T Riberi -Béridot

M G Tsoutsouva

G Regula

G Reinhart

I Périchaud

J Baruchel

N Mangelinck-Noël

Growth undercooling in multi-crystalline pure silicon and in silicon containing light impurities (C and O)

Keywords: A1. Directional solidification, A1. Characterization, A1. Growth laws, A1. Impurities, A2. Growth from melt, B2. Semiconducting silicon

Undercooling during the solidification of silicon is an essential parameter that plays a major role in grain nucleation and growth. In this study, the undercooling of the solid-liquid interface during growth of multi-crystalline silicon samples is measured for two types of silicon: pure, and containing light elements (carbon and oxygen) to assess and compare their impact on crystal growth. The solid-liquid interface undercooling is measured using in situ and real time X-ray synchrotron imaging during solidification. As a subsequent step, ex situ Electron Backscattered Diffraction (EBSD) is performed to obtain information about the crystalline structure, the grain orientation and the grain boundary character. Two main conclusions arise: i) the undercooling of the global solid-liquid front increases linearly with the growth rate which indicates uniform attachment, i.e. all atoms are equivalent, ii) the same trend is observed for pure silicon and silicon containing carbon and oxygen. Indeed, the growth law obtained is comparable in both cases, which suggests that the solutal effect is negligible as concern the undercooling in the case of a contamination with carbon (C) and oxygen (O). However, there is a clear effect of the impurity presence on the crystalline structure and grain boundary type distribution. Many grains nucleate during growth in samples containing C and O, which suggests the presence of precipitates on which grain nucleation is favored.

Introduction

Photovoltaic industry needs to produce photovoltaic material faster with always lower cost.

The classical fabrication process of silicon ingots is based on directional solidification. Solidliquid transformation under non-thermodynamic equilibrium implies issues to be tackled such as impurity segregation, grain nucleation and competition dynamics. Moreover, it is essential to develop predictive and quantitative simulations of the silicon solidification processes implying knowledge of the specific silicon growth mechanisms. As such, theoretical, experimental and modeling studies have been carried out in the past 50 years [START_REF] Voronkov | Supercooling at the face developing on a rounded crystallographic front[END_REF][START_REF] Beatty | Monte Carlo modeling of silicon crystal growth[END_REF][START_REF] Fujiwara | In situ observation of Si faceted dendrite growth from low-degree-of-undercooling melts[END_REF][START_REF] Miller | Some remarks on the undercooling of the Si(1 1 1) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF]. In particular, growth laws need to be determined to obtain reliable simulations of the growing solid-liquid interface. Theoretical growth laws have been proposed but experimental validation although difficult is strongly needed.

Growth laws are classically related to the undercooling above the solid-liquid interface.

Liquid undercooling is the phenomenon by which a material remains liquid at a temperature below the equilibrium phase diagram melting point. Undercooling is usually present at the level of the growing interface from the melt in metallurgy or crystal growth due to nonequilibrium conditions during solidification. Growth undercooling has to be distinguished from nucleation Sundercooling necessary for new grains to appear. The total undercooling of the growing solid-liquid interface can be written using the following theoretical equation which separates the main contributions to undercooling and that can be found for example in [START_REF] Herlach | Non-equilibrium solidification of undercooled metallic melts[END_REF]: [START_REF] Voronkov | Supercooling at the face developing on a rounded crystallographic front[END_REF] where , and are respectively the thermal, the constitutional, the curvature and the kinetic undercoolings:

-, the thermal undercooling, is due to the rejection of latent heat for the solid-liquid phase transformation and thus at the interface. The effect of latent heat rejection can be to slow down solidification. The dimensionless thermal undercooling can be expressed by [START_REF] Flemings | Solidification Processing[END_REF]:

(

) 2
where ΔT is the undercooling, C S is the specific heat of the solid material and H is the heat of fusion.

-, the constitutional undercooling is an important undercooling component in the case of alloys. Indeed, solutes with partition coefficient lower than 1 are rejected in the liquid at the solid-liquid interface, which changes, locally, the thermodynamic equilibrium. The rejected solute accumulates in a limited layer neighboring to the interface, the thickness of this layer can be expressed as [START_REF] Garandet | Croissance Cristalline en phase liquide: éléments théoriques[END_REF] :

(3)
where D L is the solute diffusion coefficient and the growth velocity.

The solute concentration in the liquid, for planar front, C L is given by [START_REF] Tandjaoui | Real time observation of the directional solidification of multicrystalline silicon: X-ray imaging characterization[END_REF]:

where C 0 is the nominal composition and k is the partition coefficient.

The constitutional undercooling is given by the difference between the equilibrium liquidus temperature T eq (z) and the temperature imposed by the temperature gradient T m (z).

(

) (6) 5
where T 0 is the pure silicon equilibrium melting temperature, is the liquidus slope and G L the liquidus temperature gradient.

-, The curvature undercooling is due to the Gibbs Thomson effect that implies that a smaller radius of curvature reduces melting temperature of the crystal. This contribution to the undercooling has been expressed by Herlach [START_REF] Herlach | Non-equilibrium solidification of undercooled metallic melts[END_REF] : [START_REF] Tandjaoui | Real time observation of the directional solidification of multicrystalline silicon: X-ray imaging characterization[END_REF] where is the anisotropy parameter of the interface energy, is the angle between the normal to the interface and the growth direction, the radius of curvature of the solid and  the capillary constant (Gibbs-Thomson parameter) , being the interface energy and the entropy of fusion.

-, The kinetic undercooling depends on the crystallographic orientation and on the atom kinetic attachment mechanism. This term is, in theory, predominant for the undercooling of a pure silicon macroscopically flat growing interface. The growth rate is usually related to the kinetic undercooling by an equation of the type [START_REF] Flemings | Solidification Processing[END_REF]: [START_REF] Tandjaoui | Twinning occurrence and grain competition in multicrystalline silicon during solidification[END_REF] where the function f depends on the atomic attachment mechanism as developed in the following.

 For a uniform growth regime (all sites at the interface are favorable for growth)

or for a rough interface, growth follows a linear law: [START_REF] Usami | Implementation of faceted dendrite growth on floating cast method to realize high-quality multicrsytalline Si ingot for solar cells[END_REF] where is a constant which takes into account the probability of atoms to be incorporated at the interface, lattice geometric factors and thermodynamic latent heat of solidification.

 For a growth regime by bi-dimensional nucleation on a facet (lateral growth), the growth law is exponential: [START_REF] Yang | Effect of nitrogen-oxygen complex on electrical properties of Czochralski silicon[END_REF] with A and B constants. B depends on lattice geometric factors and on the nucleation rate, A depends on a geometric factor and on the surface tension at the interface.

 For a growth regime on a facet and facilitated by the presence of screw dislocations, growth is permitted at lower undercoolings than for bidimensional growth:

where is a constant which depends on diffuseness parameter, on lattice geometric factor and on the density of dislocations.

The kinetic undercooling is thus linked to the morphology of the interface and to associated defects. A faceted morphology is associated with a high kinetic undercooling whereas a rough morphology is associated with a low kinetic undercooling because the atom attachment is easier on such surface.

Extensive theoretical, experimental and modelling works have been performed to predict the interface undercooling and in particular for silicon. Voronkov [START_REF] Voronkov | Supercooling at the face developing on a rounded crystallographic front[END_REF] conducted a theoretical study on {111} facet kinetic in silicon and derived the undercooling of silicon facets with and without dislocations. For , the undercooling at a {111} facet in a monocrystal with dislocations was estimated to be , whereas the undercooling in the case excluding the presence of dislocations was estimated to be .

Beatty and Jackson [START_REF] Beatty | Monte Carlo modeling of silicon crystal growth[END_REF] conducted Monte-Carlo simulations of the undercooling for two growth orientations <100> and <111>. They found which, for Czochralski growth rates in the [100] direction of silicon around gives .

Fujiwara et al. [START_REF] Fujiwara | In situ observation of Si faceted dendrite growth from low-degree-of-undercooling melts[END_REF] determined a relationship between and by conducting silicon growth experiments observed by scanning confocal microscopy. The melting temperature, T m , was determined when the interface was motionless. Then, the crystal was melted at a temperature lower than T m , so that, the crystal could grow in the undercooled melt. Several undercooling levels were imposed and for each of them, the solid-liquid interface growth rate was measured. Thus, they determined a relationship of the solid-liquid interface growth velocity of Si crystal as a function of the undercooling :

. For the establishment of this relationship, there was no control of the seed orientation on the contrary to the study of dendritic growth in the follow-on of the article. This relationship can be considered as the relationship applying to general silicon solid-liquid interface undercooling.

In the present paper, we determine experimentally the undercooling of the solid-liquid interface, in directional solidification conditions, as a function of the growth rate for both pure silicon and silicon containing carbon and oxygen using X-ray radiography. The in situ experiments are completed by ex situ EBSD to further characterize the influence of carbon and oxygen on the grain structure.

Experimental methodology

The experiments of directional solidification are carried out in a dedicated Bridgman-type high temperature furnace named GaTSBI (Growth at high Temperature observed by X-ray

Synchrotron Beam Imaging), installed on beamline BM05 at the European Synchrotron

Radiation Facility (ESRF). This furnace is essentially made of two resistive heating elements and the whole device is adapted to synchrotron imaging, which allows real time and in situ imaging during the solidification and growth of silicon. A detailed description of the device can be found in [START_REF] Tandjaoui | Real time observation of the directional solidification of multicrystalline silicon: X-ray imaging characterization[END_REF].

The experiments are performed on a sample of pure silicon (6N), and on a sample containing carbon and oxygen. C and O impurities constitute a case study because they are light impurities often introduced during the classical fabrication processes of silicon for PV applications. The sample size is fixed at (length width), and 300 µm in thickness. It is placed in a boron nitride crucible, which is inserted into the furnace. Then, this assembly is introduced in the two-zone furnace.

During growth, the sample is illuminated by a polychromatic (white) X-ray synchrotron beam. To perform X-ray radiography imaging, the beam is monochromated at 17.5 keV by a double Si (111) monochromator after passing through the sample, and the images are recorded on a FReLoN CCD camera. X-ray radiography imaging is based on differences of absorption of the X-ray beam by the solid and liquid phases. In the case of silicon, the density difference between solid and liquid is small ( , ), so that image post processing is necessary [START_REF] Tandjaoui | Twinning occurrence and grain competition in multicrystalline silicon during solidification[END_REF]. With this technique, we can follow the evolution of the solid-liquid interface during growth in situ and in real-time, with a large field of view (10 mm 6 mm), spatial resolution with a pixel size: 5.8 μm 5.8 μm and acquisition rate of 1

Hz. Calculations of the solid-liquid interface growth rates and its dynamic position are directly measured on the images.

Preparation of the initial solid-liquid interface

The sample/crucible assembly is initially positioned in the top zone, where the heating resistance is set to a temperature above the silicon melting point (Fig. 1.a) so that the sample is fully melted. Then a first solidification is initiated in the undercooled liquid inducing dendritic growth and few random orientations as proposed by Usami et al. [START_REF] Usami | Implementation of faceted dendrite growth on floating cast method to realize high-quality multicrsytalline Si ingot for solar cells[END_REF]. In a second step, the sample is partially melted to keep a small solid region acting as a seed at the bottom before starting controlled solidification.

Determination of the undercooling

Two types of solidification procedures are performed to measure the temperature gradient, G, and the shift towards the equilibrium position of the growing interface, h, which altogether allow determining the undercooling, T. First of all, the temperature gradient in the sample, G, is calculated during an experiment of controlled cooling (Fig. 1a). An initial temperature gradient is applied on the heaters. The solidification is triggered by applying a cooling rate on both heaters. The same cooling rate is applied on both heaters to maintain a constant temperature gradient. During cooling down experiments, the sample does not move. We measure the position of the interface which moves up during growth and determine the interface growth rate, . The corresponding temperature gradient is calculated knowing the applied cooling rate R and the measured growth rate. We earlier checked that the thermal conditions are reproducible when the same temperature gradient is imposed on both heaters for different experiments. Finally, the temperature gradient G inside the sample is given by the equation: [START_REF] Durand | Carbon solubility in solid and liquid Silicon-A review with reference to eutectic equilibrium[END_REF].

The same constant temperature gradient is applied during all the experiments studied here and as a direct consequence; the temperature gradient in the sample is constant as well. The temperature gradient inside the sample calculated with equation ( 13) is given by for all the experiments described in the following.

On the other hand, to obtain the undercooling of the solid-liquid interface, we measure the shift in position of the growth front compared to the equilibrium position. It can only be measured during pulling experiments (Fig. 1 b). In that case, the sample is pulled down from the hot zone to the cold zone at a constant pulling rate inside a constant temperature gradient.

In other words, the temperature of both resistances is maintained constant and the sample moves inside the furnace. The interface does not move in the camera frame of reference, because it is moving upwards in the sample frame of reference.

During pulling of the sample, if there is undercooling during growth, the interface is not at the position corresponding to the equilibrium temperature isotherm. If this is the case, when the pulling of the sample is stopped, the interface rises until it reaches its thermodynamic equilibrium position, which corresponds to the isotherm position of the equilibrium melting temperature. In our experiments, we check that this position does not depend on the preceding applied pulling rate which is expected because it corresponds to the phase diagram equilibrium position without any undercooling. For each experiment, we compare the stationary position during growth at a constant pulling rate with the position of thermodynamic equilibrium, as shown schematically in figure 2. The solid-liquid interface position when pulling the sample is always lower than its equilibrium position.

Fig. 2. a) Stationary position of the interface during growth for a given pulling rate, b)

Interface at its thermodynamic equilibrium position.

Then, the undercooling is obtained by multiplying the height difference between those two positions by the temperature gradient G:

(14).
This procedure is repeated several times with different sample pulling rates. Four pulling rates are applied (5, 10, 20 and 30 µm/s) on both studied samples together with a constant temperature gradient in the sample of 15 K/cm determined with the procedure described above.

Ex situ investigation

Additionally, after the last solidification, we carry out ex situ electron back-scattering diffraction (EBSD) to obtain the grain orientations in the three space directions, the twin relationships and the number of grains. In this work, Σ3

Results

Figure 3 shows the position of the interface, measured at several points along the interface for each pulling rate, during growth (in red), and after stabilization (in blue), for both samples:

pure silicon (Fig. 3 a) and silicon containing impurities (Fig. 3 b). The solid-liquid interface shape differs between both samples. This is due to the local thermal field for each particular sample.

The first most important result is that the equilibrium position of the interface after having stopped the pulling is reproducible for the same sample, confirming that we measure the equilibrium melting isotherm position. The small difference however observed is due to the unavoidable variations of thermal field between experiments. This negligible difference confirms that the thermal conditions are not significantly modified from one experiment to the other for a constant temperature gradient applied to the same sample.

Second, the growth interface has a lower position than the equilibrium melting position.

For each pulling rate, the difference between both curves increases with the pulling rate, thus, the undercooling increases with the velocity indicating immediately a kinetic effect. The undercooling is calculated with equation 14 and the growth rate is measured by using the in situ images. Finally, the growth rate as a function of the undercooling is plotted on Figure 4. Points corresponding to pure and silicon containing impurities are respectively in green and purple. The trend line was determined assuming that zero velocity implies no undercooling, which is a reasonable assumption according to the theory and to the fact that with no pulling rate applied, the interface remains at the same equilibrium position in all experiments. A linear law is justified by the alignment of the experimental points. We obtain, for pure silicon: [START_REF] Beaudhuin | Carbon reaction with levitated silicon-Experimental and thermodynamic approaches[END_REF] The error is calculated as:

(16)

where d(Δh) is the measurement error related to the pixel resolution. dG is calculated by the difference between the temperature gradient calculated for the maximal and the minimal measured growth rate during the experiment of solidification by controlled cooling in a constant applied temperature gradient. This evaluation of the error on the temperature gradient is certainly giving the highest limit of the measurement error.

The same experimental protocol is repeated with the sample voluntarily contaminated with C and O. The growth rate as a function of the undercooling shows a linear dependence similarly to pure silicon. The corresponding linear adjustment leads to the following equation:

(17)

Discussion

The undercooling during the solidification of pure silicon was also studied by Fujiwara [START_REF] Fujiwara | In situ observation of Si faceted dendrite growth from low-degree-of-undercooling melts[END_REF] as mentioned in the introduction. Our results are in agreement with Fujiwara et al. results despite that our conditions and experimental procedure are different. Indeed, the law they derived from their experiments is also linear and is given by: (18).

On top of the fact that both laws are linear, the coefficients are of the same order of magnitude. This allows us to conclude that boron doping induced by the boron nitride crucible in our experiments, has no effect on undercooling. This agreement also shows that there exists at the global solid-liquid interface of silicon a growth rate dependent undercooling. The linear behavior is consistent with the existence of a kinetic component of the undercooling.

In fact, in our experiments, the kinetic undercooling is the main component that contributes to the undercooling. Indeed, the other undercooling components are all negligible as shown in the following.

As concern the dimensionless thermal undercooling, in the case of silicon, , and equation 2 gives : which is much lower than 1 and thus it can be neglected.

Additionally, the directional solidification configuration with G > 0 ensures a morphologically stable and smooth interface. The radius of curvature approaches infinity, which means that the curvature undercooling tends toward zero and thus this term is also negligible.

At last, the constitutional undercooling component does not intervene in the pure silicon material therefore, only the kinetic component come into play.

However, in the sample containing C and O, a contribution of the constitutional undercooling could be expected due to the presence of impurities. Calculations are made to estimate the constitutional undercooling and its extension using equations (3-7) and phase diagram data using Thermo-Calc software. Even for the highest velocity, constitutional undercooling on top of the solid-liquid interface is negligible. In fact, this is consistent with the measured undercoolings in the sample containing C and O (Fig. 4). Indeed, we obtain a linear velocity dependence and the same kinetic coefficient, taking into account the error bars, as for the pure sample despite the presence of C and O, which suggests that the carbon and the oxygen do not participate to the undercooling build-up at the front.

However, although the undercooling is not modified by the presence of these impurities, the interface morphology during solidification is significantly different in both samples.

In both cases, we observe an increasing disturbance of the solid-liquid interface with increasing growth rate. Grain boundary grooves formed at the encounter between the solidliquid interface and grain boundaries can be observed and have been thoroughly studied in our previous work [START_REF] Tandjaoui | Investigation of grain boundary grooves at the solid-liquid interface during directional solidification of multi-crystalline silicon: in situ characterization by X-ray imaging[END_REF]. They reveal the multi-crystalline nature of the sample. Their occurrence is more important in the case of the sample containing impurities even for low growth rates whereas the initial stage is identical which suggests copious nucleation events during growth when C and O impurities are present.

This conclusion is confirmed by the EBSD measurements performed on the samples after the last solidification. Indeed, for both samples, the total number of grains is determined from the EBSD results. For the pure sample and for the sample containing C and O with the same processing conditions, 1221 and 2065 grains are counted respectively. Thus, more grains nucleate during the growth of the sample containing C and O.

As mentioned above, the concentrations in carbon and oxygen in this sample are measured by IGA: and . Additionally, FTIR measurements are performed. The concentration in substitutional carbon is:

and the concentration in interstitial oxygen is: . The comparison between the total C and O concentrations and the substitutional C and insterstitial O concentrations indicates that part of the carbon is in precipitate form which can be expected, due to the very low solubility limit of carbon in silicon [START_REF] Durand | Carbon solubility in solid and liquid Silicon-A review with reference to eutectic equilibrium[END_REF] . As a consequence, these impurities are involved in the formation of precipitates in the material, precipitates on which new grain nucleation is favored which is consistent with the above results. Moreover, data obtained for different processing conditions on several samples during our previous experiments also confirmed that there is a significant increase of the nucleation rate during growth in samples with C & O impurities compared to pure samples even for the concentrations considered here. There are less Σ3, Σ9 and Σ27a twin boundaries in the silicon sample containing impurities than in the pure silicon sample, which is in favor of random nucleation.

As a conclusion, C and O do not accumulate on top of the solid-liquid interface which explains their low impact on the establishment of the undercooling at the level of the solidliquid interface. Indeed, these impurities soon precipitate as the solubility limit is reached especially in the case of C [START_REF] Durand | Carbon solubility in solid and liquid Silicon-A review with reference to eutectic equilibrium[END_REF]. In the subsequent step, the driving force for new grain nucleation is the decrease of the necessary nucleation undercooling due to the presence of precipitates. Beaudhuin et al. [START_REF] Beaudhuin | Carbon reaction with levitated silicon-Experimental and thermodynamic approaches[END_REF] observed as well that the addition of carbon creates silicon carbide precipitates which favor nucleation. In pure samples, the formation of new grains is driven by the twinning phenomenon which leads to a majority of twin boundaries in the final ingot grain structure [START_REF] Riberi-Béridot | On the impact of twinning on the formation of the grain structure of multi-crystalline silicon for photovoltaic applications during directional solidification[END_REF]. When precipitates are at stake, no special relationship between the new grain and other grains is expected leading to an increase of random angle grain boundaries.

Conclusion

This work allowed determining the undercooling of the global solid-liquid interface of directionally solidified silicon for growth parameters similar to the ones used in the fabrication of silicon for photovoltaic applications. Under these conditions, the undercooling in silicon is always present and is dominated by its kinetic component. The corresponding law is linear and consistent with uniform attachment or attachment on a rough interface. The presence of grain boundary grooves and grain nucleation during growth does not modify significantly this law. Light impurities like C and O do not affect the undercooling of the global solid-liquid interface but greatly affect the final grain structure.

Fig. 1 .

 1 Fig. 1. Solidification procedures in the GaTSBI furnace: a) Solidification by cooling down of the heaters, b) Solidification by pulling of the sample inside the furnace.

Fig. 3 .

 3 Fig. 3. Measured position and shape of the solid-liquid interface at different pulling rates (V p ) for a) the pure silicon sample and b) the silicon sample containing C and O. Red curves

Fig. 4 .

 4 Fig. 4. Growth rate as a function of the undercooling. Points corresponding to pure silicon and silicon containing C and O respectively in green circles and in purple triangles.

Figure 5 Fig. 5 .

 55 Figure 5 shows the misorientation angle distribution for the two samples. The first zone

Fig. 1 .Fig. 2 .

 12 Fig. 1. Solidification procedures in the GaTSBI furnace: a) Solidification by cooling down of the heaters, b) Solidification by pulling of the sample inside the furnace.

Fig. 3 .

 3 Fig. 3. Measured position and shape of the solid-liquid interface at different pulling rates (V p ) for a) the pure silicon sample and b) the silicon sample containing C and O. Red curves correspond to the interface position during growth. Blue curves correspond to the interface at its thermodynamic equilibrium position. Error bars are smaller than the point size.

Fig. 4 .

 4 Fig. 4. Growth rate as a function of the undercooling. Points corresponding to pure silicon and silicon containing C and O respectively in green circles and in purple triangles.

Fig. 5 .

 5 Fig. 5. Misorientation angle distribution obtained by EBSD (pure silicon: green, silicon containing C and O: purple). Four particular misorientation regions are highlighted: low angle grain boundaries (LAGB), Σ27a, Σ9 and Σ3 twins.

Fig. 1 .Fig. 2 .Figure 2 Fig. 3 .Fig. 4 .

 12234 Figure Captions

Figure 4 Fig. 5 .

 45 Figure 4

Figure 5

 5 Figure 5

Acknowledgments

This work was funded by the ANR Projects SiX (N° ANR-08-HABISOL-012-01) and CrySaLID (N° ANR-14-CE05-0046-01). The ESRF (European Synchrotron Radiation Facility) BM05 team is greatly acknowledged for support during the X-ray imaging experiments. Laurent Barrallier and Fabrice Guittonneau are acknowledged for the EBSD measurements. Gildas Guillemot and Charles-André Gandin (Centre de Mise en Forme des Matériaux) are greatly acknowledged for the Thermo-Calc software data.


V Stamelou

M G Tsoutsouva

T Riberi-Béridot

G Reinhart

G Regula

J Baruchel

N Mangelinck-Noël

{111} facet growth laws and grain competition during silicon crystallization

Keywords: A1. Directional solidification, A1. Characterization, A1. Growth laws, A1. {111} facets, A2. Growth from melt, B2. Semiconducting silicon

Directional solidification from mono-crystalline Si seeds having different orientations along the growth direction is studied. Due to the frequent twinning phenomenon, new grains soon nucleate during growth. The grain competition is then characterized in situ by imaging the dynamic evolution of the grain boundaries and of the corresponding grain boundary grooves that are formed at the solid-liquid interface. To perform this study, an experimental investigation based on Bridgman solidification technique coupled with in situ X-ray imaging is conducted in an original device: GaTSBI (Growth at high Temperature observed by X-ray Synchrotron Beam Imaging). Imaging characterisation techniques using X-ray synchrotron radiation at ESRF (European Synchrotron Radiation Facility, Grenoble, France) are applied during the solidification to study the growth dynamics. Facetted / facetted grain boundary grooves only are studied due to their importance in the grain competition because of their implication in the twinning mechanism. The maximum undercooling inside the groove is calculated from the groove depth knowing the local temperature gradient. Additionally, thanks to dynamic X-ray images, the global solid-liquid interface growth rate and the normal growth rate of the {111} facets existing at the grooves and at the edges are measured. From these measurements, experimental growth laws that correlate the normal velocity of the {111} facets with the maximum undercooling of the groove are extracted and compared to existing theoretical models. Finally, the experimental laws found for the contribution to the undercooling of the {111} facets are in good agreement with the theoretical model implying nucleation and growth eased by the presence of dislocations. Moreover, it is shown that, for the same growth parameters, the undercooling at the level of the facets (always lower than 1 K) is higher at the edges so that there is a higher probability of twin nucleation at the edges which is in agreement with the grain structure development characterised in the present experiments as well as in the literature.

Introduction

The acute need for alternative sources of energy to fossil fuels is widely accepted but it is not free from economic constraints, and brings new technological and scientific challenges. In the last decades, renewable energies and more precisely photovoltaics (PV) have emerged as a source of more environmentally friendly source of energy production.

Crystalline silicon (c-Si)-based photovoltaic cell technology is well-established and currently represents about 90% of the world PV market but faces regularly new challenges due to the constantly renewed demand for higher performance together with a decrease of the cost. Among the main steps of Si material fabrication, the central one is the solidification from melted silicon. Indeed, grain structure, impurities and structural defects (the major ones being dislocations) induced during this crystal growth process have a major impact on the final PV efficiency. However, the involved fundamental mechanisms and growth laws that define the final grain structure are still not fully established preventing an efficient and reproducible control and modelling of the silicon crystallization [START_REF] Duffar | Comprehensive review on grain and twin structures in bulk photovoltaic silicon[END_REF].

In fact, the grain structure in multi-crystalline silicon largely depends on a series of phenomena related to faceting/undercooling of the solid-liquid interface during growth as for example twinning [START_REF] Gallien | Analysis of grain orientationin cold crucible continuous casting of photovoltaic Si[END_REF][START_REF] Trempa | Mono-crystalline growth in directional solidification of silicon with different orientation and splitting of seed crystals[END_REF][START_REF] Duffar | On the twinning occurence in bulk semiconductor crystal growth[END_REF].

Facets at the solid-liquid interface can be found at the edges of the samples or inside the grain boundary grooves formed at the intersection of a grain boundary with the solid-liquid interface [START_REF] Tandjaoui | Investigation of grain boundary grooves at the solid-liquid interface during directional solidification of multi-crystalline silicon: in situ characterization by X-ray imaging[END_REF]. In the case of crystalline silicon, all the crystallographic orientations are rough except the {111} planes which have a facetted morphology, as confirmed by the calculation of the Jackson criterion for the main crystallographic orientations [START_REF] Jackson | Crystal Growth Kinetics[END_REF]. The presence of {111} facets is thus an essential feature of silicon grain growth and competition. Indeed, frequent twinning nucleation take place at the level of the {111} facets in the grain boundary grooves or at the edges of the samples [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF][START_REF] Oliveira | Formation mechanism and properties of twinned structures in (111) seeded directionally solidified solar grade silicon[END_REF].

Grain boundary grooves can have one of the three following configurations, depending on the crystallographic orientation of the adjacent grains [START_REF] Tandjaoui | Investigation of grain boundary grooves at the solid-liquid interface during directional solidification of multi-crystalline silicon: in situ characterization by X-ray imaging[END_REF][START_REF] Duffar | The grain-grain-liquid triple phase line during solidification of multicrystalline silicon[END_REF]: i) rough solid-liquid interface on both sides of the groove ii) rough/facetted groove and iii) facetted/facetted groove, when the grain boundary groove consists of two facets. It is essential to study the groove dynamics as it is directly linked to the grain competition [START_REF] Nadri | Two-dimensional numerical modeling of grain structure in multi-crystalline silicon ingot[END_REF] because of its interdependence with the grain boundary orientation relatively to the solidified direction [START_REF] Riberi-Béridot | On the impact of twinning on the formation of the grain structure of multi-crystalline silicon for photovoltaic applications during directional solidification[END_REF].

In general, the normal growth velocity at the solid-liquid interface can be expressed as a function of the undercooling and is typically expressed by: [START_REF] Duffar | Comprehensive review on grain and twin structures in bulk photovoltaic silicon[END_REF] where β(T) is the kinetic coefficient and ΔΤ the kinetic undercooling. The kinetic coefficient can be a function of the undercooling and is different for different types of growth.

Indeed, the incorporation of a structural unit at the interface is not always favoured. The increase of the interfacial energy needs to be compensated by a gain in volume enthalpy. The key parameter is the geometric configuration of the system and the number of atomic neighbours. The incorporation is all the more difficult as the interface is smooth as on {111} facets [START_REF] Garandet | Croissance Cristalline en phase liquide: éléments théoriques[END_REF]. In the case of a facetted {111} Si solid-liquid interface, two theoretical laws are proposed in the literature.

First, bi-dimensional nucleation can be considered. In that situation, the proposed relation is given for example by the following equation [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF]: in [START_REF] Gallien | Analysis of grain orientationin cold crucible continuous casting of photovoltaic Si[END_REF].

Depending on the authors, the constants can differ but this does not change significantly the resulting law and typical groove undercooling values [START_REF] Jackson ; Kirk | Response to: Some remarks on the undercooling of the Si(1 1 1) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF]. The result of this approach is that a high undercooling (several degrees) is needed in the case of growth on {111} silicon facets (solid black line Figure 1) which is not what is observed during experiments [START_REF] Miyahara | Effect of twin growth on unidirectional solidification control of multicrystal silicon for solar cells[END_REF].

Fig. 1:

Theoretical growth law curves relating the facet velocity V f to the undercooling T at the level of a {111} facet. Bi-dimensional law from Miller [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF] in solid black line, quadratic law from Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] in dotted black line.

Another mechanism implying defect presence was thus theoretically proposed. If the crystal is not perfect, for example due to the presence of a screw dislocation emerging at the solid-liquid interface, the smooth facetted interface is no longer perfectly flat but shows a step [START_REF] Flemings | Solidification Processing[END_REF]. The presence of steps implies that the facet can grow at lower undercooling compared to the case of bi-dimensional growth. When atoms are added, the step persists and for a single screw dislocation, the result is a spiral. The growth rate as a function of the undercooling is then controlled by a quadratic law and can be generally written as:

(

where K is a constant parameter dependent on the dislocation density.

Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] also explained theoretically and confirmed experimentally how a single dislocation can significantly modify the facet size and undercooling. Finally, from theoretical considerations and comparison to Czochralski Si fabrication process experiments, he proposed a value for the kinetic coefficient in equation ( 3) giving the following law: in m.s -1 (4).

Importantly, growth laws from equations ( 2) and ( 4) give values of undercooling that differ by one order of magnitude as can be seen figure 1.

Abe [START_REF] Abe | The growth of Si single crystals from the melt and impurity incorporation mechanisms[END_REF] confirmed experimentally for Czochralski growth that dislocations intersecting the {111} facets in silicon decrease the growth undercooling. Indeed, this author showed that, in his experiments for pulling rates between 1.3 x 10 -5 and 2.5 x 10 -5 m.s -1 , the facet undercooling in Si crystals containing dislocations and dislocation-free <111> Si crystals is respectively T = 7 x 10 -1 K and T = 9 K. These values are in agreement with the above considerations as concern the order of magnitude when comparing dislocation free Si and Si crystals containing dislocations. However, a value of K between 2.7 x 10 -5 and 5.1 x 10 -5 m.s - 1 .K -2 is obtained with equation (3) for the experimental data provided by Abe [START_REF] Abe | The growth of Si single crystals from the melt and impurity incorporation mechanisms[END_REF]. This value is one order of magnitude smaller than the coefficient proposed by Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF].

As seen above, theoretical models of the {111} facet growth laws exist but there is a striking lack of experimental validation in particular for directional solidification used widely for the fabrication of multi-crystalline ingots and for which process modelling is needed.

In the work presented here, our main objective has been to determine the contribution to the undercooling of the {111} facets in both grain boundary grooves and at sample edges during directional solidification of silicon using X-ray imaging, to derive corresponding {111} facet growth laws and then to compare them with theoretical growth models reported in the literature.

Experimental Procedure

GaTSBI The polychromatic synchrotron radiation illuminates the Si sample during the whole crystal growth process. In the Xray radiography mode used here, after passing through the sample, the polychromatic beam is monochromated at 17.5 keV using a vertically diffracting Si (111) double-crystal monochromator. Experiments were carried out at beamline BM05 at the European Synchrotron Radiation Facility (ESRF). The Xray radiography images are recorded using a specific CCD camera developed at the ESRF named FReLoN (Fast Readout Low-Noise) with 2048×2048 image pixel size. In the experiments described in the following, an optics giving 5.8 μm pixel size and 11.9 × 11.9 mm 2 field of view was used.

The obtained X-ray radiography images are absorption contrast images due to the differences in absorption of the solid and liquid phases. However, the absorption coefficients of the solid and liquid silicon depend on the corresponding mass densities whose values are close to each other (ρ s = 2.31 g/cm 3 and ρ l = 2.56 g/cm 3 ) which means that the solid-liquid interface in the raw images is hardly distinguishable. In order to observe the shape of the interface more accurately (rough or facetted for instance), image processing which consists in dividing successive images, i.e. dividing the current image by the previous image [START_REF] Tandjaoui | Real time observation of the directional solidification of multicrystalline silicon: X-ray imaging characterization[END_REF], to enhance the solid-liquid interface contrast is performed. After the image processing is done, images of the solid-liquid interface during the solidification are obtained in which the denser liquid and the solid appear in dark and light grey respectively.

The X-ray radiography mode allows extracting information concerning the interface morphology, its dynamic evolution, to calculate its velocity and to study the grooves between two grains formed by the intersection of the grain boundary separating them with the solidliquid interface. A more detailed description of the whole equipment, of the imaging techniques, of the image processing and of the previously obtained results can be found elsewhere [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF][START_REF] Riberi-Béridot | On the impact of twinning on the formation of the grain structure of multi-crystalline silicon for photovoltaic applications during directional solidification[END_REF].

In experiments described in the following, a constant temperature gradient, G APP = 37.5 K/cm, is imposed between the two heating resistances. Three different Si samples obtained by the floating zone process (SIL'TRONIX intrinsic monocrystalline silicon with a resistivity higher than 5000  cm) are solidified. These samples are cut from the same Si <110> wafer in order to study different orientations along the growth direction. To ease the reading, the three samples are referred as FZ <100> , FZ <111> and FZ <110> with respectively <100>, <111> and <110> seed crystallographic orientations parallel to the growth direction. Initially, the samples are partially melted to be able to initiate growth from the seed then, the crystal is solidified first taking up the initial orientations of the seed. Directional solidification of the remaining liquid is obtained by applying a cooling rate R = 1 K/min on both heaters. Several melting / solidification cycles are carried out with each sample to obtain a better statistic on the measurements.

Experimental results

As a first step, the average solid-liquid interface velocity is calculated from the radiograph successive images for the three different samples submitted to the same applied processing conditions (temperature gradient and cooling rate). In order to do so, the height of the interface is measured in different snapshots and the progress of the interface from one snapshot to another is calculated. It is important to notice that as the solid-liquid interface is not perfectly flat, to improve the accuracy, measurement are carried on at the same point of the solid-liquid interface and some averaging is performed. As the precise time difference between the images is known, the solid-liquid interface growth rate (V g ) can be calculated.

The results of these measurements for the three different samples are depicted in figure 2. The FZ <100> (red point), FZ <111> (green point), and FZ <110> (blue point) samples have respectively <100>, <111> and <110> seed crystallographic orientation parallel to the growth direction. Bars are given by the standard deviation of the growth rate distribution. Processing conditions: applied temperature gradient G AP = 37.5 K/cm, cooling rate R = 1 K/min.

As can be seen in the graph of figure 2 showing also the standard deviation of the distribution, the growth rate of the solid-liquid interface of the three samples is comparable for the same processing conditions ensuring reproducibility of our experiments as was also verified in previous experiments.

Second, in order to determine {111} facet growth laws, the grain boundary grooves observed at the solid-liquid interface have been geometrically characterized by their angle and depth as can be seen in figure 3 in the case of an ideal facetted/facetted groove [START_REF] Dash | Dynamics of Faceted Grain Boundary Grooves[END_REF].

Fig. 3: Sketch of a facetted/facetted grain boundary groove at the solid-liquid interface with its characteristic parameters [START_REF] Dash | Dynamics of Faceted Grain Boundary Grooves[END_REF]: groove angle (), normal facet growth rate (V f ), groove depth (h), solidliquid interface growth rate (V g ).

These geometrical parameters of the grain boundary grooves are measured directly on the radiography images collected during solidification. Moreover, the kinetic parameters which are the growth velocity of the interface (V g ) and the normal growth velocity of the groove facets (V f ) are also measured thanks to in situ and real-time X-ray imaging of the solid-liquid interface. It is worth noting that in all experiments, the crystallographic orientation of the facets was checked after the experiments using EBSD measurements. Facets observed in our experiments are always {111} facets as expected in silicon [START_REF] Jackson | Crystal Growth Kinetics[END_REF].

The evolution during solidification of a typical facetted / facetted grain boundary groove observed at the solid-liquid interface is shown in figure 4. In figure 4, we follow a grain boundary groove at the solid-liquid interface (Figure 4a).

The groove gets deeper (Figure 4b-d) while keeping the same angle during growth and at last, there is a nucleation inside the groove (Figure 4e). This sequence is typical for the majority of the grooves detected in our samples.

From the depth measurement, the thermal undercooling inside the grain boundary groove can be calculated knowing the local temperature gradient.

First of all, the local temperature gradient is determined experimentally making the assumption that the cooling rate is the same in the whole sample and that it corresponds to the imposed cooling rate applied to the heaters. So, the local temperature gradient is calculated with the following equation:

(

) 5
where R (K/min) is the cooling rate imposed to the heaters and V g is the measured growth rate of the solid-liquid interface.

Then, the undercooling ΔΤ (Κ) inside the groove is determined by the following equation: [START_REF] Jackson | Crystal Growth Kinetics[END_REF] where h is the depth of the groove. As the groove depth is evolving during growth, we calculate the maximum undercooling inside the grooves using equation ( 6) taking the grain boundary groove maximum depth for h in the equation.

Additionally, from the radiograph images obtained during our experiments, the normal velocity of the {111} facets in the grooves is measured.

Eventually, the results of the mean facet velocity measurements as a function of the undercooling, inside the grain boundary grooves, can be seen in figure 5. Polynomial fits are also plotted with the reasonable hypothesis that the facet velocity is equal to zero when there is no undercooling. The black dotted curve in this graph represents the Voronkov quadratic law (equation 4) and the bi-dimensional exponential law (equation 2) is shown in solid black line. Finally, the adjustment of a quadratic growth law (equation 3) to our experimental points results in the following equations for the contribution of the facets to the undercooling:

. FZ <100> : V f (m/s) = 2.8 x 10 -4 ΔT 2 (K),

. FZ <110> : V f (m/s) = 1.0 x 10 -4 ΔT 2 (K).

If all experimental points are considered together, the resulting quadratic law obtained is:

V f (m/s) = 1.2 x 10 -4 ΔT 2 (K). . The black dotted curve corresponds to the Voronkow law [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the black solid line curve to the bi-dimensional law [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

Apart from the formation of grooves at the solid-liquid interface, facets at the edges of the samples are also frequently observed as shown in figure 6. Indeed, {111} facets are generally existing at the edges of the solid-liquid interface in the ingots and lead to the occurrence of twinning ultimately competing with the central grain growing from the seed [START_REF] Trempa | Mono-crystalline growth in directional solidification of silicon with different orientation and splitting of seed crystals[END_REF][START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF]. As a consequence, the normal velocity has also been measured at the level of edge facets and points are gathered as a function of the undercooling in figure 7. The projection effect due to the radiography method was corrected to obtain the normal velocity V f that we want to measure.

Fig. 7:

Mean facet normal velocity at the sample edges for FZ <100> (red points and adjusted curve), FZ <111> (green points and adjusted curve), FZ <110> (dark blue points and adjusted curve) samples as a function of the maximum undercooling. Error bars are calculated by . The black dotted curve corresponds to the Voronkow law [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the solid black line to the bi-dimensional law [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

For {111} facets at the edges, the experimental laws are given by:

. FZ <100> : V f (m/s) = 2.0 x 10 -5 ΔT 2 (K),

. FZ <111> : V f (m/s) = 3.0 x 10 -5 ΔT 2 (K),

. FZ <110> : V f (m/s) = 3.0 x 10 -5 ΔT 2 (K).

If all experimental points are considered together, the resulting quadratic law obtained is:

V f (m/s) = 3.0 x 10 -5 ΔT 2 (K).

For the three seed growth orientations, the kinetic coefficient of the quadratic growth law for {111} facets at the edges is one order of magnitude smaller than the one found for {111} facets in a groove.

Discussion

From the observation of the solid-liquid interface during solidification by X-ray radiography of three samples with initial seeds oriented <100>, <111> and <110> along the growth direction, we can first conclude that the solid-liquid interface is morphologically stable for our processing conditions.

As can be seen on figure 2, the mean global interface growth rate depends solely on the processing conditions applied, that are identical for the three samples, and this proves that thermal conditions are reproducible and well controlled.

Locally, grooves are observed at the solid-liquid interface due to the presence of grain boundaries in the sample and highlight the existence of several grains even though the initial seed is monocrystalline. This is due to twin nucleation inducing subsequent grain competition as already discussed in our previous work [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF].

From the measurement of the global interface growth rate, we can also conclude that as expected, the pristine orientation of the seed does not affect the mean velocity of the solidliquid interface during directional solidification.

Moreover, when comparing the growth rate of the global solid-liquid interface (Figure 2)

for the three samples to the {111} facet growth rates (Figures 5 and7), it appears that the growth rates of the {111} facets either inside the grooves (Figure 5: V f = 7.2 x 10 -6 m/s in average, standard deviation = 1.9 x 10 -6 m/s) or at the edges (Figure 7: V f = 8.8 x 10 -6 m/s in average, standard deviation = 2.2 x 10 -6 m/s) are in any case smaller than the one of the global solid-liquid interface (Figure 2: V g = 1.5 x 10 -5 m/s in average, standard deviation = 6.2 x 10 -7 m/s). This can be expected because of the slower kinetics of the {111} planes. Indeed, the {111} facets have a slower kinetics than the other orientations so that they are lagging behind them. Consequently, they appear below the global solid-liquid interface on the images.

Recently, we showed that there exists undercooling of the global solid-liquid interface of a few kelvins for the processing parameters used in our experiments [START_REF] Riberi-Béridot | Growth undercooling in multi-crystalline pure silicon and in silicon containing light impurities (C and O)[END_REF]. Thus, the measured undercooling inside the grain boundary groove or at the level of the edge facets is an additional component to the undercooling. A major consequence is that the undercooling is higher in the groove and at the level of the edge facets compared to the one at the level of the global solid-liquid interface. This favours nucleation events inside grooves and at the edge facets that are indeed often observed in real time during our experiments.

Inside a facetted / facetted groove as can be seen for example in figure 4, until nucleation can take place, both facets grow at the same growth rate as can be concluded from the constant angle and orientation of the groove in all observed facetted/facetted grain boundary grooves studied. This conclusion is also sustained by the X-ray radiography images contrast as can be seen in figure 4d and in our previous work [START_REF] Tandjaoui | Investigation of grain boundary grooves at the solid-liquid interface during directional solidification of multi-crystalline silicon: in situ characterization by X-ray imaging[END_REF]. Indeed, in figure 4d, the white areas on the facets evidence the new grown solid between two successive images because of the image processing performed. These white regions have the same thickness on both facets; this is due to the concomitant growth at the same rate of both facets seen in figure 4d. Inside a particular facetted / facetted grain boundary groove, the growth kinetics is thus equivalent on both facets. Moreover, according to the images, it seems that growth is homogeneous on the facets along the depth of the groove even though the undercooling should not be constant from the bottom to the top of the grain boundary groove. Moreover, inside the grain boundary grooves, the measured maximum undercooling is always lower than 1 K ranging from 1 x 10 -1 to 4 x 10 -1 K.

At the level of the edge facets, the measured maximum undercooling is also always lower than 1 K as for facets inside grain boundary grooves. However, higher values (ranging from 2

x 10 -1 to 8 x 10 -1 K) compared to the undercooling inside grain boundary grooves (ranging from 1 x 10 -1 to 4 x 10 -1 K) are measured at the edges. We consider that this result is significant because the same evolution is obtained for several samples and for both grooves at the edges independently from possible sample particularities.

In solidification experiments, the solid-liquid interface is generally not perfectly planar because of isotherm shape which in our experiments is slightly convex. Due to the latter, a greater undercooling of the facets observed at the edges can indeed be expected in comparison with the one of the facets in the grain boundary grooves. Part of the undercooling of the facets at the sample edges close to the crucible walls can be attributed to the Gibbs-Thomson effect at the corner of the solid-liquid interface [START_REF] Trempa | Mono-crystalline growth in directional solidification of silicon with different orientation and splitting of seed crystals[END_REF]. The Gibbs-Thomson effect in crystal growth refers to the fact that the melting point of a crystal is decreased with decreasing radius of curvature of the interface. Additionally, the interface with vacuum at the edges modifies the equilibrium shape balance adding vacuum to the solid-liquid-crucible equilibrium and might induce an increase of the facet height at the edges of the sample [START_REF] Epure | Comparison between analytical and numeric determination of the interface curvature during dewetted Bridgman crystal growth[END_REF].

The higher undercooling measured at the level of the facet at the edges has a significant impact on the grain structure obtained at the end of the solidification by increasing the nucleation probability during growth. This is clearly confirmed by the grain structure obtained in the samples at the end of the experiments (Figure 6) for which twin nucleation is frequent at the far edges of the facets. This major contribution of twins nucleating on edge facets to grain competition and final grain structure was previously reported [START_REF] Trempa | Mono-crystalline growth in directional solidification of silicon with different orientation and splitting of seed crystals[END_REF][START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF].

At last, growth rate was also measured at the level of grain boundary grooves and edge facets. According to the large difference of the order of magnitude of the characteristic undercooling obtained for the corresponding {111} facet growth velocity, bi-dimensional nucleation growth mechanism can be excluded. Indeed, as seen above, undercooling inside the grain boundary grooves and at the level of edge facets is always lower than 1K. As a consequence, the experimental results concerning {111} facets kinetics in our experiments can only be compared to the theoretical law corresponding to a growth mechanism eased by the presence of dislocations (Figures 5 and7). This is in agreement with the fact that dislocations are expected to be easily generated during silicon growth and found emerging at the level of facets as shown for example in [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF][START_REF] Oliveira | Formation mechanism and properties of twinned structures in (111) seeded directionally solidified solar grade silicon[END_REF][START_REF] Ervik | Dislocation formation in seeds for quasi-monocrystalline silicon for solar cells[END_REF]. On the one hand, the kinetic quadratic law coefficient obtained in the case of grain boundary grooves is consistent with the one obtained by Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF]. On the other hand, the coefficient obtained for the quadratic law at the edges is one order of magnitude lower than the one determined for the grain boundary grooves which is a direct consequence of the higher undercooling measured at the edges for the same growth rate. It is worth noting that the obtained quadratic law coefficient is then very comparable to what was measured by Abe [START_REF] Abe | The growth of Si single crystals from the melt and impurity incorporation mechanisms[END_REF].

However, the most important result remains that the undercooling either inside grain boundary grooves or at the level of facet edges is in any case lower than 1 K.

Conclusion

In situ radiography imaging was successfully used to characterise the {111} facets kinetics during seeded silicon growth. As a first step, from the successive image videos created for each sample during solidification, the growth velocity of the global solid-liquid interface is measured. For the same process conditions, the velocity is identical disregarding of the sample seed orientation from which we conclude that the orientation of the starting seed does not play any role on the establishment of the general solid-liquid interface growth rate and that the thermal conditions are reproducible for identical process parameters applied.

The angle and depth of the grain boundary grooves are measured directly and the velocity of the facets that form the groove is calculated from the successive images recorded during solidification. The maximum undercooling for each groove is calculated from the groove maximum depth and calculated local temperature gradient. The first main result is that the facet undercooling value is in all cases limited to less than one Kelvin. Moreover, from these measurements, an experimental law that correlates the normal velocity of the {111} facets with the contribution to the undercooling of the groove is extracted. This experimental law is adjusted to a quadratic law with the reasonable assumption of a zero value for the growth rate in the case of no undercooling. The quadratic law has been chosen because it is the only theoretical law available leading to undercooling with an order of magnitude corresponding to the one measured in our experiments i.e. lower than 1 K. Indeed, the other available theoretical law developed for {111} facet growth, namely bi-dimensional growth law, cannot be compared successfully with the experimental results. As quadratic theoretical laws were derived for the situation of nucleation and growth eased by the presence of dislocations, we can conclude that we are most likely in this situation in our experiments. This can be expected because dislocations, twins or stacking faults are always present in PV cast silicon and in our samples as observed regularly and previously in our experiments [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of <110> seeded growth Si[END_REF]. More precisely, quadratic growth law coefficient for {111} facets inside grain boundary grooves can be compared successfully to the value determined by Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] in the case of Czocralski process for {111} facets containing dislocations.

During Si growth, facets at the edges of the samples are also observed in the radiography images. At the level of the edge facets, the angle that they form with the edge of the crucible, the depth and the {111} facet normal velocity is measured as well. An experimental law correlating the facet normal velocity with the contribution to the undercooling at the edges is extracted in this case too. As for the {111} facets in the grooves, the measured undercooling value (lower than 1 K) can only correspond to theoretical laws derived for the situation of nucleation and growth facilitated by the emergence of dislocations on facets. However, the results show that the experimental undercooling at the edges is higher than inside the grooves.

The origin of this difference is attributed to the effect of the shape of the interface and to complex equilibrium balance of surface energies at the edges including vacuum and crucible.

As a consequence, the coefficient of the quadratic growth law extracted for facets at the edges is lower by an order of magnitude compared to the one measured inside the grain boundary grooves. The major impact of this measured higher undercooling although always lower than 1 K is that a higher nucleation probability is expected at the edges. This is in perfect agreement with the repeated nucleation of twins observed at the edges of the sample during the experiments.

Fig. 1: Theoretical growth law curves relating the facet velocity V f to the undercooling T at the level of a {111} facet. Bi-dimensional law from Miller [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF] in solid black line, quadratic law from Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] in dotted black line.

Fig. 2:

Mean growth rate (V g ) of the solid-liquid interface for the three different samples analysed. The FZ <100> (red point), FZ <111> (green point), and FZ <110> (blue point) samples have respectively <100>, <111> and <110> seed crystallographic orientation parallel to the growth direction. Bars are given by the standard deviation of the growth rate distribution. Processing conditions: applied temperature gradient G AP = 37.5 K/cm, cooling rate R = 1 K/min. Fig. 3: Sketch of a facetted/facetted grain boundary groove at the solid-liquid interface with its characteristic parameters [START_REF] Dash | Dynamics of Faceted Grain Boundary Grooves[END_REF]: groove angle (), normal facet growth rate (V f ), groove depth (h), solid-liquid interface growth rate (V g ).

Fig. 4: Image sequence showing the evolution of a typical facetted/facetted groove at the solid-liquid interface during directional solidification of the FZ <100> Si sample. Applied temperature gradient: G APP = 37.5 K/cm, cooling rate R = 1 K/min at t 0 , a) t 0 + 352 s, b) t 0 + 355 s, c) t 0 + 358 s, d) t 0 + 361 s, e) t 0 + 364 s.

Fig. 5: Mean facet normal velocity (V f ) in grain boundary grooves for FZ <100> (red points and adjusted curve) and FZ <110> (blue points and adjusted curve) samples as a function of the maximum undercooling. Error bars are calculated by . The black dotted curve corresponds to the Voronkow law [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the black solid line curve to the bidimensional law [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF]. . The black dotted curve corresponds to the Voronkow law [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the solid black line to the bi-dimensional law [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF]. Fig. 1: Theoretical growth law curves relating the facet velocity V f to the undercooling T at the level of a {111} facet. Bi-dimensional law from Miller [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF] in solid black line, quadratic law from Voronkov [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] in dotted black line.

T (K)

Fig. 2: Mean growth rate (V g ) of the solid-liquid interface for the three different samples analysed. The FZ <100> (red point), FZ <111> (green point), and FZ <110> (blue point) samples have respectively <100>, <111> and <110> seed crystallographic orientation parallel to the growth direction. Bars are given by the standard deviation of the growth rate distribution. Processing conditions: applied temperature gradient G AP = 37.5 K/cm, cooling rate R = 1 K/min. Vg (m/s)

Fig. 3: Sketch of a facetted/facetted grain boundary groove at the solid-liquid interface with its characteristic parameters [START_REF] Dash | Dynamics of Faceted Grain Boundary Grooves[END_REF]: groove angle (), normal facet growth rate (V f ), groove depth (h), solid-liquid interface growth rate (V g ). . The black dotted curve corresponds to the Voronkow law [START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the solid black line to the bi-dimensional law [START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

Fig. 2 :

 2 Fig. 2: Mean growth rate (V g ) of the solid-liquid interface for the three different samples analysed.

Fig. 4 :

 4 Fig. 4: Image sequence showing the evolution of a typical facetted/facetted groove at the solidliquid interface during directional solidification of the FZ <100> Si sample. Applied temperature gradient: G APP = 37.5 K/cm, cooling rate R = 1 K/min at t 0 , a) t 0 + 352 s, b) t 0 + 355 s, c) t 0 + 358 s, d) t 0 + 361 s, e) t 0 + 364 s.

Fig. 5 :

 5 Fig. 5: Mean facet normal velocity (V f ) in grain boundary grooves for FZ <100> (red points and adjusted curve) and FZ <110> (blue points and adjusted curve) samples as a function of the maximum undercooling. Error bars are calculated by. The black dotted curve corresponds to the Voronkow law[START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the black solid line curve to the bi-dimensional law[START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

Fig. 6 :

 6 Fig. 6: FZ <100> Si sample: a) Crystallographic orientation of the grains along the growth direction obtained by EBSD after sample solidification (Inverse pole figure map). b) Radiograph image at the level of the solid-liquid interface showing {111} edge facets.

Fig. 6 :

 6 Fig. 6: FZ <100> Si sample: a) Crystallographic orientation of the grains along the growth direction obtained by EBSD after sample solidification (Inverse pole figure map). b) Radiograph image at the level of the solid-liquid interface showing {111} edge facets.

Fig. 7 :

 7 Fig. 7: Mean facet normal velocity at the sample edges for FZ <100> (red points and adjusted curve), FZ <111> (green points and adjusted curve), FZ <110> (dark blue points and adjusted curve) samples as a function of the maximum undercooling. Error bars are calculated by. The black dotted curve corresponds to the Voronkow law[START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the solid black line to the bi-dimensional law[START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

  Figure Captions

Fig. 4 :Figure 4 Fig. 5 :Figure 5 Fig. 6 :

 44556 Fig.4: Image sequence showing the evolution of a typical facetted/facetted groove at the solid-liquid interface during directional solidification of the FZ <100> Si sample. Applied temperature gradient: G APP = 37.5 K/cm, cooling rate R = 1 K/min at t 0 , a) t 0 + 352 s, b) t 0 + 355 s, c) t 0 + 358 s, d) t 0 + 361 s, e) t 0 + 364 s.

Fig. 7 :

 7 Fig. 7: Mean facet normal velocity at the sample edges for FZ <100> (red points and adjusted curve), FZ <111> (green points and adjusted curve), FZ <110> (dark blue points and adjusted curve) samples as a function of the maximum undercooling. Error bars are calculated by. The black dotted curve corresponds to the Voronkow law[START_REF] Voronkov | Supercooling at the facet developing on a a rounded crystalization front[END_REF] and the solid black line to the bi-dimensional law[START_REF] Miller | Some remarks on the undercooling of the Si(111) facet and the "Monte Carlo modeling of silicon crystal growth[END_REF].

5 Figure 7

 57 Figure 7

Acknowledgments

This work was funded by the ANR Project CrySaLID (N° ANR-14-CE05-0046-01). The ESRF (European Synchrotron Radiation Facility) BM05 team is acknowledged for support during the X-ray imaging experiments. Laurent Barrallier and Fabrice Guittonneau are greatly acknowledged for the EBSD measurements.


Youcef A Bioud

Abderraouf Boucherif

Etienne Paradis

Ali Soltani

Dominique Drouin

Richard Arès
email: richard.arès@usherbrooke.ca

Low cost Ge/Si virtual substrate through dislocation trapping by nanovoids

A low-cost method to reduce the threading dislocations density (TDD) in relaxed germanium (Ge) epilayers grown on silicon (Si) substrates is presented. Ge/Si substrate was treated with post epitaxial process to create a region with a high density of nanovoids in Ge layer which act as a barrier for threading dislocations propagation.

Introduction

Multi-junction III-V concentrator cells have the most important energy conversion efficiency, competing with any other solar cell technology [1]. Germanium appears to be the material of choice for the bottom cell to cover the IR part of the solar spectrum. A high efficiency of 40.7% was measured for a metamorphic three junctions GaInP/GaInAs/Ge cell at 240 suns [2]. This architecture has achieved system efficiency of 30% [3]. A road map based on the integration of Ge on Si wafers can be a solution to replace bulk Ge substrates and would result in a significant cost reduction. However, due to the important 4.2% lattice mismatch between Si and Ge, strain is introduced in epitaxial Ge layers on Si which leads to the generation of a high density of misfit dislocations. Dislocations act as recombination centers for carriers and strongly degrade the solar cell performance. Several techniques of dislocation reduction were reported in recent years such as graded SiGe buffer layers, cyclic annealing, aspect-ratio-trapping (ART), selective area depositions (SAD) and deep substrate patterning (DSP) [4]. However, the treading dislocations density (TDD) in the Ge layer remains higher than 10 6 cm -1 , reducing its usefulness for many device applications. In this talk, we will discuss a new strategy for making cost effective Ge/Si virtual substrate which is based on a simple and cost effective technique that uses nanovoids as a barrier for dislocation propagation. Then, we will present detailed optimization of the porosification and the annealing of Ge on Si and study the effect of different process parameters on TDD and on surface roughness.

Results and Discussion

Approach

As sketched in Fig 1, Ge/Si substrate was treated with post epitaxial process to create a region with a high density of nanovoids within the Ge layer on Si is expected to act as a barrier for threading dislocations propagation. Nanovoids are formed by electrochemical porosification followed by thermal annealing. The process is completed by CMP (chemical mechanical planarization) step to obtain an epi-ready virtual substrate.

Electrochemical etching of dislocated Ge/Si layer

The porous morphologies obtained from the electrochemical etching of Ge/Si layer are quite different from that of the bulk Ge. For bulk Ge, the layer structure shows uniformly distributed mesopores with controllable nanostructure and size [5], [8]. However, for dislocated Ge/Si, non-uniformly distributed mesopores are obtained (see Fig 2a and4b). The porous layer consists of three zones: (i) a thin PGe layer formed at the surface, (ii) a thick PGe layer with weakly branched pores that trend to follow threading dislocation cores and (iii) porous Si layer obtained by HF crossing dislocations. Calibration of etching rates has been done by varying the current density in the electrochemical process. Fig 2b illustrates the dependence of the etching rate on the current density of bulk and dislocated Ge. The etch rate ratio between dislocated and bulk regions shows very high values up to 6. The preferential etching via dislocations can be explained by their high electrical conduction and their role in the transport properties with the electrolyte.

Nanovoid layer formation

The morphological change of porous Ge during annealing is based on thermally activated jumps of surface atoms to stable positions following to the mass transport mechanism, for which atoms have a maximum of covalent bonds until their outer energy level is full. In terms of geometry, atoms diffuse from the larger curvature surface toward the lower curvature surface down a gradient potential [9]. Therefore, the pores in the volume are enlarged and the pores on the surface are closed. This reveals the formation of a crystalline region in the top surface and a voided region in the volume (as shown in Fig. 4d).

Depending on annealing temperature (400-700°C), random pores obtained after electrochemical etching become spherical voids. The nearest small voids are then coalesced to form larger voids, as shown in Fig 3 . Keeping activation energy, surface atoms of spherical voids jump to form faced voids that are bounded by low-energy facets. The measured angle between the two prominent orientations is in perfect agreement with the angle value of ~54° between the lowest surface energy (100) and (111) oriented facets in crystalline Ge [10], [11].

Dislocation-void interactions

Fig. 4 shows STEM images obtained at the different steps of our process. A dislocation-free region is observed at the upper part of the voided region. Fig. 5 revealed that threading dislocations were terminated at the nanovoid area. Similar effect was observed by Myers et al. he suggests that this overlap is due to an attractive force created by a population of nanovoids with a binding energy that depends upon the surface tension of the void and the relative positions of the interacting dislocation [12]. Also, during thermal annealing, TDs were stopped moving at the void probably because their gliding distance to the voids side-walls were very short [13], [14]. Therefore, the formation of surface-free regions located between the top surface of Ge and Ge/Si interface could be an effective tool to reduce significantly the TDD in Ge/Si layer. The TD free top layers offer a unique potential and a technologically attractive path for overgrowth of high quality Ge based optical devices.

Fig. 1

 1 Fig. 1 Schematic illustration of dislocation trapping by nanovoids approach.

Fig. 2

 2 Fig. 2 Cross-section SEM images of porous Ge/Si layer formed at 1.5mA/cm 2 (a) and (b) etch rates versus current density of dislocated and bulk regions in Ge/Si layer.

Fig. 3 .

 3 Fig. 3. Different types of the void shapes observed during the annealing of Ge/Si layer (400-700°C) in hydrogen ambient.

Fig. 5

 5 Fig. 5 STEM images shows the termination of a dislocation segment in the nanovoid area.

Fig. 4

 4 Fig. 4 STEM mixed Dark Field and Bright Field images obtained from irradiated 80 nm Ge/Si sample, (a) reference sample, (b) porous Ge/Si sample, (c) voided Ge/Si sample and (d) higher magnification of (c) indicates the formation of a dislocation-free region at the upper part of the voided region.

ConclusionsIn this work, we proposed a new architecture of low cost virtual substrate engineering. This concept is promising for the fabrication of dislocation-free Ge on Si substrate. The resulting epi-ready surface will be used as the starting point for the deposition of the III-V junctions in the 3J PV cell. Using a simple, inexpensive process through electrochemical etching and thermal annealing will be crucial in cutting manufacturing costs and enabling market penetration of Ge based devices.