END_REF][START_REF] Milakovich | Growth and Characterization of GaAsP Top Cells for High Efficiency III-V/Si Tandem PV[END_REF] and thus suggests that the challenge of these structures relies more on the optimization the epitaxial growth conditions for porous silicon substrates rather than the use of reverse graded buffers.

On the wafer surface, some cracks are still visible to the naked eye, although the crack density has been qualitatively reduced as compared with growths on standard Si substrates with equally thin bottom subcells [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF]). The precise quantitative assessment of the cracking density is still an open question. We observe that the cracking is very irregular amid wafers of the same batch and even in different areas of the same wafer. Even more so, we have evidence that apparently clean areas have buried cracks that only appear when the upper layers are etched-off. For these reasons, the precise crack density could not be quantified and remains under investigation. cell with the same structure -i.e. also with a thin 1 µm SiGe bottom cell-but manufactured on standard Si wafers is included for reference. The final tail of the EQE (Eg ~1.64 eV) agrees well with the GaAs0.75P0.25 composition determined from the reciprocal space maps, which should correspond to around Eg = 1.66 eV. However, the EQE reaches a maximum of ~50% whereas other works in the literature for similar devices grown by MBE report clearly higher EQEs, as can be observed in the green line corresponding to GaAsP solar cell grown by Grassman and coworkers on Si with GaP nucleation layers and GaAsP graded buffers [START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF]. We choose to benchmark our results with [START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF] since the GaAsP top cell has a similar structure -though with a slightly higher bandgap -also grown by MBE and underwent a reactor transfer prior to the growth of the top cell. There are better results in the literature [START_REF] Fan | 20%-Efficient Epitaxial GaAsP/Si Tandem Solar Cells[END_REF][START_REF] Diaz | Tandem GaAsP/SiGe on Si Solar Cells[END_REF][START_REF] Grassman | Toward >25% Efficient Monolithic Epitaxial GaAsP/Si Tandem Solar Cells[END_REF]) that we do not use as a benchmark simply because they were either grown with notably different structures or techniques (MOVPE), or lack sufficient structural information. 245 Fig. 5.b shows in a black dashed line the expected (i.e. simulated) EQE from the SiGe bottom cell that could not be actually measured. In our previous work, the EQE of 1 µm thick SiGe bottom subcells could not be measured either. However, as shown in a blue line in Fig. 5.b, the EQE of thicker SiGe subcell designs (5 µm) could be measured and 250 thus a functional GaAsP/SiGe tandem solar cell on silicon was demonstrated [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF]. However, in the structure grown on porous Silicon only the top cell EQE could be measured, even though many solar cells were manufactured and tested. The lack of bottom cell response will be discussed later. In order to gain some insight into the performance differences of the top cells, EQE simulations were made using an analytic model [START_REF] Algora | Handbook on Concentrator Photovoltaic Technology[END_REF]. Figure 6 present the results of these simulations for GaAsP subcells grown on: standard Si (Fig. [START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF]). Round symbols correspond to the experimental data. The thick black line is the total simulated EQE whereas the thin black line corresponds to total absorptance. The thick colored lines correspond to the contributions of each layer: orange for the window, blue for the emitter, green for the space charge region and red for the base. The thin colored lines are the absorptance in each layer with the same color code. The JSC obtained from the integration of the experimental EQE over the AM1.5G spectrum is included in each graph.

Solar Cell Results

the literature (Fig. 6.c) [START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF]. In these figures, circles correspond to the experimental data of Fig. 5.a and lines account for the modelling. The thick black line in each panel is the calculated total EQE whereas each thin black line corresponds to the total absorptance in the top cell (i.e. the ideal EQE with unity collection efficiency).

Thick colored lines correspond to the contributions of each layer with the color code as indicated in the figure caption, whereas their thin counterparts again correspond to the absorptance in each layer. Therefore, the difference between thick and thin lines of the same color gives a visual indication about how close (or far) a layer is from perfect collection efficiency. Comparing the calculated EQE with the absorptance, it can be concluded that the response of all GaAsP top cells clearly shows room for significant improvement. In the case of the emitter (blue lines), very similar responses are found in the three designs. In all cases it could be fitted with a diffusion length of ~100 nm, which coincides with the emitter thickness in the three cases. In fact, this yields a reasonable blue response of the three devices. However, the base and space charge region are clearly different in the three designs. The response of the base is some way below the corresponding absorptance curve (thin red lines). In addition, the three designs behave quite differently: we see a remarkably low response in the cells grown on standard Si (Fig. 6.a), which could be fitted with a diffusion length of ~200 nm; there is a very low response in the cells grown on porous Si (Fig. 6.b), which could be fitted with a diffusion length of ~350 nm; and we find a low response in the cells from [START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF] (Fig. 6.c), which could be fitted with a diffusion length of ~550 nm. In all three cases, the diffusion lengths are notably smaller than the base layer thickness (2 µm) and thus the collection efficiency is deleteriously affected. Regarding the space charge region, some differences in collection efficiency are observable too. In each case, it was necessary to simulate the space charge region (SCR) with a collection probability similar to that of the quasi-neutral base in order to obtain a reasonable fit of the experimental curve. If unity collection efficiency was assumed for the SCR (as in Hovel's equations (Algora and Rey-Stolle 2016)), no fit was possible.

Since the three GaAsP top cell structures are very similar, these simulation results can be understood as progress towards a better crystallographic quality in each design, though still reaching far from perfect minority carrier collection in the SCR and base.

Other than the evident effect of TDD, it has been reported that the GaAsP epitaxial growth conditions influence material quality in a poorly understood way [START_REF] Fan | 20%-Efficient Epitaxial GaAsP/Si Tandem Solar Cells[END_REF]. We therefore believe that there is an ample margin for improvement of these results by tuning the growth conditions, considering that this is our first attempt to grow tandem cells on porous Si substrates.

Coming back to the lack of response in the SiGe bottom cell, it should be noted that the situation here resembles that of a GaInP/Ga(In)As/Ge triple junction solar cell (3JSC)

where the EQE of the germanium bottom cell is frequently unmeasurable for a number of reasons [START_REF] Meusel | Spectral Response Measurements of Monolithic GaInP/Ga(in)as/Ge Triple-junction Solar Cells: Measurement Artifacts and their Explanation[END_REF], Barrigón et al. 2015a). The emitter of our SiGe cell is formed during the growth of the GaInP nucleation layer -as in Ge subcells in 3JSCs-by the concurrent in-diffusion of P, but also of Ga and In, and out-diffusion of Ge from the capping layer. Such a strong atom exchange at the III-V/IV heterointerface, in a material with a high TDD, may produce a p-n junction exhibiting both a low shunt resistance and a low breakdown voltage, which are typical causes for an unmeasurable bottom cell (Barrigón et al. 2015b). I-V curves under one-sun illumination (AM1.5d ASTM G173 spectrum) of the GaAsP/SiGe tandem cells are presented in Fig. 7.top for the design on porous silicon (red and pink curves) and standard silicon wafers (blue). Again, the red line represents the best device, whilst the thin pink lines are the I-V curves of other cells in the same wafer and thus give an idea of device-to-device variability. The short-circuit current density (JSC = 8.9 mA/cm 2 ) is in agreement with the integral of the top cell EQE (Fig 5.b). However, as a result of its deliberate limited thickness, we would expect the thin SiGe cell to limit the JSC of the tandem. The absence of this limitation together with the fact that we could not measure the EQE of the SiGe subcell underpin the hypothesis that this subcell is severely shunted, and such level of shunting is quite uniform across the devices manufactured and wafers processed. The magnitude of the JSC measured is obviously low, which is a result of the collection problems discussed around the EQE curves and also of the lack of an ARC. Looking at the shape of the curve around VOC, the impact of a parasitic diode in reverse bias is also evident, which we address below.

Apart from this parasitic diode the most remarkable features in the I-V curves are the limited FF and low VOC in both designs. In the case of tandem cells grown on conventional Si substrates, the low shunt resistance is very evident in the I-V curve, causing deleterious effects in both FF and VOC. In [START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF] it was argued that the shunting was caused by a high crack density in the cell, which behave both as efficient recombination centers for minority carriers and as electrical shorts. However, 325 in the samples grown on porous Si substrates, clear improvements in both FF (from 26.5% to 48.6%) and VOC are observed (from 0.48 to 0.67 V), as a result of lower crack densities. Values are still low but the improvement is evident. In Fig. 7 (bottom) the dark I-V curves of the GaAsP/SiGe tandems are plotted following the same color code as in Fig. 7 (top). The shunt resistances can be visually compared between both structures, 330 being higher in the porous silicon design.

Porous silicon has been reported to have a low electrical conductivity [START_REF] Zheng | Highly Sensitive Photodetector using Porous Silicon[END_REF][START_REF] Smestad | Photovoltaic Response in Electrochemically Prepared Photoluminescent Porous Silicon[END_REF][START_REF] Tsai | Rapid-Thermal-Oxidised Porous Si Photodetectors[END_REF][START_REF] Jiménez-Cruz | Porous Silicon Infiltration with Advanced Materials for their use in Third Generation of Solar Cells[END_REF], so its impact on series resistance is a possible concern. However, the presence of the parasitic junction complicates the assessment of the series resistance from the J-Vs in Fig. 7.

In order to confirm the presence of a parasitic diode, C-V measurements were performed on the samples (Fig. 7, bottom inset). The detailed understanding of C-V curves in multijunction solar cells is not straightforward, since only the voltage across the device terminals is known and no information is available about the particular bias at each subcell [START_REF] Ruiz | Capacitance Measurements for Subcell Characterization in Multijunction Solar Cells[END_REF][START_REF] Rutzinger | Measurement of Subcell Depletion Layer Capacitances in Multijunction Solar Cells[END_REF][START_REF] Hoheisel | Determination of the Subcell Photovoltage in Multijunction Solar Cells Via Voltage Dependent Capacitance Analysis[END_REF]. However, in this case we simply intend to detect the signature of a parasitic diode in reverse bias.

The inverse of the total capacitance of the multijunction solar cell (CT) equals the sum of the inverse of the capacitance of each junction in the device (Ci):

(1)

In principle, in well-behaved cells, it is reasonable to assume that neither the metal/semiconductor contacts nor the tunnel junctions contribute to CT. In particular, tunnel junctions, being nominally ultra-highly doped, should have a very high junction capacitance and therefore should play a negligible effect in the sum of inverses, which is dominated by the smallest contributions [START_REF] Ruiz | Capacitance Measurements for Subcell Characterization in Multijunction Solar Cells[END_REF]. So in Eq. (1) only capacitance contributions from p/n junctions should be present. As reported by [START_REF] Ruiz | Capacitance Measurements for Subcell Characterization in Multijunction Solar Cells[END_REF], the C-V curve of a dual junction solar cell with a good tunnel junction would show two humps in the first quadrant -one for each subcell-, a roll-off at high positive voltages, whereas at low positive voltages and under reverse bias the capacitance should steadily decline to its lowest value. We performed C-V measurements at 300K under dark conditions and obtained the curve in the inset of Fig. 7 (bottom). In the positive voltage range, we find the two humps expected followed by the fall of capacitance at high voltages (V>1.3V). These two peaks correspond to the GaAsP and SiGe junctions and, in the second quadrant, we would expect to observe a decline in the capacitance associated with the extension of their space charge regions at negative bias [START_REF] Ruiz | Capacitance Measurements for Subcell Characterization in Multijunction Solar Cells[END_REF]. Instead, we find a third peak that we interpret as a third junction in reverse polarity since it shows the expected behavior for the first quadrant but in the second, namely, a fall in capacitance at high voltages, then a peak and finally a decline as we approach 0 volts. This is the signature of a p/n junction in reverse bias, namely, a parasitic junction, as anticipated from the lighted I-V curves in Fig. 7 (top).

We hypothesize that this diode is probably related to the tunnel junction or an adjacent layer. In conventional multijunction cells on Ge substrates, it has been reported that Ge can diffuse over 300 nm into the GaInP nucleation layer, producing strong n-type background doping [START_REF] Barrutia | Effect of Ge Autodoping during III-V MOVPE Growth on Ge Substrates[END_REF]. In our structure, we have an analogous situation in which a 100 nm n-type GaInP nucleation layer is grown on a Ge cap layer (see Fig. 1). It is possible that Ge diffusion during the growth of the GaAsP top cell may reach the tunnel junction p-side -or its cladding layer-partially compensating its nominally high doping, thereby losing the tunneling properties. This would produce a diode in reverse polarity affecting both the series resistance and the fill factor. The growth of a thicker GaInP nucleation layer could avoid this issue in the future.

In order to get rid of the influence of the parasitic diode and assess the performance limits of the device, we measured the I-V curve at different irradiances and performed a JSC-VOC analysis as shown in Fig 8 .a. The JSC-VOC line follows a clear logarithmic trend corresponding to an ideality factor of n=3.6 and a reverse saturation current density of J0 = 8.5•10 -3 A/cm 2 . An ideality factor of 3.6 is in agreement with a device being limited by recombination either in the space charge region or at the junction perimeter, or most likely a combination of both. A high perimeter recombination may be linked to exposed facets in cracks. A high recombination rate within the space charge region is also consistent with the abundance of defects and with the simulations of the EQE, where limited collection in this region had to be assumed. The reconstructed I-V from JSC-VOC measurements is shown in Fig. 8.b together with the experimental I-V. The FF of the reconstructed curve is 63.2%, much better than the real I-V, though still far from the ~80% reported for GaAsP top cells in the best results from the literature [START_REF] Fan | 20%-Efficient Epitaxial GaAsP/Si Tandem Solar Cells[END_REF][START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF]. This indicates that, even leaving aside resistive and parasitic diode effects, recombination losses still impact severely the performance of the solar cells. This is another argument for the improvement in the crystallographic quality of the GaAsP top cell as an imperative. In this regard, the GaAsP material quality may be compromised due to the fact that it was grown by MBE. There is evidence that III-V nucleation on Ge -note that the III-V growth takes place on the thin Ge cap (see Fig. 1)is much more challenging when using MBE vs MOVPE [START_REF] Li | Growth of Device-Quality GaAs Layer Directly on (001) Ge Substrates by both Solid-Source and Gas-Source MBE[END_REF]. We are confident that improvements in GaAsP material quality and top cell performance will be demonstrated using III-V material grown by MOVPE.

The above JSC-VOC analysis is not a direct proof of a working bottom cell. To verify the existence of a working SiGe bottom cell, spectral photovoltage (SPV) measurements were made. The advantage of SPV over EQE is its higher signal to noise ratio and its 400 immunity to series resistance effects. energies for the GaAsP top cell and SiGe bottom cell, respectively. These graphs can be interpreted as formally equivalent to Tauc plots in the wavelength range covered. In the case of the GaAsP (Fig. 9.b), as a direct bandgap material, the SPV signal squared at energies close to the bandgap has been assumed to be proportional to the bandgap energy. In the case of the SiGe (Fig. 9.c), as an indirect bandgap material, it is the square root of SPV, which is proportional to Eg for energies near the bandgap energy. The values obtained from the fits agree well with the target design values as well as with the fit obtained from the EQE of the top cell.

As a concluding point for this study, the responses in the SPV signal of Fig. 9 demonstrate that a truly tandem GaAsP/SiGe solar cell has been grown on a Silicon substrate with a Si porous layer using reverse graded buffers. With this new design, clear improvements in device performance over the results obtained with devices grown on standard Si wafers have been presented. Certainly, these results are still far from demonstrating that porous silicon layers can be used to solve the problems of reverse graded buffers. However, our findings provide a few evidences in this direction and point towards strategies for further gains in performance.

Conclusions

The great impulse in the use of SiGe alloys in the silicon microelectronic industry makes them attractive candidates for the development of tandem solar cells on silicon substrates. As a matter of fact, the combination of III-V compounds with Si1-xGex alloys provides a wide palette of materials to reach optimum bandgap combinations for dual, triple and even quadruple-junction solar cells on silicon. Despite having reached efficiencies over 20%, III-V/SiGe tandem solar cells need to be grown on Si using thick buffer layers (>10m) which are expensive and eventually lead to the formation of cracks, hindering the development of large area devices. In a search to mitigate these two problems -costly thick buffers and cracking-and move forward in the integration III-V compounds on silicon for photovoltaic applications, GaAsP/SiGe tandem solar cells have been grown on porous silicon substrates through group IV reverse graded buffer layers. Reverse buffers show promise to reduce the threading dislocation density and allow the growth of thinner buffers, although they have been shown to be also prone to cracking. To mitigate this problem, in this study we have used a porous silicon layer incorporated in the substrate to increase its flexibility. In comparison with similar solar cell structures grown on standard substrates, the porous silicon layer has (i) decreased though not eliminated the number of visible cracks, (ii) increased shunt resistance of the structure, (iii) improved the top cell spectral response and (iv) improved the VOC and FF of the cells. However, the overall performance of this design still remains below similar architectures grown on forward graded buffers or on GaP/Si templates. From the results presented it cannot be unequivocally concluded that porous layers solve the problems of reverse graded buffer layers. However, we have presented evidence of partial improvements and proposed a pathway for further gains. We are confident that improvements in GaAsP material quality and cell performance will be demonstrated using III-V material grown by MOVPE.

FIGURE 1 .

 1 FIGURE 1. Structure of the GaAsP/SiGe tandem with porous silicon buffer layer, where the lattice constants are schematically represented. Layers with thicknesses and doping levels are labeled.

Fig. 3

 3 Fig.3.a shows the porous layer with a range of pore sizes, delimited by a line of large pores at the substrate interface. As discussed below, this may impact the series resistance

FIGURE 2 .

 2 FIGURE 2. Cross-sectional bright-field transmission electron microscope image of the as-grown GaAsP/SiGe wafer, with key layers identified. The layer labeled GaAsP cell includes the tunnel junction and nucleation layer. The labels on the right indicates which layers are shown in figure 3.

Figure 4

 4 Figure4shows X-ray RSMs from symmetric 004 (a) and asymmetric 224 (b) reflections of the complete structure. The different layers in the structure can be identified and their lattice mismatch and composition quantified. All the layers were found to be fully relaxed. The silicon substrate peak is clearly visible in the upper left part of the graph. The Ge buffer peak can be observed with a mismatch of 4.36%, as

FIGURE 4 .

 4 FIGURE 4. Reciprocal space maps (RSM) of the structure, including (004) reflections (a) and (224) reflections (b)

Fig 5 .

 5 Fig 5.a shows in red the experimental External Quantum Efficiency (EQE) of the top cell in the GaAsP/SiGe solar cells grown on porous Si substrates. The EQE of a solar

FIGURE 5 .

 5 FIGURE 5. Experimental EQEs of the GaAsP/SiGe tandem solar cells on porous silicon in comparison with other results from the literature. (a) Experimental EQE of the best GaAsP cell grown on porous silicon (thick red line); thin pink lines represent the EQE of other devices in the same wafer to give an idea of performance homogeneity; a linear fit to obtain the bandgap has been included as a dashed line; EQE of the best GaAsP top cell grown on standard silicon wafers (blue line) (Caño, Pablo et al. 2020); EQE of a GaAsP top cell taken from (Grassman, T. J. et al. 2016) as a benchmark (green line). (b) Experimental EQE of the SiGe subcell grown on standard silicon wafers (blue line) (Caño, Pablo et al. 2020); expected EQE of SiGe bottom cell grown on porous Si substrates estimated from previous simulations (Caño et al. 2020) and Hovel's equations (Algora and Rey-Stolle 2016).

  6.a)[START_REF] Caño | Hybrid III-V/SiGe Solar Cells Grown on Si Substrates through Reverse Graded Buffers[END_REF]; porous Si (Fig.6.b) (this work); and our reference device from

FIGURE 6 .

 6 FIGURE 6. Simulated EQEs of the GaAsP/SiGe tandem solar cells on porous silicon in comparison with other results from the literature. (a) Simulation of the EQE of the GaAsP to cells in GaAsP/SiGe tandems grown on standard Si substrates, (taken from (Caño et al. 2020)). (b) Simulation of the EQE of the GaAsP to cells in GaAsP/SiGe tandems grown on porous Si substrates (this work); (c) Simulation of the EQE of the GaAsP to cells in GaAsP/Si tandems (taken from[START_REF] Grassman | GaAs0.75P0.25/Si Dual-Junction Solar Cells Grown by MBE and MOCVD[END_REF]). Round symbols correspond to the experimental data. The thick black line is the total simulated EQE whereas the thin black line corresponds to total absorptance. The thick colored lines correspond to the contributions of each layer: orange for the window, blue for the emitter, green for the space charge region and red for the base. The thin colored lines are the absorptance in each layer with the same color code. The JSC obtained from the integration of the experimental EQE over the AM1.5G spectrum is included in each graph.

FIGURE 7 .

 7 FIGURE 7. (Top panel) Lighted I-V curve of the best GaAsP/SiGe tandem cell grown on porous silicon (thick red line); the thin pink lines represent the I-V curves of most of the devices measured and give an idea of the device-to-device variability; as reference, the illuminated I-V curve of the best GaAsP/SiGe tandem cell grown on standard silicon substrates has been included (blue line) (Caño et al. 2020). (Bottom panel) Dark I-V curves of the solar cells following the same color coding. Inset: C-V curve of a representative device obtained at 300 K under dark conditions. Labels indicate the signature of the different p-n junctions. The thin dashed line qualitatively shows the expected trend without a parasitic diode.

  FIGURE 8. (a) JSC-VOC curve of a GaAsP/SiGe tandem cell grown on a porous Si substrate and subsequent fit. (b) Reconstructed illuminated J-V from the JSC-VOC data and measured lighted curve.

FIGURE 9 .

 9 FIGURE 9. (a) Spectral photovoltage measurements of the GaAsP/SiGe tandem cells grown on porous substrates. The red curve corresponds to the bottom cell responses whereas the yellow curve is that of the GaAsP top cell. (b) Fit for obtaining the GaAsP top cell bandgap energy. (c) Fit for obtaining the SiGe bottom cell bandgap energy.

Acknowledgements

The technical assistance of Mr. Luis Cifuentes and Mr. Jesús Bautista for solar cell manufacturing and characterization are greatly appreciated. This work was supported by the Spanish Ministerio de Ciencia e Innovación through project VIGNEMALE, with grant number RTI2018-094291-B-I00.


J P Tessonnier

G Winé

C Estournès

C Leuvrey

M J Ledoux

C Pham-Huu

Carbon nanotubes as a 1D template for the synthesis of air sensitive materials: About the confinement effect

Keywords: Nanowires, Multi-walled carbon nanotubes, Confinement effect, Nanoreactor 1. Introduction 2. Experimental 3. Results and discussion 4. Conclusion References

Cobalt ferrite and cobalt iron nanowires with an average diameter of 50 nm and lengths up to several micrometers were synthesized inside multi-walled carbon nanotubes (MWNTs) under mild reaction conditions, i.e. 100 °C and atmospheric pressure, using an aqueous nitrate precursor salt filling the tubes. The concept of a confinement effect inside carbon nanotubes has been advanced to explain the formation of CoFe 2 O 4 under such mild reaction conditions. The formation of caps near the tube tips at the beginning of the nitrate decomposition meant that each nanotube was considered as a closed nanoreactor, in which the reaction conditions could be very different to the macroscopic conditions outside the tube. The subsequent reduction of the CoFe 2 O 4 allowed to obtain CoFe nanowires cast in the carbon nanotubes. These nanowires exhibit a high resistance towards oxidation, whereas bulk CoFe is known to undergo oxidation at room temperature and atmospheric pressure. This phenomenon was attributed to oxygen diffusion problems due to the confinement effect of the carbon nanotubes.

Introduction

Carbon nanostructures, nanotubes and nanofibers, the smallest organized form of carbon, have received an increasing scientific interest, from a fundamental point of view as well as for their physical properties, and for several of their potential applications ranging from living matter structure manipulation to nanometer-sized computer circuits [1], [2], [3] and [4], all this since their discovery at the beginning of the last decade by Iijima as a by-product in the arc-discharge synthesis [5]. The inner diameter of the multi-walled carbon nanotubes (MWNTs) can vary from few nanometers to several dozens depending on the nature of the synthesis method, i.e. arc-discharge, laser ablation or chemical vapor deposition [4], [6] and [7]. Both the single-walled carbon nanotubes (SWNTs) and the MWNTs exhibit an extremely high aspect ratio (length-todiameter ratio) ranging from 30 to more than many thousands [8]. It was expected that the tubular morphology and the high aspect ratio of MWNTs could induce a confinement effect on the gas or the liquids trapped inside the tube leading to completely different physical behavior when compared to conventional bulk material. Several reports recently dealt with the behavior of water trapped inside MWNTs [9], [10], [11], [12], [13], [14] and [15]. The most detailed study was reported by . They reported that water and gas trapped inside the tubes rapidly reached the supercritical state under heating because of the confinement effect induced by the surrounding wall. Nanotubes can also be used to protect sensitive products against aggressive media. Dravid [16] has reported that nickel particles introduced inside the nanotubes were not damaged even after storage for several months in an aqua regia solution.

In addition, introduction of foreign elements inside the carbon tubules can give rise to peculiar effects either on the properties of the filling element or of the filled carbon nanotube. Ajayan et al. [17] used the surface-tension effect to fill carbon nanotubes with vanadium oxide, which can be used in the catalytic oxidation process, at relatively high filling temperature, i.e. 1023 K. However, it should be stressed that the filling selectivity remained relatively low, filled and externally coated carbon nanotubes both being present in the final sample. Recently, Xu et al. [18] have reported the filling of single-walled carbon nanotubes with lanthanide halide by heating an intimate mixture of carbon nanotubes and halide above the melting point of this halide. These encapsulated material can be subsequently used in several potential domains including magnetism and catalysis fields.

Metallic single elements or alloys of iron and cobalt made up of nanoclusters are extremely useful owing to their peculiar properties compared to the bulk metal or oxide counterparts either for magnetic or catalysis applications [19], [20], [21] and [22]. However, the use of metallic nanocluster materials was hampered by the high oxidation behavior of these materials in the presence of air.

The aim of the present work is to report the use of multi-walled carbon nanotubes as a template with a confinement effect for the synthesis of CoFe 2 O 4 nanowires under mild conditions and their subsequent transformation into CoFe alloy nanowires. Thanks to the high aspect ratio of the MWNT template, the metallic alloy exhibits a relatively high resistance towards oxidation in air, allowing an easy handling and storage. This work is also motivated by the possible application of these metallic alloy nanowires directly as catalyst.

Experimental

Purified open MWNTs with mean outer and inner diameters of 100 and 60 nm, respectively, and lengths up to several micrometers (Applied Surface Science, OH, USA) were used as template. The CoFe 2 O 4 nanowires were synthesized as follows: the iron and cobalt precursors salts (14.47 g Fe(NO 3 ) 3 •9H 2 O and 4.94 g of Co(NO 3 ) 2 •6H 2 O) were first dissolved in a volume of water and ethanol (60 ml, 90% distilled water and 10% ethanol) corresponding to the inner volume of the carbon tubule, previously determined by analysis of a statistical sample of microscopy images. Carbon nanotubes (10 g) were then added without further purification. The metal loading was set to be ca. 30%. The filling of the carbon nanotubes was possible thanks to capillary forces, the surface tension of the solution (<72 mN m -1 ) being lower than the critical surface tension reported in the literature for the carbon nanotubes (190 mN m -1 ). The filled material was then dried in air in a oven (Nabertherm model L3/11/P320) at 100 °C for 2 h in order to remove the excess of water.

The calcination of the sample at 450 and 600 °C was performed in a Thermolyne 21100 furnace equipped with a tubular quartz reactor, respectively, under flowing air and flowing helium (50 cm 3 min -1 , heating rate 10 °C min -1 ), during 2 h. The reduction of the CoFe 2 O 4 into CoFe was performed in the same furnace under flowing hydrogen (50 cm 3 min -1 ) at 400 °C during 2 h (heating rate 10 °C min -1 ).

The crystallinity of the sample was characterized by powder X-ray diffraction technique (XRD) using a non-monochromatic Co Kα source. The location of the sample with respect to the carbon nanotubes and the microstructure was followed by means of transmission electron microscopy (TEM) with a Topcon 002B microscope working at 200 kV accelerating voltage with a point-topoint resolution of 0.17 nm.

Results and discussion

The drying of the filled material in air at 100 °C for 2 h induced the formation of a needle-like CoFe 2 O 4 with a short-range crystallinity (Fig. 1A). Increasing the heating temperature from 100 to 450 °C led to the modification of the nanowire microstructure, i.e. the hair-like CoFe 2 O 4 was transformed into a more crystallized material (Fig. 1B). The nanowire microstructure again drastically changed when the heating temperature was increased up to 600 °C (Fig. 1C). The CoFe 2 O 4 particles of the sample after heat treatment at 600 °C became big enough to be detected by the XRD technique. The mean particle size deduced from the Scherrer formula was ca. 35 nm, which was still relatively small owing to the reduction temperature. The details of the synthesis procedure have already been published elsewhere [23].

The formation of the CoFe 2 O 4 hair-like structure after heat treatment at 100 °C of the impregnated sample was attributed to the confinement effect of the surrounding MWNT. After 10 min of heating at 100 °C, part of the water was vaporized leading to the formation of a semi-permeable amorphous CoFe 2 O 4 solid cap at the tube tip, which in turn modifies the pressure inside the tube (Fig. 2A). The non-decomposed nitrate precursor remains trapped inside the tubes and continues to decompose under a higher local pressure leading, after 30 min at 100 °C, to the formation of a low ordered CoFe 2 O 4 nanowire with a hair-like microstructure (Fig. 2B). Fig. 2. In situ TEM images of the permeable solid cap, i.e. amorphous CoFe 2 O 4 phase, formation from the nitrate solution under electron beam heating during 10 min (A) and 30 min (B). The length of the permeable solid cap formed as a function of heating time was schematized by an arrow with increasing length. Such a pressure increase would be similar to that involved in hydrothermal synthesis of zeolites and probably enhances the rate of CoFe 2 O 4 formation from its mother nitrate solution under mild synthesis conditions, i.e. 100 °C. Mössbauer spectrum of 57 Fe nuclei was recorded at 4.2 K and confirmed the nature of the phase present in the MWNT tubule after the 30 min heating at 100 °C (Fig. 3). The formation of solid caps at the tube tips led to consider the inside of each nanotube like a closed nanoreactor. The non-decomposed nitrate precursor remaining trapped and blocked by the caps inside the tubes could see local synthesis conditions, different from those outside the tubes.

The local pressure inside the tubes, i.e. a nanoscopic pressure, could be greater than the macroscopic pressure which remained unchanged outside the tubes. The strength of MWNTs could allow this pressure increase inside the nanotubes without being destroyed. It has recently been reported by Gogotsi et al. [15] that fluids pressurized around 30 MPa at room temperature could be trapped inside MWNTs with an outer diameter of 100 nm, without having destroyed the tubes. This confinement effect, in terms of increase in the reactant partial pressure inside the tubes, has already been advanced to explain kinetically the gain in catalytic performance when using silicon carbide nanotubes as catalyst support instead of the analogue grain-based material for supporting the phase active for the hydrogen sulfide selective oxidation into elemental sulfur [24].

The TEM image of the CoFe alloy cast inside carbon nanotube after reduction of the parent CoFe 2 O 4 at 400 °C is presented in Fig. 4. The reduction temperature was deliberately kept lower than 600 °C in order to avoid any gazification of the surrounding carbon nanotubes by reaction with hydrogen, which could lead to the formation of methane and thus carburization of the CoFe alloy. Separate experiments carried out (not reported) have shown that the carburization starts at temperature of about 680 °C. The relatively low-temperature reduction allowed the complete retention of the 1D morphology of the material. A high-magnification TEM image (not shown) revealed that the alloy nanowire was made up of small particles with an average diameter of ca. 30-40 nm. Higher magnification in order to obtain further insight into the microstructure of the material was not possible due to the thickness of the material. The XRD pattern (Fig. 5) confirmed the complete reduction of the parent CoFe 2 O 4 into its corresponding metallic alloy while the nanowire morphology was completely retained during the reduction process. The average metallic particle size deduced from the X-ray line broadening was around 36 nm, which was in good agreement with that observed by TEM. It is worth noting that the XRD pattern indicates the complete absence of any other phases, i.e. single oxide or carbide, in the samples reduced at temperature up to 400 °C which was not the case for high temperature encapsulation where a significant amount of carbide was observed due to the solid-solid reaction between the encapsulated material and the carbon material or through a gas-phase reaction during the preparation step [25], [26] and [27]. The oxidative resistance of the alloy was evaluated using an in situ high-temperature X-ray technique. The sample was heated in air from room temperature to 500 °C and a XRD pattern was recorded every 50 °C in order to check the phase transformation during heating. The results obtained are presented in Fig. 6. It is worth noting that the same CoFe alloy without carbon nanotubes surrounding was completely oxidized back to its corresponding parent oxide after exposure to air at room temperature. The observed results were in good agreement with those reported in the literature about the extremely high air sensitivity of the metallic alloy nanoclusters. On the other hand, the carbon cast nanoclusters metallic alloy showed a relatively high resistance towards air oxidation as no trace of CoFe 2 O 4 was observed even after heating up to 200 °C. At temperatures higher than 200 °C, an oxidation process started to occur and a diffraction line corresponding to the CoFe 2 O 4 phase was observed in the XRD pattern. At 350 °C, the complete oxidation occurred and the XRD pattern only showed a diffraction line corresponding to the CoFe 2 O 4 phase. The results observed clearly indicate that most ferromagnetic nanoclusters casted inside the multi-walled carbon nanotubes were well protected by the surrounding graphene planes. The high oxidative resistance of the CoFe cast samples could also be attributed to the high aspect ratio of the MWNT, which hindered oxygen diffusion inside the tube under atmospheric pressure and thus prevented the complete oxidation of the cast CoFe nanowire encountered with the naked sample.

Conclusion

In summary, 1D ferromagnetic nanoclusters can be efficiently prepared using multi-walled carbon nanotubes as a controlling size template by a simple and reproducible method working at relatively low temperature. The as-synthesized sample exhibits a relatively high oxidative resistance as no oxidic peaks were detected even after heating in air up to 200 °C. Apparently, a combination of the high aspect ratio and the small free space between the surrounding carbon nanotube wall and the ferromagnetic alloy hindered oxygen diffusion inside the tube, thus avoiding the excessive oxidation of the sample as observed for its naked metallic counterpart. This example shows the exciting possibility of using such carbon nanomaterials as nanoscale reactors and opens new perspectives for preparing highly anisotropic nanomaterials for catalytic applications. Work is ongoing to check these 1D CoFe metallic alloy as catalyst and will be presented soon.

Fig. 1 .

 1 Fig. 1. In situ TEM micrographs recorded as a function of the heating temperature of the CoFe 2 O 4 cast inside MWNT: (A) 100 °C; (B) 450 °C; and (C) 600 °C.

Fig. 3 .

 3 Fig. 3. Mössbauer spectrum recorded at 4.2 K on the nanowires filling the sample of carbon nanotubes after air treatment at 100 °C. Experimental points (○), irons ions in octahedral (...) and tetrahedral (-) sites and the resulting fit ( ).

Fig. 4 .

 4 Fig. 4. The CoFe 2 O 4 sample after heat treatment at 600 °C and after reduction in flowing hydrogen at 400 °C. The contrast generated inside the reduced sample was attributed to the density change when going from CoFe 2 O 4 to CoFe, which lead to the formation of voids inside the nanowire structure.

Fig. 5 .

 5 Fig. 5. XRD pattern of the CoFe alloy reduced in flowing hydrogen at 400 °C for 2 h showing the high selectivity of the reduction process.

Fig. 6 .

 6 Fig. 6. In situ high-temperature XRD patterns of the CoFe alloy during heating in air at different temperatures. The diffraction line corresponding to the Mo element was originated from the sample holder. It is worth noting that the same sample without carbon nanotube steadily combusted back to the CoFe 2 O 4 phase after air exposure at room temperature.


Antoine Pierquin

Thomas Henneron

Nonlinear data-driven model order reduction applied to circuit-field magnetic problem

As in most of the domains in physics, finite element formulation is a very common method for electromagnetic fields computation. Since many years both proper orthogonal decomposition and empirical interpolation method are also often used in a model order reduction context. If these methods are efficients, their application is intrusive because it requires an access to the matrices and the assembly step. To avoid such an aspect, a data-driven model order reduction based on proper orthogonal decomposition with an approximation of the nonlinear terms by radial basis functions interpolation is applied to a magnetostatic problem coupled with circuit equations. The nonlinear reduced order model only needs solutions of a finite element analysis to be generated. Index Terms-nonlinear magnetic problem, model order reduction, data driven

I. INTRODUCTION

M ODEL order reduction (MOR) approaches are widely used in all the engineering sciences to decrease simulations durations of complex systems [START_REF] Hesthaven | Certified Reduced Basis Methods for Parametrized Partial Differential Equations[END_REF], [START_REF] Chinesta | Separated Representations and PGD-Based Model Reduction. Fundamentals and Applications[END_REF]. In electromagnetic fields computation, MOR methods are often applied to finite element (FE) problems using proper generalized decomposition [START_REF] Barroso | A staggered high-dimensional Proper Generalised Decomposition for coupled magneto-mechanical problems with application to MRI scanners[END_REF]- [START_REF] Henneron | Application of the Proper Generalized Decomposition to Solve MagnetoElectric Problem[END_REF], Padé expansion [START_REF] Sato | Generation of Equivalent Circuit from Finite-Element Model Using Model Order Reduction[END_REF], proper orthogonal decomposition [START_REF] Seoane | A combined reduced order-full order methodology for the solution of 3D magnetomechanical problems with application to magnetic resonance imaging scanners[END_REF], [START_REF] Paquay | Reduced Order Model for Accounting for High Frequency Effects in Power Electronic Components[END_REF], [START_REF] Schmidthausler | Reduction of linear subdomains for non-linear electro-quasistatic field simulations[END_REF],... Among these methods the proper orthogonal decomposition (POD) is probably one of the most popular. Based on the solutions (called snapshots) of the FE simulation for different values of parameters, the POD enables to approximate the solution of the FE problem in a reduced basis. Then the initial FE system is projected onto the reduced basis decreasing the number of degrees of freedom to find for new parameter values. However this method is intrusive in the sense that an access to the matrix of the FE system is needed. Moreover in the nonlinear case, POD is classically combined with empirical interpolation method (EIM) [START_REF] Sato | Adaptive Subdomain Model Order Reduction With Discrete Empirical Interpolation Method for Nonlinear Magneto-Quasi-Static Problems[END_REF], [START_REF] Henneron | Comparison of DEIM and BPIM to Speed up a POD-based Nonlinear Magnetostatic Model[END_REF] which is even more intrusive since an access to the assembly process of the nonlinear terms is required. Indeed this method consists in interpolating the nonlinear terms of the full model by calculating only some of their entries. Consequently inference based or data-driven MOR strategies [START_REF] Peherstorfer | Data-driven operator inference for nonintrusive projection-based model reduction[END_REF]- [START_REF] Pierquin | Data-Driven Model Order Reduction for Magnetostatic Problem Coupled with Circuit Equations[END_REF] are interesting because they are non intrusive and allow to obtain a reduced model only from full model solutions. Nevertheless, if datadriven strategy is well defined when the non-linear expression is clearly identified [START_REF] Peherstorfer | Data-driven operator inference for nonintrusive projection-based model reduction[END_REF], it could be more complex to deal with if the nonlinear behaviour is not known. In this article we investigate a data-driven strategy for model order reduction of an electromagnetic device simulated by FE analysis with no a priori knowledge of the nonlinear expression, this expression being treated in a more general way using an interpolation method.

We consider the Maxwell equations in magnetostatics low frequency case: displacement current density is neglected, no eddy current is considered and a current density source is Manuscript received ???; revised ???. Corresponding author: A. Pierquin (email: antoine.pierquin@univ-nantes.fr).

imposed through a winding. The common A-formulation is appplied to express magnetic induction from a vector potential. Moreover Maxwell equations are coupled with circuit equations where the magnetic flux is linked with winding's resistance, winding's current and a voltage source, introducing a time dependancy. From this problem a data-driven method is developped. The data-driven method is based on POD with an approximation of the nonlinear terms based on radial basis functions (RBF) interpolation [START_REF] Jakobsson | Rational radial basis function interpolation with applications to antenna design[END_REF]. The reduced order model (ROM) is obtained from snapshots of classical tests, at no load and in short circuit, usually used in electrical engineering to characterize an electromagnetic device. The POD is applied on the snapshots, after what the reduced matrices are identified using operator inference [START_REF] Peherstorfer | Data-driven operator inference for nonintrusive projection-based model reduction[END_REF], [START_REF] Peherstorfer | Dynamic data-driven model reduction: adapting reduced models from incomplete data[END_REF]. The key point of the proposed method is to be able to finally approximate the reduced nonlinear terms without any a priori knowledge or hypothesis about it, using RBF interpolation.

The contents of this article is as follows. Firstly the problem statement and the variationnal formulation are introduced. Secondly the model order reduction method is developped with details about proper orthogonal decomposition, operator inference and nonlinear terms interpolation. Finally in an offline/online context [START_REF] Henneron | Model-Order Reduction of Multiple-Input Non-Linear Systems Based on POD and DEI Methods[END_REF], [START_REF] Montier | Transient simulation of an electrical rotating machine achieved through model order reduction[END_REF] the ROM generated offline should be able to simulate a different case online. Consequently the reduction strategy is applied on a 3D transformer simulation with validations on different load connected to the device.

II. NONLINEAR CIRCUIT-FIELD MAGNETIC PROBLEM

Let consider the circuit-field magnetic problem on a domain Ω with a magnetic subdomain Ω m and n i windings (figure 1) such that

∇ × H = ni k=1 J k (1)
with H the magnetic field and J k the current density vector source in the k-th winding. From the conservative law ∇.B = 0, the magnetic induction B is expressed as B = ∇×A with A the vector potential. Moreover in the ferromagnetic material, the permeability µ is nonlinear depending on the magnetic field H in the constitutive law B = µ(H)H; otherwise the law is B = µ 0 H with µ 0 the vacuum permeability. In the case of A formulation, it is necessary to express H as a function of B.

Then, magnetic field is espressed as H = ν(B)B with ν(B) the reluctivity. The boundary condition B.n = 0 is imposed on ∂Ω, with n the outgoing normal vector. Consequently the Ampere's law (1) rewrites

∇ × [ν(B)∇ × A] = ni k=1 J k . (2)
The current density vector is imposed in the n i inductors by

J k = T k i k S k D k with
(t) dt + R k i k (t) = v k (t) (3)
with φ k the magnetic linkage flux, R k the winding's resistance and v k the source voltage for the k-th inductor. Applying a variationnal formulation (1) leads to [START_REF] Gyselinck | Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations: Application of the Newton-Raphson method[END_REF]

(∇ × [ν(B)∇ × A] , A ) Ω = ni k=1 (J k , A ) Ω k ⇒ (ν(B)∇ × A, ∇ × A ) Ω +(A , n × H) ∂Ω = (4) ni k=1 (J k , A ) Ω k ,
where (., .) is the dot product and A the test function. The contour integral onto ∂Ω will vanish by imposing A × n = 0 on ∂Ω to satisfy B.n = 0, with n the outgoing normal vector.

The nonlinear term can be transfered only on the right-handside such that

(ν 0 ∇ × A, ∇ × A ) Ω = ni k=1 (J k , A ) Ω k (5) -((ν(B) -ν 0 )∇ × A, ∇ × A ) Ω ,
with ν 0 the slope of ν in 0.

Adding to these expressions the circuit equations (3) in each winding where the magnetic linkage flux φ k can be expressed as a function of A by [START_REF] Piriou | A non-linear coupled 3D model for magnetic field and electric circuit equations[END_REF], [START_REF] Piriou | Finite element analysis in electromagnetic systemsaccounting for electric circuits[END_REF]

φ k = T k S k D k , A Ω , (6)
Finally after a Ritz-Galerkin discretization we obtain for linear case the system

MX(t) -FI(t) = 0, F t dX(t) dt + RI(t) = V(t), (7)
and for nonlinear case

   M X(t) -F Ĩ(t) = G( X(t)), F t d X(t) dt + R Ĩ(t) = V(t), (8)
with X ∈ R nx and X ∈ R nx the circulations of potential vector A for all edges of the mesh, I ∈ R ni and Ĩ ∈ R ni the vector of winding currents, V ∈ R ni the source voltage vector. M ∈ R nx×nx is the "curl-curl" matrix such that

M ,k = Ω ν 0 ∇ × w • ∇ × w k dΩ, (9)
with (w k ) k=1,...,nx the edge shape functions. The nonlinear vector is such that

G ( X(t)) = nx k=1 Ω (ν(B) -ν 0 )∇ × w • ∇ × w k Xk (t) dΩ, (10)
R ∈ R ni×ni is the diagonal matrix of resistor and F ∈ R nx×ni the matrix

F ,k = Ω T k S k D k • ∇ × w dΩ, (11)
such that the magnetic fluxes can be expressed by

Φ(t) = F t X(t) ∈ R ni (12)
and

Φ(t) = F t X(t) ∈ R ni , (13)
discrete forms of equation [START_REF] Henneron | Application of the Proper Generalized Decomposition to Solve MagnetoElectric Problem[END_REF].

III. NONLINEAR DATA-DRIVEN MODEL ORDER REDUCTION

Both linear and nonlinear problems ( 7)-( 8) are solved using a time stepping scheme to obtain n s solutions X(t k ), I(t k ) and X(t k ), Ĩ(t k ) for time steps t k with k = 1 to n s . The associated flux Φ(t k ) and Φ(t k ) are also computed. The solutions are stored in snapshot matrices with

S X = X(t 1 ), X(t 2 ), . . . , X(t ns ) ∈ R nx×ns (14)
S I = I(t 1 ), I(t 2 ), . . . , I(t ns ) ∈ R ni×ns (15)
S Φ = Φ(t 1 ), Φ(t 2 ), . . . , Φ(t ns ) ∈ R ni×ns (16)
S X = X(t 1 ), X(t 2 ), . . . , X(t ns ) ∈ R nx×ns (17)
S Ĩ = Ĩ(t 1 ), Ĩ(t 2 ), . . . , Ĩ(t ns ) ∈ R ni×ns (18)
S Φ = Φ(t 1 ), Φ(t 2 ), . . . , Φ(t ns ) ∈ R ni×ns (19)
The matrix R is known and the number of currents is small so the reduction will be applied only on the vectors X and X.

A. Proper orthogonal decomposition

A first reduced basis Ψ is computed using proper orthogonal decomposition (POD) from S X and a second ones Ψ from S X. The POD consists in applying a singular value decomposition (SVD) to S X and S X to obtain the decompositions

S X = U X Σ X W t X with S X = U XΣ XW t X, (20)
with U X , U X ∈ R nx×nx and W X , W X ∈ R ns×ns orthonormal matrices and Σ X , Σ X ∈ R nx×ns diagonal matrices of singular values σ X,k and σ X,k , k = 1 to n s . The projection matrices Ψ and Ψ are made of the first columns of U X and U X respectively, the number of columns being chosen most of the time using the singular values by the following energetic criterion

r X k=1 σ X,k ns k=1 σ X,k ≤ 1 -ε and r X k=1 σ X,k ns k=1 σ X,k ≤ 1 -ε, (21)
with ε a given tolerance.

Once Ψ and Ψ are computed then the final basis P ∈ R nx×nr is obtained by a third SVD on the concatenated matrix S = Ψ, Ψ ∈ R nx×r S with r s = r X + r X in order to optimize the size of the basis. Once again S = UΣW t with U ∈ R nx×nx and W ∈ R rs×rs orthonormal matrices and Σ ∈ R nx×rs a diagonal matrix of singular values σ k , k = 1 to r s . Finally P is made of the n r first columns of U, the choice of n r being based on the same criterion

nr k=1 σ k ns k=1 σ k ≤ 1 -ε. (22)
The reduced basis is used to perform projection of the solutions onto the reduced space by

s x = P t S X = x 1 , x 2 , . . . , x ns ∈ R nr×ns (23)
for the linear solutions and

s x = P t S X = x1 , x2 , . . . , xns ∈ R nr×ns (24)
for the nonlinear ones.

B. Operators inference

The first aim is to obtain the reduced matrices f ∈ R nr×ni and m ∈ R nr×nr , reduced forms of M and F by Galerkin projection, respectively

m P t MP, (25)
f P t F. ( 26
)
To do so we consider that the reduced vectors x k from s x must verify [START_REF] Peherstorfer | Data-driven operator inference for nonintrusive projection-based model reduction[END_REF], [START_REF] Peherstorfer | Dynamic data-driven model reduction: adapting reduced models from incomplete data[END_REF], [START_REF] Pierquin | Data-Driven Model Order Reduction for Magnetostatic Problem Coupled with Circuit Equations[END_REF]

f t x(t k ) = Φ(t k ) ∀k ∈ {1, . . . , n s } (27) and mx(t k ) = fI(t k ) ∀k ∈ {1, . . . , n s }, (28)
reduced expressions of [START_REF] Peherstorfer | Data-driven operator inference for nonintrusive projection-based model reduction[END_REF] and the first equation of [START_REF] Sato | Generation of Equivalent Circuit from Finite-Element Model Using Model Order Reduction[END_REF]. Equation ( 27) can be written with snapshots matrices as f t s x = S Φ and by applying transpose operator s x t f = S Φ t . Consequently a minimization problem allows to obtain the

matrix f because f k , k-th column of f , is solution of the problem min y∈R nr s x t y -c k , (29)
with c k the k-th column of S Φ t which is

c k = [φ k (t 1 ), φ k (t 2 ), . . . , φ k (t ns )] t . (30)
In the same way (28) can be written ms x = f S I or s x t m t = S I t f and so a minimization allows to obtain the matrix m by solving a problem as in (29) with c k the k-th column of S I t f . The least square problem (29) is basically solved by computing the pseudo-inverse of the matrix s x t .

C. Nonlinear terms interpolation

Now that the matrices of the reduced system are known, the snapshot matrix of the reduced nonlinear term is approximated by s g = ms x -fS Ĩ ∈ R nr×ns , containing the reduced term g of the nonlinear vector G at each time step t k , k = 1 to n s . These reduced terms will be interpolated for each component during the solving of the reduced problem to obtain an approximation of the nonlinear term depending on the vector x using radial basis functions (RBF). Consequently the reduced nonlinear term g is approximated for a reduced vector x by

g(x)       α 11 α 12 . . . α 1ns α 21 α 22 . . . α 2ns . . . α nr1 α nr2 . . . α nr,ns            r x -x 1 , a 1 r x -x 2 , a 2 . . . r x -x ns , a ns      , (31)
with α k , = 1 to n s , the interpolation coefficients associated with the k-th component of g(x) for k = 1 to n r . Function r xx , a is the RBF centered on the point x , a a hyperparameter of the function and . the 2-norm. The hyperparameter is usually chosen and imposed by the user and the same for all the functions but multiple hyperparameters are possible and their computation can be done by cross-validation [START_REF] Urquhart | Surrogate-based optimisation using adaptively scaled radial basis functions[END_REF]. The function r can classically be [START_REF] Jakobsson | Rational radial basis function interpolation with applications to antenna design[END_REF] • Gaussian r(x, a) = exp -

x 2 a 2 . • Inverse quadratic r(x, a) = 1 1 + x 2 a 2 . • Multiquadric r(x, a) = 1 + x 2 a 2 . • Inverse multiquadric r(x, a) = 1 1 + x 2 a 2 .
• Thin plate spline r(x) = x 2 ln x.

The coefficients α k are computed by solving the n s × n s system

    r ( x1 -x1 ) r ( x2 -x1 ) . . . r ( xn s -x1 ) r ( x1 -x2 ) r ( x2 -x2 ) . . . r ( xn s -x2 ) . . . r ( x1 -xn s ) r ( x2 -xn s ) . . . r ( xn s -xn s )         α k1 α k2 . . . α kns     =     g k1 g k2 . . . g kns     (32)
Once the coefficients α k are known, the interpolation is fast because it just requires to compute the distance x -x k for the n s points then the matrix-vector product (31). Always from (31), we see that n r interpolations are performed to evaluate g(x). Among the interpolation methods, interpolation based on RBF has the advantages to be used whatever the dimension of the input coordinates and does not require a mesh. Indeed the interpolation uses the euclidian distance between the interpolated points and functions with a continuous support.

D. Reduced system solving

Finally, the system to solve is

   mx(t) -f ĩ(t) = g(x(t)), f t dx(t) dt + R ĩ(t) = V(t). (33)
An implicit Euler time-scheme is applied then at each time-step a nonlinear problem is solved using a fixed-point strategy to deal with the nonlinear behaviour of the magnetic reluctivity. During the fixed-point iterative process the reduced vector g(x) is approximated by the RBF interpolation (31). It is known that fixed-point strategy is less efficient than a Newton method, however solving reduced system is very fast so the low convergence of fixed-point is not prohibitive. Moreover the RBF interpolation does not guarantee a good approximation of the derivative of the interpolated function consequently the use of the jacobian could be disadvantageous.

E. Procedure summary

Now that all the steps are detailed, the order reduction procedure can be summarised as following:

1) Solve the linear and nonlinear problems ( 7) and (8) for n s time-steps. 2) Create the snapshots matrices.

3) Solve the minimization problem (29) with c k the k-th column of Φ t . 4) Solve the minimization problem (29) with c k the k-th column of S I t f . 5) Compute the nonlinear term snapshot matrix s g = ms x -fS Ĩ. 6) Find the interpolation coefficients of the RBF by solving (32). 7) Solve the reduced system (33) on the whole timedomain.

IV. APPLICATIONS

The data-driven strategy is applied to 3D single-phase and a three-phase 50 Hz transformers simulated using finite element solver Gsmh/GetDP [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF], [START_REF] Geuzaine | GetDP: a general finite-element solver for the de Rham complex[END_REF]. Considering symetries of the device, only one quarter of the transformers is simulated. The solver used an implicit Euler scheme in time with a time-step equal to 0.25 milliseconds and a Newton strategy to solve the nonlinear problem at each time step. The simulations are done on a cluster allowing a parallel solving of the matrix systems using MUMPS solver with 8 cores. The magnetic cores of both transfomers are similar. The nonlinear magnetic law is interpolated from a set of measured points [START_REF] Frǎt | Contribution à la prise en compte des pertes fer dans la modlisation des machines électriques par éléments finis[END_REF] as shown on figure 2. Fig. 2. Nonlinear magnetic law [START_REF] Frǎt | Contribution à la prise en compte des pertes fer dans la modlisation des machines électriques par éléments finis[END_REF].

In order to obtain a reduced model valid on the whole operating range of transformers (i.e. for any load connected to the secondary windings), the solutions are computed for two typical configurations corresponding to extreme operating points: short-circuit to have a linear response; at no-load to have a strongly nonlinear response. Based on previous experimentations [START_REF] Henneron | Surrogate Model Based on the POD Combined With the RBF Interpolation of Nonlinear Magnetostatic FE Model[END_REF] the inverse multiquadric functions are used as RBF and the hyperparameter chosen is 1 σ 2 with σ the standard deviation of the distances between the vectors xk , k = 1 to n s . The different POD are done with the tolerance ε = 10 -12 in equations ( 21)- [START_REF] Piriou | Finite element analysis in electromagnetic systemsaccounting for electric circuits[END_REF].

Once the reduced model is obtained, it is simulated coupled with different loads (resistance R c or/and inductance L c ) on a different time-window then the results are compared with a reference that is the same initial system with addition of the loads in circuit equation

   M X(t) -F Ĩ(t) = G( X(t)), F t d X(t) dt + L c d Ĩ(t) dt + (R + R c ) Ĩ(t) = V(t) (34)
with L c a diagonal matrix. The aim is to highlight that once the reduced model is obtained, it can be used for different configurations.

The mesh is the same for both transformers and it is made of 7808 nodes and 41224 tetrahedrons leading to 40766 unknowns in the finite element formulation. Figure 3 presents the mesh of the magnetic core and of the windings.

A. Single-phase transformer

First the current responses are plotted figure 4 at no-load and in short-circuit. These simulations are done one the time window [0,0.06] seconds corresponding to three periods of the voltage source. The reduced order model is extracted from the solutions as explained in section III. Then the reduced order model allows to simulate the transformer coupled with a resistive load R c equal to 30 Ω on the time window [0,0.1] seconds and a second case with resistive R c and inductive L c loads equal to 30 Ω and 0.1 H respectively, currents being plotted figure 5. The reduced system has 66 unknowns and the RBF interpolation is done with 160 vectors.

To compare the currents we define a temporal norm

• t such that i(t) t = i(t 1 ) i(t 2 ) . . . i(t n ) 2 , (35)
with • 2 the Euclidian norm.

To compare the solution vectors we use the 2-norm on a matrix

X(t) 2 = X(t 1 ), X(t 2 ), . . . , X(t n ) 2 .
(36) 100.

i ref 1 -i mor 1 t i ref 1 t 100. i ref 2 -i mor 2 t i ref 2 t 100. X ref -X mor 2 X ref 2
Error (%) 0.69 0.35 0.39

The errors are compiled in tables I and II: the errors are inferior to 1% on the primary current and even inferior to 0.5% on the secondary current, and less than 1% on the solution vector. The reduced model is able to approximate correctly both global and local values. For that matter the magnetic induction are compared at time t = 9 ms corresponding to the maximum nonlinear behaviour onto the current: the fields for full and reduced models along with the difference between the fields are presented figures 6 and 7. Once again we see that the reduced order model is able to simulate the nonlinear behaviour of different load configurations and to be accurate over the local fields.

The magnetic induction B is computed from potential vector A with B = ∇×A. Vector solution X contains the circulations of A over the edges of the mesh consequently B is obtained on each volume element v by

B| v = e∈v (∇ × w e ) X e , (37)
with w e the edge shape functions associated with the edge e.

For the reduced case the formula becomes

B| v e∈v (∇ × w e ) (Px) e . (38)
To illustrate the contribution of the ROM modes we compute the contribution B (k) of a single vector basis k from P, thus defined

B (k) | v = e∈v (∇ × w e ) (P :k ) e , (39)
with P :k the k-th column of P. Figure 8 presents the first four modes of B. The first mode has a physical sense, B (1) gives the main contribution on the magnetic induction. For the next ones, the contribution is local with a maximal magnitude localised in the internal corners where the saturation of the magnetic core is strong. Table III shows simulation durations which are: 24 minutes, 36 minutes, 63 minutes and 85 minutes for the short-circuit, no-load and loaded full simulations respectively. Regarding the reduced model, less than 13 seconds are needed to read the datas, less than four seconds to compute the basis and the interpolation coefficients and about one minute fifteen seconds to solve the reduced system. Considering only the solving part, the speed-up is between 40 and 60.

B. Three phase transformer

As in previous section the current responses are plotted figure 9 at no-load and in short-circuit. Then the transformer is coupled with balanced resistive loads (R c = 30 Ω) on the time window [0,0.12] seconds then with unbalanced resistive loads (R c = 30 Ω but R c = 10 Ω on secondary winding), the currents being available figure 10. The reduced system has 166 unknowns and the RBF interpolation is done with 240 vectors.

The errors are compiled in tables IV and V: the errors are inferior to 10% on the primary current and even inferior to 5% on the secondary current, and less than 2% on the solution vector. The reduced model is able to approximate correctly both global and local values. Table VI shows simulation durations which are: 24 minutes, 48 minutes, 86 minutes and 88 minutes for the short-circuit, no-load and loaded full simulations respectively. Regarding the reduced model, less than 13 seconds are needed to read the datas, less than four seconds to compute the basis and the interpolation coefficients and about thirty seconds to solve the reduced system. Considering only the solving part, the speedup is here between 130 and 180. V. CONCLUSION

In order to reduce simulation duration for the modeling of a physical phenomena, two main families of methods exist: model order reduction and metamodel. However most of the reduced order model are intrusives notably in the nonlinear case: on acces to the matrix of the system or to the assembly process is required. To avoid this problem metamodels can be used instead of reduced order model but they are not based on physical formulations so their usage is less adapted in a offline/online strategy or for a multiphysic simulation. On the opposite the data-driven philosophy is to create a reduced model only from the solutions of a problem called snapshots, based on the knowledge of the physical and mathematical formulation. From the solutions a reduced basis is obtained using proper orthogonal decomposition allowing to project the full solutions onto a reduced space. Then through least square problems and radial basis function interpolations a precise and fast reduced order model can be simulated even for a configuration not used for the snapshots. The method is described for magnetostatic problems with a nonlinear Fig. 10. Three-phase transformer: primary currents obtained with full and reduced models with balanced resistive loads (top) and unbalanced resistive loads (bottom). magnetic law coupled with time-dependant electric equations. From both application examples, the proposed approach is efficient to simulate fastly a configuration that is not used for the snapshots. In order to improve the solving of the nonlinear reduced order model, development of an approach allowing to use a Newton method could be considered.

Fig. 1 .

 1 Fig. 1. Domain and subdomains of the problem.

  T k the number of turns, i k the current, S k the equivalent surface and D k the normalised current direction for the k-th stranded inductor. The magnetic flux allows to couple the local equation (1) with circuit equation by adding the n i equalities dφ k

Fig. 3 .

 3 Fig. 3. Mesh of the one-phase (top) and three-phase (bottom) transformers.

Fig. 4 .

 4 Fig. 4. Single-phase transformer: currents obtained with full and reduced models for no-load and short-circuit cases.

Fig. 5 .

 5 Fig. 5. Single-phase transformer: currents obtained with full and reduced models with resistive load (top) and resistive+inductive load (bottom).

Fig. 6 .

 6 Fig.6. Single-phase transformer: magnetic induction B (T) obtained with full (Bref) and reduced (Bmor) models with resistive load, and the error between the fields (bottom) at time t = 9 ms.

Fig. 7 .

 7 Fig. 7. Single-phase transformer: magnetic induction B (T) obtained with full (Bref) and reduced (Bmor) models with resistive and inductive loads, and the error between the fields (bottom) at time t = 9 ms.

Fig. 8 .

 8 Fig. 8. Single-phase transformer: contribution of the 4 first modes (i.e. the 4 first columns) of the reduced basis P .

Fig. 9 .

 9 Fig.9. Three-phase transformer: currents obtained with full and reduced models for no-load and short-circuit cases.

TABLE I SINGLE

 I -PHASE TRANSFORMER: ERRORS BETWEEN REDUCED AND FULL SOLUTIONS FOR RESISTIVE LOAD SIMULATION.

	100.	i ref 1 -i mor 1 i ref 1 t	t	100.	i ref 2 -i mor 2 2 t i ref	t	100.
	Error (%)	1.10			0.42		0.51

X ref -X mor 2 X ref 2

TABLE II SINGLE

 II -PHASE TRANSFORMER: ERRORS BETWEEN REDUCED AND FULL SOLUTIONS FOR RESISTIVE-INDUCTIVE LOAD SIMULATION.

TABLE III SINGLE

 III -PHASE TRANSFORMER: COMPUTATION DURATIONS FOR THE DIFFERENT SIMULATIONS.

	Simulation	Duration
	Full model short-circuit	24 min
	Full model no-load	36 min
	Full model resistive load	63 min
	Full model resistive-inductive load	85 min
	ROM resistive load	1 min 27 s
	ROM model resistive-inductive load 1 min 30 s

TABLE IV THREE

 IV -PHASE TRANSFORMER: ERRORS BETWEEN REDUCED AND FULL SOLUTIONS FOR BALANCED RESISITIVE LOAD SIMULATION.

				Error (%)			Error (%)
	100. 100. 100. 100.	i ref 11 -i mor 11 t i ref 11 t i ref 21 mor 21 t i ref 21 t i ref 31 -i mor 31 t i ref 31 t X ref -X mor X ref 2	2	7.94 2.45 2.52 1.53	100. 100. 100.	i ref 12 -i mor 12 t i ref 12 t i ref 22 -i mor 22 t i ref 22 t i ref 32 -i mor 32 t i ref 32 t	0.43 0.21 0.56

TABLE V THREE

 V -PHASE TRANSFORMER: ERRORS BETWEEN REDUCED AND FULL SOLUTIONS FOR UNBALANCED RESISTIVE LOAD SIMULATION.

			Error (%)			Error (%)
	100. 100. 100.	i ref 11 -i mor 11 t i ref 11 t i ref 21 -i mor 21 t i ref 21 t i ref 31 -i mor 31 t i ref 31 t	3.03 2.50 2.02	100. 100. 100.	i ref 12 -i mor 12 t i ref 12 t i ref 22 -i mor 22 t i ref 22 t i ref 32 -i mor 32 t i ref 32 t	0.35 0.18 0.49
	100.

X ref -X mor 2 X ref 2 1.71

ACKNOWLEDGEMENT

Antoine Pierquin would like to thank the CCIPL (Nantes, France) for providing access to its high-performance computing facilities.