Presentation of the Electrical Molten Zone (EMZ) Technique

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Presentation of the Electrical Molten Zone (EMZ) Technique João C. C. Henriques Faculty of Sciences of the University of Lisbon - Physics Department

Abstract This article presents a new zone melting crystallization technique for photovoltaic silicon ribbon production. It starts by showing some background research on related methods concerning, in particular, to ways of achieving and moving electrical molten zones (EMZ). It presents the fundamental mechanism responsible for the electric current concentration, which allows zone formation. Demonstrates the possibility of zone melting recrystallization (ZMR) through this technique and shows growth rates and energy consumptions that can be expected of it. It exposes ideas and results of the first attempts to obtain silicon ribbons using commercial (granular) feedstock. The electrical characteristics and control methods of the system are shown as well as the related temperature distributions in the ribbons. It reveals the existence of interfacial instability phenomena, which is thought to be of magnetohydrodynamic (MHD) origin, thus (tentative) explanations for those are illustrated by analogy with typical occurrences observed (in other contexts) in tubes of electrically conducting fluids.

Keywords: Magnetic fields; Morphological stability; Recrystallization; Electrical molten zone technique; Ribbon growth; Semiconducting silicon.

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1. Review of EMZ Concepts The work presented here shows the state of the art of a project aiming an experimental demonstration of principle of a new zone melting crystallization technique, for silicon ribbon production to photovoltaic applications, following the example of others in the industry like EFG [1], String-Ribbon [2] and Dendritic-Web [3]. This growth method allows, in principle, an increment of purity and structural perfection of the base materials, while offering a significant cost reduction by: a) avoiding the use of expensive consumables like crucibles; b) lowering the process energy consumption and c) suppressing the ingot slicing operation to obtain silicon wafers. The concept of zone melting (re)crystallization of silicon materials through direct application of electric current, either longitudinal (i.e. in the growth direction) or transversal, is an old and relatively straightforward idea, without any special equipment requirements. Worth mentioning, however, is the precursor work of W. G. Pfann [4] with electrical molten zones and feedstock supplying pools, as he was probably the first one to realize them, having even registered several patents on the process (e.g. [5]). Pfann suggested several possibilities for moving the molten zones and thus recrystallize the base materials. This is usually made by creating an asymmetry in the temperature distribution, with adequate thermal gradients in the material, imposed by the current injecting electrodes themselves, by using thermal shields or insulation and by modulating the meniscus shape (variation of the current passage cross-section). The last possibility can be implemented through: a) The gravity effect, making the meniscus thinner in the upper part in relation to the lower, thus originating higher Joule dissipation near the former solid-liquid interface, which induces a upwards zone movement; b) The meniscus mechanical constriction (with an insulating piece) in order to increase the resistance locally and cause higher dissipation, the translation of the piece induces zone movement.

3 It is also possible to move the zone even in the absence of any thermal gradient, which constitutes a very interesting alternative from the point of view of reducing the internal stresses of the crystals, an especially important problem in ribbon growth techniques due to the limitation that imposes to their growth rate. This can be done by: a) The Peltier effect between the material and its own melt (with longitudinal current). The Peltier heat is absorbed in one of the solid-liquid interfaces and released on the other, depending on the current direction. It may be seen as if it would inject the heat of fusion in an interface and extract it in the opposite one with both at the same temperature. For example, in silicon with a current density of 500 Acm-2 (cf. sec. 5) the interface advance speed may be about 7 mm⋅min-1. b) Electrodiffusion of impurities with high ionic mobility. Transfer of these from one interface to another (segregation) originates the solidification of the first one and the fusion of the second one (according to the solidus curve in the phase diagram). The application of an external magnetic field, perpendicularly to the current direction in the material, may not only generate a temperature distribution capable of moving the zone, but also suspend it by magnetic levitation. An interesting way of accomplish this is through the interaction between the current in the zone and the one that passes in an external electrical conductor positioned parallel to the zone, that is by the Ampere’s force action. For example, with currents in the order of 50 A and a conductor placed at 1 mm from a silicon molten zone, with characteristics similar to the ones in the present study, the Ampere’s force is some 70 times the weight of the melt. Some of the aforementioned concepts were rediscovered and used in this work, however if high growth rates are desirable, without compromising material purity and system simplicity, only external radiative pre-heating seems interesting, that is why it was the chosen solution in the present study. On the other hand in the EMZ technique the current passes transversally in the ribbon (i.e. in a direction perpendicular to that of the growth, but in the plane of the ribbon), which is a more efficient direct heating method, since it results in a much more localized temperature distribution in

4 the material. There are several alternatives to do this, for example using an electric arc between an electrode, of a suitable shape and material, which sweeps the recrystallizing charge. Depending on the specific configuration, the current may or may not disperse throughout the charge (i.e. the counter-electrode may or may not be the growing crystal), thus allowing a supplementary degree of freedom in the temperature distribution of the material. Kuhlmann-Schäfer [6] patented in 1976 an electrical molten zone process, which presents remarkable similarities of principle with the one proposed here. In that, two or more electrodes of the same material as the charge are disposed laterally in contact with the crystallizing material (fig. 1). These electrodes or the inferior part of the charge may serve as feedstock source to the zone. In the second case, the electrodes must be at a lower temperature in order to avoid being consumed in the process, allowing only for movement in relation to the charge, which may have a cylindrical or plane shape. The cross sections of the electrodes in contact with the charge should be small in order to achieve the necessary current concentration and, therefore, the desired temperature. However, they should have higher thickness than the zone width in order to provide the necessary confinement to it.

Figure 1 Kuhlmann-Schäfer method for generation of cylindrical (a) or ribbon (b) shaped crystals by electrical molten zone. Electrodes 2 are fixed to supports 4 around charge 1 delimitating the molten zone 3 (the lower drawings represent top views).

5 A more recent (1993) German patent also shows a similar electrical molten zone process (initiated by radiative heating – fig. 2) [7]. The method allows the production of ribbons (up to 10 cm in width) or thin walled tubes and is destined specifically to the photovoltaic market.

Figure 2 T. Wolfgang method for ribbon generation by electrical molten zone. The silicon electrodes 2 delimitate the molten zone 3 and the silicon piece 4 molds the meniscus for the extracting ribbon 1. Underneath the zone is the feeding material 5. The zone is initiated by (laser) radiation 6.

2. Electrical Molten Zone The positive dependence of the electrical conductivity with temperature, characteristic of semiconductor materials (in contrast to what happens in metals), allows the current concentration phenomenon which is fundamental in the EMZ technique. It should be noted that in silicon the electrical conductivity rises exponentially from about 4×10-4 Sm-1 at room temperature to nearly 5×104 Sm-1 (1.25×106 Sm-1 in the liquid) at fusion temperature, 1687 K [8, 9]. On the other hand, in the same temperature interval, the thermal conductivity falls from 156 Wm-1K-1 to 22 Wm-1K-1 [10].

6 Comparatively in a metal like copper the electric conductivity decreases slightly from 6.5×107 Sm-1 at ambient temperature to 5.5×106 Sm-1 at fusion temperature, 1385 K, and the thermal conductivity decreases linearly only from 400 Wm-1K-1 to 325 Wm-1K-1 [11]. Consider, then, the consequences of the abovementioned characteristics in the behavior of a system comprising a certain material volume of arbitrary shape in which 2 electrodes were placed and through which electric current was injected. In these conditions, the current density and temperature distribution functions are naturally variable from one point to another in the volume and dependent of, besides the intrinsic material properties, the volume geometry and its losses to the environment. In the case of the metal, the current density will be lower in the hotter regions of the material and these have large thermal conduction losses. This causes a dispersion of energy, thus the volume temperature tends to become homogeneous, with smooth thermal gradients, determined mainly by losses to the surroundings. In the case of the semiconductor, the situation is symmetric, the current density will be higher in the hotter regions of the material and these have smaller thermal conduction losses. This originates a concentration of energy and generates very steep thermal gradients, being the losses to the surroundings relatively unimportant. The mentioned energy concentration mechanism eventually results in the phase transition of the material and the formation of a molten zone (fig. 3). For an isotropic material, with heat transfer losses approximately symmetric in relation to the straight line that passes through the electrodes, the molten zone will form itself along that line, as that is the shortest electrical resistance path. The zone is at every instant in an equilibrium position, although this might not be a stable one. The zone displays some mobility and may have non-rectilinear trajectories. The causes for this behavior should be sought mainly in environmental conditions disturbances, like convective turbulence or alterations in radiative transfers (e.g. oxide deposits). In addition, certain material geometries are more stable than others.

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Figure 3 Closing of a linear electrical molten zone in a silicon plate with 100 × 20 × 0.45 mm3 (inter-electrode distance of ~90 mm). The process starts at the electrodes with 20 A and ends at centre with 40 A having, in this case, a duration of 55 s. Notice the alteration on the temperature distribution of the plate as the extremities of the zone advance to the centre.

The possibility of obtaining very thin linear electrical molten zones and their inherent stability problems were also observed by Pfann (in longitudinal current configurations). He noticed that in

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materials for which the ratio of the electrical resistivities of the solid by that of the liquid is ρs/ρl > 1 (ρs/ρl ≅ 25 for silicon), the electrical zones are unstable unless there is a sufficiently large

temperature gradient in the adjacent solid. The reason for this is that any protuberance in the liquid zone is amplified by the tendency of current lines to concentrate on it, thus propagating the zone. Through this effect Pfann was probably the first researcher to attain electrical molten zones inside 10 cm long germanium crystals (ρs/ρl ≅ 17) [12]. If one of the dimensions of the aforementioned material volume is much smaller than the others, the molten zone may have free surfaces. This makes its profile determined by the solidliquid-gas interface equilibrium (surfaces free energies), situation which results in an increase of complexity of the system, originating very relevant surface tension phenomena.

Figure 4 One of the first linear electrical molten zones made in the present study (silicon plate with 100 × 15 × 0.45 mm3).

In the present study the material used consisted in thin solar-grade silicon plates (fig. 4) with typical dimensions of 100 × 30 × 0.35 mm3 and a resistivity of 0.5-5 Ωcm. The choice of an adequate size for the silicon plates allows for a passive form of zone stabilization. In plates with more than 30 mm width (in a direction transversal to that of the zone), any perturbation capable of driving the zone towards one side of it tends to increase the temperature of that side which is compensated by higher thermal dissipation of that side. Experience shows that this asymmetry in the temperature profile is unstable, hence the zones remain very straight and immobile at the centre of the plate. On the other hand in narrow plates that deviation is susceptible to be amplified, as the

9 side towards which the zone drifted becoming more conductive increases its current density, thus heating even more (the opposite happens on the other side, which cools down strongly). This may be the driving force for zone drifting to the edge, where it usually collapses (fig. 5; see also sec. 7).

Figure 5 Image series showing the rupture of a zone (closing at 41.3 A) through drift to the edge of the plate. Time from closing to rupture: 220 s.

10 Another passive approach for zone stabilization consists in the use of optical feedback from the zone itself, with the plate positioned in a reflective cylindrical cavity. An active approach to the problem may consist in the imposition of external thermal gradients, for example by optical concentration in the centre of the plate and / or cooling of its edges. In either case, if the plate is to survive its internal strains, excessively steep thermal gradients should be avoided. It was also observed that it is possible to define the trajectory and even extinguish locally the zone (e.g. near the graphite contacts, in order to avoid contamination of the melt) by placing shield plates over the main one (which is equivalent to a local thickness increment). These tend to homogenize the temperature throughout the width of the plate, forcing a current redistribution and consequently preventing its concentration.

3. Zone Melting Recrystallization As a preliminary step towards a full crystallization process from feedstock, and in order to gain some insight into the involved stability and control matters, some recrystallizations of silicon ribbons were carried through by sweeping the electrical molten zone through the base plate. The first attempts in this direction faced some difficulties, particularly in the electrodes interfaces, for that reason a substantial number of different configurations for those were tried (fig. 6). It was found that graphite electrodes were very wetted by silicon (contact angle of 12º [13]) which originated mass extraction from the zone and accumulation of it in the extremities of those. This generates strong solid bridges between the electrodes and the ribbon, which makes the movement of the zone impossible. It was sought to alleviate the problem – without success – using rotating electrodes on the edges of the ribbon (fig. 6d). Finally, a demonstration of the process viability was achieved by edge stabilization with small graphite or silicon plates – frames – (fig. 6e) and with electrodes of identical material. In this configuration, the electrodes slide smoothly over the frame and there is no relative movement between this and the ribbon. The graphite frames are easily

11 removable at the end of the process. The silicon frames, however, become heavily welded to the edges of the ribbon, which is an important disadvantage of its use, given that it is unfeasible to remove them without fracturing the ribbon.

Figure 6 Evolution of the electrode configuration.

The graphite frames generate the thermal gradient necessary to recrystallization, but only in narrow ribbons and at low speeds. High recrystallization rates may originate zone curvature, delaying it in the central portion of the plate in relation to the electrical contact position on the edge. The silicon frames do not generate the gradient necessary to a purely electrical recrystallization; hence, this is only possible with optical assistance (fig. 7). However, this also facilitates considerably the beginning of the process and makes irrelevant the electrode geometry, given that the zones are sharply localized at the optical centre of the furnace. It was also observed that, in some operating regimes, it is possible to have just surface recrystallization of the ribbon. In this configuration the maximum growth rate depends strongly on the optical component, falling from 12.5 mm⋅min-1 with the internal lamps at 1200 W to 3-5 mm⋅min-1 without optical component (for a plate with 100 × 30 × 0.35 mm3). The energy consumption per unit of recrystallized area, for an optical furnace with a global efficiency of about 28% and with an

12 adequate power supply, is around 38 kWh⋅m-2, which is about 50% higher than what can be attained with optical recrystallization solely.

Figure 7 Electrical molten zone recrystallization with auxiliary optical concentration (current injection perpendicular to the plane of the figure, on the optical centre of the furnace).

4. Silicon Ribbon Pulling Despite the relative success of the recrystallization processes, the objective of the study was the crystallization of silicon ribbons from commercial feedstock (in granular form). This can be implemented through a small pool of molten silicon made at one of the extremities of the zone in which the silicon granules are introduced. Interesting enough it was also Pfann who suggested the use of these pools, confined in support plates of the same material in order to avoid melt contamination by foreign matter [14]. The pools made in the present work are just liquid silicon films with a diameter up to 10 mm and a thickness up to 1 mm, suspended by the plate (i.e. held only by their own surface tension) or confined in a crucible in the plate itself (when this one does not melt throughout its thickness). The silicon granules transport is made via a vibratory system. At

13 any given instant the total melted mass in the system is very small, 100-200 mg for the pool and 1020 mg for a 30 mm long zone. Several alternatives were considered for making the silicon pool, among which the possibility of doing it electrically, in the same way as the zone itself, with an array of electrodes disposed along a circumference and coupled to a current switching device. There were also made some experiments with electric arcs and electromagnetic induction, but the final choice rested upon the method of optical concentration due to its relative simplicity, ease of coupling to the existing apparatus and previous experience with this solution. For this purpose it was used a 2 kW xenon arc lamp with an ellipsoidal reflector, which is however, a low efficiency choice and requires the use of an auxiliary nonimaging concentrator [15, 16] (conical internal mirror) in order to attain the necessary radiation density on the pool. The first idea for pulling ribbon from the system, consisted in the configuration of figure 8a, that is with direct extraction from a zone made in a horizontal silicon plate, with the edges stabilized by fibers (quartz, carbon, etc.), similarly to the String-Ribbon technique. This idea was abandoned very early due to the problems that presents, namely of surface flatness (dependent on the zone trajectory) and of possibility of solid bridge formation (between the ribbon and the horizontal plate) along the entire perimeter of the zone. A few experiments performed in this configuration showed that the zone tends to rupture near the fibers or to deviate from them. The alternative configuration of figure 8b, has a similar topology but with edge stabilization by small intermediate silicon plates. This geometry offers better guaranties of flatness though without solving the problems of solid bridge formation at the intermediate plates, and of mass transfer from the pool to the growing ribbon through those.

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Figure 8 Ideas for ribbon pulling by EMZ: (a) Ribbon R is fed by the adjacent pool L and stabilized laterally by fibers F. The horizontal plate P with electrodes E supports the assembly; (b) the fibers were replaced by two intermediate plates I and the zone is now supported by the fixed lower plate S.

The outstanding difficulties found in the demonstration of effective mass transfer in this configuration in stable conditions, manifested themselves in a great number of experiments through the rupture of the zone due to thickness reduction of the ribbons until critical values. These troubles even cast some doubt over the existence of a hydraulic link, between the pool and the zone in the ribbon. The confirmation of this was obtained, nevertheless, through the observation of surface oscillations in the zone with the fall of granules in the pool, and by pool draining (reduction of its convexity) when ribbon was drawn without adding feedstock. In unstable conditions, which precede

15 several rupture modes, mass transfers from the zone to the pool and vice-versa are frequently observed. It was noticed, for example, that when the completion of the zone precedes that of the pool, this one does not have a circular contour rather it shows itself only as a local widening of the zone, which is an extremely unstable situation, susceptible to rupture by zone draining (fig. 9). These rupture modes are common with high currents (≥ 40 A) and are consistent with the effects of magnetohydrodynamic (MHD) forces (v. sec. 7) and / or surface tension present in the system. To avoid them the optical pre-heating should be augmented and the pool approached to the growing ribbon as much as feasible.

Figure 9 Zone draining to the pool at about 40 A, with rupture of this one – typical well-preserved rupture mode. The pool has roughly 12 mm in diameter and the minimum zone width is 0.35-0.5 mm. It can also be observed a characteristic delta at the pool exit and plate edge.

The severe restrictions that this technique imposes to melt flow, which limit the maximum growth rate that can be expected of it, specially in thin and wide ribbons, is due primarily to the small melt flow cross section, given that all mass is injected through one of the edges (or eventually both). In order to solve this problem a number of options were considered (but not implemented yet): a) Create a hydrostatic gradient between the pool and the zone in the ribbon.

16 b) Pressurize the region of the furnace that contains the pool relatively to the one that contains the zone (particularly difficult to implement). c) Apply a magnetic field perpendicularly to the plane of the ribbon, along with longitudinal current (i.e. in the direction of growth), thus implementing a MHD pump. The longitudinal current could be applied by means of a switching device, alternating with the transversal current. It should be mentioned still that the process has an important disadvantage, the impossibility of impurity segregation during growth, due to the small dimensions of the zone and the unidirectional character of the melt flow. Unless, of course, some form of melt extraction is implemented in the plate on the opposite side to that of the pool which, nonetheless, results in a further decrease in the maximum possible growth rate.

5. Electrical Characteristics Consider again the hypothetical system mentioned in section 2, consisting of a certain volume of material of arbitrary shape in which two electrodes were placed and through which electrical current was injected. Observe now the behavior of this system in transient conditions, when controlled in voltage as in the present work. If in instant t0 a potential difference were applied between the electrodes, the material reaches equilibrium distributions for current density J and temperature T at the end of a certain thermal relaxation time Δt, during which the current varies. At instant t0+Δt, in the case of a metal, the current would be inferior to the initial one, since the electrical conductivity decreases with temperature, in the case of a semiconductor it would be superior to the initial one since the electric conductivity increases with temperature. Thus, during the transient, one has dT/dJ < 0 for the metal and dT/dJ > 0 for the semiconductor. This means that in the case of the metal the system exhibits negative feedback, which makes it particularly stable and easily controlled. In the case of the semiconductor, the intrinsic positive feedback makes the system evolution towards equilibrium, from about 700 K in heating or cooling, being abrupt. If

17 sufficient power is available, temperature transients in the order of 1400 Ks-1 are possible during heating, which leads to the systematic fracture of the silicon plates. The high initial plate resistance forces high starting voltages, however once a critical value is reached – designated in this work as ignition voltage – the control system should reduce the applied voltage smoothly in order to avoid sudden temperature variations, thus implementing metastable equilibrium conditions. It should be noticed that at this stage the system presents a negative dynamic (or differential) resistance, Rd = dV/dI < 0. The transient response time is very short, for this reason a control in current, with a programmable power supply, besides being much more efficient, greatly facilitates the work and, by avoiding the operator manual control, makes the results more reproducible. Nevertheless, in the absence of such power supply the control problems may be solved in alternative ways: a) By placing ohmic resistances in series with the silicon plate, adapted to the internal load, like plates dimensions and contact resistances, in order to guarantee stability during ignition (high voltage) without penalizing the operation regime (high current). b) By optical pre-heating, making it possible to have a positive Rd during startup, which makes the system very stable and easy to control, although with higher energy consumption. Notice that even so, for higher currents (with molten zone), the dynamical resistance becomes again slightly negative, thus persists the tendency towards the elevation of the current by itself, for the applied voltage. This approach has the additional advantage of attenuating the thermal gradients in the material and consequently its internal stresses. c) By a hybrid solution, coupling the electrical to the optical circuit, thus making the resistance increase of the internal lamps compensate the resistance decrease of the silicon plate with the current. This configuration behaves similarly to option a) until zone formation and is much more energy efficient than option b).

18 The V(I) curves (and particularly P(I) – fig. 10) show a characteristic step at the point of effective zone closing due to the significant alteration of the plate temperature distribution (cf. fig. 3) and, therefore, of its resistance. A curious aspect of the V(I) characteristics is, apparently, the appearance of hysteresis phenomena whenever there are fast temperature transients, as in the instance of zone closing. This is probably due to heat of fusion absorption and release from the material, which seems to be corroborated by the observation that the areas of the cycles vary from 3.5-4 W in the situation of low optical heating to about 0.5 W in the situation of high optical heating.

Figure 10 Variation of electrical power with current for various optical powers Pl in a silicon plate with 100 × 30 × 0.35 mm3 (inter-electrode distance: 30 mm). Pl = Variable refers the hybrid configuration (v. text).

19 The electrical functions FE(L,e) of current, voltage and power vary, in first approximation, linearly with width (L) and thickness (e) of the silicon plates, but in a strongly non-linear way with the optical power PO(L,e), and may be well adjusted by the relation

⎛ FE ⎜⎜ ⎝ FEo

⎞ P ⎟⎟ + O = 1 POo ⎠ n

(1)

in which FEo(L,e) is the electrical function for PO = 0 and POo (L,e) is the optical power for FE = 0, the exponent n is 6 for current, 3 for voltage and 2 for power. Figure 11 shows a family of curves for various plate widths (with average thickness of 0.336 mm), obtained in the configuration of figure 6b and in typical conditions of zone closing. The measurements include all losses in both the optical and electrical circuits.

Figure 11 Variation of electrical power with optical power for various plate widths (L).

20 The zone width is a function of current and, in stable conditions for a plate 0.35 mm thick, has values below 1 mm (fig. 12), being able to reach close to 2 mm near the electric contact in conditions approaching rupture. A typical zone current density is in the order of 125 Amm-2 and the power dissipated per unit of length is 1.4 Wmm-1. This means that, due to the low resistivity of liquid silicon, less than half the power in the plate is being dissipated in the zone.

Figure 12 Zone width (at the centre of the plate) as a function of current for a plate with cross-

section of 30 × 0.35 mm2.

6. Temperature Distribution The temperature profiles in the plate on the configuration of figure 6b and the conditions of table 1 may be seen in figure 13. In the absence of optical component (Pl = 0 W) the profile is reasonably well approximated by a simple function like

T ( x ) = Ta +

T f − Ta 1 + kx

21 (2)

with Ta the ambient temperature (300 K), Tf the temperature at the edges of the zone (1687 K) and k suggests the profile curvature, with values of 125 and -140 m-1 for the upper and lower parts of the plate respectively. All curves show some asymmetry between the upper and the lower halves of the plate, especially in conditions of high optical power, attributable to (argon) convection currents inside the furnace.

Figure 13 Temperature profiles for several optical powers Pl (the positive abscissa indicates the

upper part of the plate, which has 100 × 30 × 0.35 mm3).

Pl / W 0 900 1700 I/A 32.5 ± 0.5 20.0 ± 0.1 V/v 14.2 ± 0.7 10.9 ± 0.7 0.34 0.74 2.5 t / mm Table 1 Current (I) voltage (V) and zone width (t) for the temperature profiles of figure 13.

22 The analysis of these profiles suggests a thermal gradient at the solidification interface of 250350 Kmm-1 (Pl = 0 W), thus growth rates up to 100 mm⋅min-1 seem theoretically possible. This advantage of the electrical molten zones it is not, however, likely to be utilizable given that the rate of change of the gradients (∇2T), which determine internal stresses and plastic strain of the ribbons, is also very high [17, 18]. Regarding the temperature transients, it was observed that with only 1 A (~90 W) applied the temperature near the electrical contact at the edge rises to 858 K and at the centre of the plate to 638 K. Thus to prevent its fracture, particularly below 993 K (the silicon brittle-ductile transition temperature [19]), the heating rate should be low, which is rather difficult in the absence of optical pre-heating due to the very negative dynamic resistance at this stage. The temperature difference between the centre and the edge (at contact level) reaches 435 K, immediately before fusion of the edge (~300 W), generating a transversal gradient in the plate of almost 30 Kmm-1, which constitutes the driving force for zone progression.

7. Zone Stability Problems The passage of high electrical current densities directly through conducting fluids generates complex magnetohydrodynamic (MHD) stability problems. These have been the object of numerous studies concerning crystal growth and especially plasma physics (nuclear fusion, astrophysics, etc.), therefore only some features of those, with importance in the present experimental work, will be sketched here. Some (tentative) explanations for the instabilities are sought, by analogy with typical phenomena observed (in other contexts) in tubes of electrically conducting fluids. It should be remembered that the electrical molten zone is nothing more than a conducting fluid capillary in equilibrium by the forces of gravity, surface tension and magnetodynamic. Chandrasekhar [20] made an excellent analysis of these problems, showing the conditions for development of hydrodynamic instabilities and its settling as stationary convection

23 patterns in fluids, giving various examples for the cases of systems with thermal gradients, rotation and presence of magnetic fields. The Lorentz force over a fluid volume due to the presence of a magnetic induction B may be written in the form ⎛ B2 ⎞ 1 ⎟⎟ + (B ⋅ ∇)B F = −∇ ⎜⎜ ⎝ 2μ ⎠ μ

(3)

(with μ the magnetic permeability of the medium) thus being composed by a (1º) term of hydrostatic pressure transverse to the field lines (electromagnetic pinch), and a (2º) term of tension

along the field lines, both of magnitude B2/2μ. Integrating F in an infinite conducting fluid cylinder

of radius R and with a constant axial current density (I/πR2) one obtains the pinch pressure (directed radially inwards) [21]

μ I2 p(r ) = 4π 2 R 2

⎡ ⎛ r ⎞2 ⎤ ⎢1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ R ⎠ ⎥⎦

(4)

Figure 14 illustrates two of the most characteristic instabilities in conducting fluid tubes [22]. In the sausage instability type, the magnetic field pressure increases in the constricted region (since the current density increases – v. eq. 4) and decreases in the expanded region, thus generating a tube of variable cross section without altering its curvature, situation that tends to get worse until its collapse. In the kink instability type, the magnetic field lines compress themselves in the concave part and expand themselves in the convex part of the tube surface, thus generating a larger curvature without changing its cross section, which accounts for the tendency of the distortion to propagate in the direction of the convexity until it collapses. The characteristic surface undulation, mainly of the first instability, is analogous to the one that occurs in the well known Rayleigh-Taylor hydrodynamic instability, which in the presence of a magnetic field is also known as the Parker instability. Characterizes itself through the periodic deformation of the equilibrium surface between

24 two moving fluid layers (of different densities), under the action of inertial forces (gravitational or centrifugal).

Figure 14 Instabilities in a conductive fluid cylinder: (a) Unstable equilibrium (B0 is the maximum

field – at the surface of the conductor); (b) Sausage instability; (c) Kink instability.

The susceptibility for both types of instability may be minimized by applying an axial magnetic field. The field lines tension tends to keep them straight opposing deflections (mainly of short period), thus stabilizing the tube. However, the presence of an axial field Bz (in addition to the azimuthal Bφ) leads to helicoidal instabilities (fig. 15) [23, 24, 25]. When the magnetic flux lines are sufficiently twisted (but not enough to originate a kink) – by analogy with the torsion of an elastic line – the tube spontaneously assumes a helicoidal form, thus relaxing the magnetic energy accumulated as field line tension. The total magnetic energy decreases converting itself into kinetic energy and this in turn into internal energy via Joule heat and viscous dissipation. For an incompressible inviscid fluid the instability occurs when the field lines helical pitch q exceeds a critical value qcr

25 q≡

Bφ rB z

≥ q cr

(5)

Figure 15 Helicoidal instability for q = 5.

All instabilities mentioned here occur between fluids or between a fluid (usually a plasma) and the vacuum, for this reason its applicability to the phenomena observed in the present work, that is, between a liquid and a solid boundary in phase transition conditions, is only tentative as this is clearly a very complex phenomenon. Solidification interface instabilities capable of originating faceted or cellular structures are common in crystal growth systems, mainly due to constitution supercooling of the melt. It is also well known that electromagnetic fields have a strong affect in the distribution coefficient of many impurities, primarily due to its effect on the liquid convection currents [26, 27, 28]. However, in the present case, given that the instability manifests itself exclusively in electrical molten zones, even in stationary conditions, seems to indicate that this is a different phenomenon, one of magnetohydrodynamic origin. The electrical current in the zone and its inherent magnetic field seem to originate a convection current pattern that generates periodic temperature fluctuations along the solid-liquid interface, as those in figures 16 and 17, regardless of plate orientation (e.g. horizontal or vertical). The sinusoidal undulation of the zone edges has an amplitude and period that increase, in first approximation, linearly with the zone width (fig. 18). Alongside this, it is possible to observe zone surface vibrations, through light reflections on it. Not only the interface but also the

26 melt surface itself does not seem flat which, by the way, is confirmed by the appearance of striations in the areas recrystallized in these conditions.

Figure 16 Image series showing the zone undulation pattern. The instabilities appear around 40 A

and increase their amplitude and period as the zone widens until its rupture (shortly after the last image of the series) at about 52 A. A liquid protuberance formation generally precedes collapse.

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Figure 17 Undulation in a zone with only superficial fusion. Rayleigh-Taylor type of vortices.

Figure 18 Variation of the undulation amplitude and period as a function of zone width (without

optical component) for several plate thicknesses (e). The amplitude should be understood as the difference between maxima and minima on the same zone edge. Results for e = 0.45 mm correspond to the zone of figure 16.

28 The applicability of the MHD model for the formation of helicoidal tubes, to the present results – with which presents remarkable resemblances – requires the demonstration of the existence of an axial field in this system. This must certainly be sought in the internal convection currents, observed by the movement of small oxide particles on the surface of the zone. In very thin zones stabilized by surface tension, the velocity field is probably dominated by Marangoni (thermocapillary) currents, which may have speeds up to 100 times the crystals growth rate in the EFG and FZ techniques [29, 30]. These currents might generate a longitudinal vortex sheet, like a long solenoid throughout the zone length, susceptible of generating a longitudinal field. Another possibility worth considering is the fact that convection currents have radial components, which means that the Lorentz force should have a small axial component, thus driving a helicoidal movement of the particles. The apparently discontinuous character of the instability, as evidenced by the fact that it only manifests itself above a certain current density, seems to find a parallel in the MHD model in the critical field line torsion parameter (eq. 5) for the helicoidal instability appearance. This limit seems to occur near (3/2)I0 (I0 is the zone closing current dependent, naturally, of the optical power) and represents an operational limit to the working current of the system. Rupture occurs generally around 2I0. In this current interval, the zone doubles its width. Certain characteristic zone deformation and rupture modes suggest also instabilities of the kink and sausage types and represent a great source of experimental problems. For example, zones of variable width are unstable since the pinch generates mass transport from the thinner regions to the wider, which eventually may originate its collapse. This effect may, however, be compensated by the zone surface tension, through variation of the liquid surfaces curvature, which adjust themselves at every instant and in every point, to the electrical current and zone / pool width. Thus, it is possible in certain conditions, in the previously mentioned current operating interval, the

29 establishment of an equilibrium of null flow, which explains the extraordinary difficulties found in the ribbon growth process.

Conclusions It was demonstrated the possibility of silicon ribbon recrystallization by EMZ, with edge stabilization by graphite or silicon plates. Growth rates up to 12.5 mm⋅min-1 were achieved, and the analysis of the solidification temperature gradient suggests the possibility of increments up to 100 mm⋅min-1. The growth rate and electrical power involved depend strongly on the auxiliary optical power. Regardless of that, the process energy consumption is in the order of 38 kWh⋅m-2. The difficulties of zone stabilization and mass transfer to the growing ribbon have shown to be a greater challenge than anticipated, which is why, up to the moment, it was not possible to surpass transient crystallization conditions, being yet to demonstrate the possibility of sustained ribbon growth. Some ideas for the causes of the instabilities and resolution of the mass transfer problems were presented, however a deeper analysis of this subjects is out of the scope of the present study. Still many problems remain to be explained and solved before this technique may be considered a true alternative in the generation of silicon ribbons for photovoltaic application.

Acknowledgements The author gratefully acknowledges the helpful discussions and valuable suggestions of Prof. A. G. Vallêra, Prof. J. M. Alves and Dr. R. M. Gamboa from the Semiconductors Laboratory (FCUL), which greatly contributed for the advancement of the experimental work. Also deeply appreciated was the support, through grant BD / 11228 / 97, provided by the Foundation for Science and Technology (MCT) under the PRAXIS XXI program. This work was performed under EU project THIMOCE (contract JOR-CT98-0287).

30

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