Patent Publication Number: US-2012027996-A1

Title: Mold shape to optimize thickness uniformity of silicon film

Description:
FIELD 
     The disclosure relates to methods of making an article of semiconducting material, and more particularly to exocasting methods whereby an article of semiconducting material is formed over an external surface of a shaped mold. 
     BACKGROUND 
     Semiconducting materials are used in a variety of applications, and may be incorporated, for example, into electronic devices such as photovoltaic devices. Photovoltaic devices convert light radiation into electrical energy through the photovoltaic effect. 
     The properties of semiconducting materials may depend on a variety of factors, including crystal structure, the concentration and type of intrinsic defects, and the presence and distribution of dopants and other impurities. Within a semiconducting material, the grain size and grain size distribution, for example, can impact the performance of resulting devices. By way of example, the electrical conductivity and thus the overall efficiency of a semiconductor-based device such as a photovoltaic cell will generally improve with larger and more uniform grains. 
     For silicon-based devices, silicon may be formed using a variety of techniques. Examples include silicon formed as an ingot, sheet or ribbon. The silicon may be supported or unsupported by an underlying substrate. However, such conventional methods for making supported and unsupported articles of silicon have a number of shortcomings. 
     Methods of making unsupported thin semiconducting material sheets, including silicon sheets, may be slow or wasteful of the semiconducting material feedstock. Unsupported single crystalline semiconducting materials can be produced, for example, using Czochralski or Bridgman processes. However, such bulk methods may disadvantageously result in significant kerf loss when the material is cut into thin sheets or wafers. Additional methods by which unsupported polycrystalline semiconducting materials can be produced include electromagnetic casting and direct net-shape sheet growth methods such as ribbon growth processes. However, these techniques tend to be slow and expensive. Polycrystalline silicon ribbon produced using silicon ribbon growth technologies is typically formed at a rate of only about 1-2 cm/min. 
     Supported semiconducting material sheets may be produced less expensively, but the semiconducting material sheet may be limited by the substrate on which it is formed, and the substrate may have to meet various process and application requirements, which may be conflicting. 
     Methods for producing unsupported polycrystalline semiconducting materials are disclosed in commonly-owned U.S. patent application Ser. No. 12/466,143, filed May 14, 2009, and commonly-owned U.S. patent application Ser. No. 12/394,608, filed Feb. 27, 2009, the disclosures of which are hereby incorporated by reference. 
     As described herein, the inventors have now discovered additional methods by which supported and unsupported articles of semiconducting materials may be made. The disclosed methods may facilitate formation of exocast semiconducting materials having desirable attributes such as uniform thickness while reducing material waste and increasing the rate of production. 
     SUMMARY 
     In accordance with various exemplary embodiments, an exocasting method of making a solid layer of a semiconducting material comprises selecting a mold having a leading edge thickness, a trailing edge thickness and a length separating the leading edge from the trailing edge such that in respective plots of solid layer thickness versus effective submersion time for submersion of the leading and trailing edges into molten semiconducting material for respective first and second submersion times, a thickness of the solid layer adjacent to the leading and trailing edges is substantially equal to a target thickness. The mold is then submersed into and withdrawn from the molten semiconducting material to form a solid layer of semiconducting material over an external surface of the mold, wherein the leading edge of the mold is submersed for the first submersion time and the trailing edge of the mold is submersed for the second submersion time. 
     As used herein, the term “semiconducting material” includes materials that may exhibit semiconducting properties, such as, for example, silicon, alloys and compounds of silicon, germanium, alloys and compounds of germanium, gallium arsenide, alloys and compounds of gallium arsenide, and combinations thereof. In various embodiments, the semiconducting material may be pure (such as, for example, intrinsic or i-type silicon) or doped (such as, for example, silicon containing at least one n-type or p-type dopant, such as phosphorous or boron, respectively). 
     As used herein, the phrases “article of semiconducting material,” “exocast article,” and variations thereof include any shape or form of semiconducting material made using the disclosed methods. Examples of such articles may be smooth, textured, flat, curved, bent, angled, dense, porous, symmetric or asymmetric. Articles of semiconducting materials may comprise forms such as, for example, sheets, wafers or tubes. 
     The term “mold” means a physical structure having an external surface upon or over which the article of semiconducting material can be formed. Molten or solid semiconducting material need not physically contact an external surface of the mold, although contact may occur. 
     The term “external surface of a mold” means a surface of the mold that may be exposed to molten semiconducting material upon submersion of the mold into the molten semiconducting material. 
     The term “supported” means that an article of semiconducting material is integral with a mold. The supported article of semiconducting material may optionally remain on the mold for further processing. 
     The term “unsupported” means that an article of semiconducting material is not integral with a mold. The unsupported article of semiconducting material may be supported by a mold while it is being formed, but is then separated from the mold. 
     The phrase “form a solid layer of a semiconducting material over an external surface of a mold” and variations thereof mean that at least some of the semiconducting material from the molten semiconducting material solidifies on or over an external surface of the mold. 
     The term “crystalline” means any material comprising a crystal structure, including, for example, single crystal and polycrystalline semiconducting materials. 
     The term “polycrystalline” includes any material comprised of a plurality of crystal grains. For example, polycrystalline materials may include micro-crystalline and nano-crystalline materials. 
     The terms “temperature of the molten semiconducting material,” “bulk temperature of the molten semiconducting material,” and variations thereof mean the average temperature of the molten semiconducting material contained within a vessel. Localized temperatures within the molten semiconducting material may vary spatially at any point in time, such as, for example, near melt-vessel or melt-atmosphere boundaries, or in areas of the molten semiconducting material proximate to the mold while the mold is submersed. In various embodiments, the average temperature of the molten semiconducting material is substantially uniform despite any localized temperature variation. 
     As used herein, the term “undercooling” refers to a process by which a material is cooled below a transformation temperature without obtaining the transformation. The amount of undercooling of a liquid, for example, is the temperature difference between a measured temperature and a solidification temperature of the liquid. The amount of undercooling may be measured in degrees Celsius (° C.) or degrees Fahrenheit (° F.). 
     As used herein, the term “average submersion time,” unless otherwise indicated, refers to the average time that a mold is submersed in molten semiconducting material. For a mold having length L, and assuming no acceleration or deceleration during submersion and withdrawal, the average submersion time is equal to L/2V in +L/2V out /+t dwell , where V in  and V out  are the submersion and withdrawal velocities, respectively, and t dwell  is an optional dwell time (e.g., hold time) between submersion and withdrawal. In embodiments where the submersion velocity is equal to the withdrawal velocity and the dwell time is zero, the average submersion time is simply equal to L/V. For a mold having a length L, a “first submersion time” corresponding to a leading edge of the mold is equal to L/V in +L/V out +t dwell , while a “second submersion time” corresponding to a trailing edge of the mold is equal to t dwell . 
     Methods of affecting the thickness, thickness variability and/or morphology of a solid layer formed during an exocasting process are described herein. In the description that follows, certain aspects and embodiments will become evident. It should be understood that the invention, in its broadest sense, could be practiced without having one or more features of these aspects and embodiments. It should be understood also that these aspects and embodiments are merely exemplary and explanatory, and are not restrictive of the invention as claimed. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The following figures, which are described below and which are incorporated in and constitute a part of the specification, illustrate exemplary embodiments and are not to be considered limiting of the scope of the invention. The figures are not necessarily to scale, and certain features and certain views of the figures may be shown exaggerated in scale or in schematic in the interest of clarity and conciseness. 
         FIGS. 1A-1L  illustrate an exemplary exocasting method for making an article of semiconducting material; 
         FIG. 2  is a theoretical graph of a solid layer thickness versus submersion time; 
         FIG. 3  is a graph of solid layer thickness versus submersion time according to an example; 
         FIG. 4  is a series of graphs of solid layer thickness versus submersion time for various mold thicknesses; 
         FIG. 5  is a schematic of example mold geometries according to various embodiments; and 
         FIG. 6  is a graph of maximum solid layer thickness versus initial mold temperature for different mold thicknesses. 
     
    
    
     DETAILED DESCRIPTION 
     In an exocasting process, a solid mold is submersed into and then withdrawn from a volume of molten semiconducting material. Due in large part to heat loss to the mold and the surroundings, a portion of the molten semiconducting material undergoes a liquid-to-solid phase transformation, which results in the formation of a solid layer of the semiconducting material over an external surface of the mold. In the process, the mold acts as both a heat sink and a solid form for the solidification to occur. By controlling the submersion time, both a solid layer thickness and the variability in the solid layer thickness can be affected. 
     It will be appreciated that during the acts of submersion and withdrawal, a leading edge of the mold enters the molten semiconducting material before a trailing edge of the mold and therefore is submersed for a longer total time. This time dispersion along a length of the mold, i.e., a length in the direction of submersion and withdrawal, can introduce variability in the properties of the resulting solid layer, including solid layer thickness. As disclosed herein, the effects of the time dispersion can be minimized by appropriately adjusting the heat transfer kinetics throughout the exocasting process. By modifying the geometry of the mold, the effect on solid layer thickness of the residence time difference between the leading and trailing edges of the mold can be minimized. In embodiments, a mold is designed for exocasting where one or both of the thickness and the width of the mold are not constant along its length. 
     In an embodiment, a target thickness is selected for the solid layer. Plots of solid layer thickness versus submersion time are then calculated for a mold having particular attributes, including mold composition, mold thickness and initial mold temperature in order to determine for a particular mold the submersion times that result in a solid layer on an external surface adjacent to the leading edge of the mold having a thickness that is substantially equal to a thickness of a solid layer on an external surface adjacent to the trailing edge of the mold. With the foregoing, intentional variations in the mold geometry (i.e., mold thickness) can be used to offset the difference in total residence time between the leading and trailing edges in order to minimize the total thickness variability of the solid layer. According to various embodiments, by adjusting the mold geometry, variations in residence time over the external surface of the mold will not lead to corresponding variations in solid layer thickness. 
     As shown in cross-section in  FIG. 1A , solid mold  100  having an external surface  102  is suspended above a vessel  110  containing a molten semiconducting material  120 . Mold  100  may be in any form suitable for use in the disclosed methods. For example, mold  100  may be in the form of a monolith or wafer. Mold  100  may comprise a porous or a non-porous body, optionally having one or more porous or non-porous coatings. Mold  100  may comprise one or more flat external surfaces  102  or one or more curved external surfaces. A curved external surface may be convex or concave. The mold and its external surface(s) may be characterized by features including shape, dimension, surface area, surface roughness, etc. One or more of these features may be uniform or non-uniform. It will be understood that the features of the mold  100  and its external surface  102  may affect the properties of resulting exocast article. 
     It will be appreciated that although mold  100  and external surface  102  are illustrated in two-dimensional cross-section, mold  100  is a three-dimensional body and the solid layer  140  that forms over the external surface  102  of the mold is also a three-dimensional body having a length, a width, and a thickness. As disclosed in additional detail hereinafter, the exocast solid layer  140  is formed during different stages of the exocasting process and comprises solid material formed during at least three stages of solidification. 
     In embodiments, mold  100  is formed from a material that is compatible with the molten semiconducting material  120 . For example, the mold  100  may be formed from a material that does not melt or soften when submersed. As a further example, the mold  100  may be thermally stable and/or chemically inert to the molten semiconducting material  120 , and therefore non-reactive or substantially non-reactive with the molten semiconducting material. 
     By way of example, the mold  100  may comprise or consist of refractory materials such as fused silica, graphite, silicon nitride, single crystal or polycrystalline silicon, as well as combinations and composites thereof. In at least one embodiment, mold  100  is made of vitreous silicon dioxide or quartz. The mold can have a thickness ranging from about 0.1 to 100 mm (e.g., 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50 or 100 mm). A length and width of the mold can independently vary from about 1 cm to 100 cm or greater. 
     The molten semiconducting material  120  may be provided by melting a suitable semiconducting material in vessel  110 . Vessel  110  may be made from a high temperature or refractory material chosen from vitreous silica, graphite, and silicon nitride. Alternatively, vessel  110  may be formed from a first high temperature or refractory material and provided with an internal coating of a second high temperature or refractory material where the internal coating is adapted to be in contact with the molten semiconducting material. The semiconducting material may be silicon. In addition to silicon, the molten semiconducting material may be chosen from alloys and compounds of silicon, germanium, alloys and compounds of germanium, gallium arsenide, alloys and compounds of gallium arsenide, and combinations thereof. 
     The molten semiconducting material may comprise at least one non-semiconducting element that may form a semiconducting alloy or compound. For example, the molten semiconducting material may comprise gallium arsenide (GaAs), aluminum nitride (AlN) or indium phosphide (InP). 
     According to various embodiments, the molten semiconducting material  120  may be pure or doped. Example dopants, if present, may include boron, phosphorous, or aluminum, and may be present in any suitable concentration, e.g., 1-100 ppm, which may be chosen based on, for example, the desired dopant concentration in the resulting article of semiconducting material. 
     To form an article of a semiconducting material, mold  100  is at least partially submersed into molten semiconducting material  120  and then withdrawn. During the acts of submersion and withdrawal, the molten semiconducting material  102  solidifies and forms a solid layer  140  of semiconducting material over an external surface  102  of the mold. 
     Without wishing to be bound by theory, solidification occurs in three principal stages. The exocasting process, including a more detailed description of solidification in Stages I-III, can be understood with reference to  FIGS. 1A-1L , which portray a series of sequential schematic illustrations according to various embodiments. The submersion of the mold  100  into molten semiconducting material  120  is illustrated schematically in  FIGS. 1A-1F , while withdrawal of the mold  100  from the molten semiconducting material  120  is illustrated schematically in  FIGS. 1G-1L . 
     In one exemplary embodiment, using any suitable heating device or method, mold  100  may be brought to a temperature, T M , and the molten semiconducting material  120  may be brought to a bulk temperature, T S , which is greater than or equal to a melting temperature of the semiconducting material. 
     At least one heating element (not shown) may be used to heat mold  100 , vessel  110  and/or maintain the molten semiconducting material  120  at a desired temperature. Examples of suitable heating elements include resistive or inductive heating elements, infrared (IR) heat sources (e.g., IR lamps), and flame heat sources. An example of an inductive heating element is a radio frequency (RF) induction heating element. RF induction heating may provide a cleaner environment by minimizing the presence of foreign matter in the melt. 
     The composition of the atmosphere  190  above the molten semiconducting material  120  can be controlled before, during, and after submersion. It is believed that the use of vitreous silica for the mold  100  and/or vessel  110  may lead to oxygen contamination of the article of semiconducting material. Thus, in various embodiments, oxygen contamination may optionally be mitigated or substantially mitigated, by melting the semiconducting material and forming the article in a low-oxygen environment, comprising, for example a dry mixture of hydrogen (e.g., less than 1 ppm water) and an inert gas such as argon, krypton or xenon. A low-oxygen environment may include one or more of hydrogen, helium, argon, or nitrogen. In at least one exemplary embodiment, the atmosphere may be chosen from an Ar/1.0 wt % H 2  mixture or an Ar/2.5 wt % H 2  mixture. 
     Prior to submersion ( FIG. 1A ) the temperature of the mold T M  and the temperature of the molten semiconducting material T S  each can be controlled such that T M &lt;T S . In embodiments where the molten semiconducting material comprises silicon, the bulk temperature of the molten silicon, T S , may range from 1414° C. to 1550° C., such as, for example, from 1450° C. to 1490° C., e.g., 1460° C. The initial temperature of the mold, T M , may range from −50° C. to 1400° C. (e.g., from −35° C.-0° C., 20° C.-30° C., 300° C.-500° C., or 600-900° C.) prior to submersion in the molten semiconducting material  120 . In addition to controlling the mold and molten semiconducting material temperatures, the temperature of the radiant environment, T E , such as a wall  112  of the vessel  110 , may also be controlled. 
     Referring to  FIGS. 1B and 1C , as the mold  100  is brought closer to and then submersed into the molten semiconducting material  120 , a temperature of the mold, e.g., a temperature of the mold  100  at leading edge  104 , will increase due initially to radiative and then conductive and convective heat transfer from the molten semiconducting material  102  to the mold  100 . As is evident from  FIG. 1 , in the illustrated embodiment a thickness of the mold  100  at the leading edge  104  is greater than a thickness of the mold  100  at the trailing edge  106 . 
     In embodiments where the mold  100  comprises silica and the molten semiconducting material  120  comprises silicon, a convex meniscus  124  will form at the point of entry of the mold into the molten silicon because silicon does not readily wet to the mold&#39;s silica surface. 
     Initially, the temperature of the mold  100  will remain less than the temperature of the molten semiconducting material  120 . As the mold is submersed further into the molten semiconducting material ( FIGS. 1D and 1E ), a relatively large temperature difference between the mold  100  and the molten semiconducting material  120  will induce a liquid-to-solid phase transformation that results in the formation of a solid layer  140  of the semiconducting material over the external surface  102  of the mold. 
     The magnitude of the temperature difference between the mold  100  and the molten semiconducting material  120  can affect the microstructure and other properties of the solid layer  140 . The combination of a relatively large temperature gradient between the mold  100  and the molten semiconducting material  120 , which may be on the order of 800° C., results in the formation of a Stage I solid layer  142  over the external surface of the mold. The Stage I solid layer may comprise a relatively fine grain size. 
     As shown in  FIGS. 1C-1E , as the mold  100  is submersed, molten semiconducting material  120  is first solidified at the leading edge  104  of the mold  100 . As the mold is further submersed, a thin Stage I solid layer  142  forms over the exposed surface  102  of the mold. The growth front of the Stage I solid layer  142  is continuously fed during immersion by molten material from the convex meniscus  124 , and the growth direction of the Stage I solid layer  142  is substantially parallel to the relative direction of motion between the mold and the melt (i.e., the growth direction of the Stage I solid layer is substantially parallel to the exposed surface  102  of the mold). 
     According to embodiments, mold  100  may be rotated or vibrated as it is submersed. In other embodiments, however, the mold is maintained essentially stationary in the transverse dimensions as it is lowered into and raised out of the molten semiconducting material  120 . It will be appreciated that in addition to the foregoing, the mold may be held stationary and the vessel containing the molten semiconducting material may be moved (i.e., raised) in order to submerse the mold within the molten semiconducting material. In embodiments, the entire mold may be submersed or substantially all of the mold may be submersed into the molten semiconducting material. For instance, with respect to its length, 90% or more of the mold may be submersed (e.g., 90, 95, 99 or 100%). 
     As shown in  FIGS. 1D-1F , with the mold  100  at least partially submersed in the molten semiconducting material  120 , the Stage I solid layer  142  (formed via a growth interface having a growth direction substantially parallel to the external surface of the mold) becomes the template for the formation of a Stage II solid layer, where molten semiconducting material  120  from the melt solidifies at the exposed surface of the Stage I solid layer. Initial formation of a Stage II solid layer  144 , which typically occurs at a lower temperature differential than Stage I growth, can increase the thickness of the solid layer  140 . Thus, in contrast to Stage I growth, the Stage II solid layer  144  is formed via a growth interface having a growth direction that is substantially perpendicular to the external surface of the mold. Experimental data reveal that the solid layer growth rate during Stage II growth can be on the order of 100 μm/sec. 
     The microstructure of the solid layer  140  (including the Stage I and Stage II solid layers), in addition to its dependence on the temperature gradient between the mold and the melt, is a function of the rate at which the relative position of the mold  100  is changed with respect to the molten semiconducting material  120 . At relatively slow submersion velocities (e.g., on the order of about 1 cm/sec), the temperature differential between the mold  100  and the molten semiconducting material  120  is reduced due to heating of the mold, which generally results in a solid layer  140  having relatively large grains but a relatively small total thickness. On the other hand, at submersion velocities on the order of about 50 cm/sec, the relatively high velocity can disturb the shape of the convex meniscus  124 , which can disrupt continuous grain growth and result in a discontinuous solid layer  140  having relatively small crystal grains. In embodiments, the submersion rate can be from about 0.5 to 50 cm/sec, e.g., 1, 2, 5, 10 or 20 cm/sec. 
     In further embodiments, the submersion rate may be changed (i.e., increased or decreased) during the act of submersion such that the mold is accelerated or decelerated. In one example, during submersion the mold velocity is decreased from about 10 cm/sec to 0 cm/sec at 100 cm/sec 2  over 7.5 cm of submersed mold. 
     Quiescent growth of the solid layer during Stage II is a function of the submersion time (i.e., residence time), which, due to the dynamic nature of the exocasting process, will vary spatially over the external surface of the mold  100 . The leading edge of the mold will be in contact with the molten semiconducting material for a longer time than the trailing edge of the mold. This leads to an excess residence time for the leading edge equal to L/V in +L/V out , compared to the trailing edge, where L is the length of the mold and V in  and V out  are the submersion and withdrawal velocities. Because the leading edge  104  of the mold is the first part of the mold to be submersed, initial growth of the Stage II solid layer  144  can be fastest at or near the leading edge  104  where the temperature differential is the greatest. On the other hand, because the leading edge of the mold is the last part of the mold to be withdrawn, remelting of the Stage II solid layer  144  near the leading edge  104  can decrease the thickness of the solid layer  140  near the leading edge  104 . 
     Mold  100  may be submersed in the molten semiconducting material  120  for a period of time sufficient to allow a solid layer  140  of the semiconducting material to solidify over a surface  102  of the mold  100 . The mold  100  may be submersed in the molten semiconducting material  120  for up to 30 seconds or more (e.g., from 0.5 to 30 seconds). By way of a further example, the mold  100  may be submersed for up to 10 seconds (e.g., from 1 to 4 seconds). The submersion time may be varied appropriately based on parameters known to those of skill in the art, such as, for example, the temperatures and heat transfer properties of the system, and the desired properties of the article of semiconducting material. 
       FIG. 2  shows a calculated graph of solid layer thickness measured from the external surface  102  of mold  100  as a function of submersion time. Over an initial time period, the solid layer grows rapidly to a maximum thickness. The thickness then decreases over a subsequent time period. During the initial time period, solidification of the molten semiconducting material commences at the interface between the Stage I solid layer  142  and the melt, and the Stage II layer  144  advances into the molten semiconducting material, which results in a positive rate of growth for the solid layer  140 . During the subsequent time period, as the temperature of the mold increases and the heat capacity of the mold is exhausted, remelting of the Stage II solid layer  144  takes place, which results in a negative rate of growth. If the mold were left in the molten semiconducting material  120  indefinitely, eventually the entire solid layer  140  (Stage I and Stage II solid layers) would remelt and dissipate as the mold thermally equilibrates with the molten semiconducting material. 
     The time where the transition from solidification to remelting takes place is defined as the “transition time.” The thickness of the Stage II solid layer  144  attains its maximum value at the transition time. According to embodiments, the mold can be removed from the molten semiconducting material after a predetermined time that corresponds to the desired thickness of the solid layer. 
     The dynamics of both the growth and the remelting of the Stage II layer  140  can also be seen with particular reference to  FIGS. 1E and 1F . In  FIG. 1E , as the mold  102  is near the full extent of its immersion into the molten semiconducting material  120 , the Stage II layer  144  can have a non-uniform thickness. Near the leading edge  104  of mold  100 , where the average mold temperature is greatest due to its longer submersion time, the Stage II layer  144  begins to remelt as the direction of the local heat flux is outward from the mold. The remelting causes a local thinning of the Stage II layer  144  near the leading edge  104 . At the other end of the mold, which has a lower average mold temperature, the direction of the local heat flux is still into the mold. Absorption of heat by the mold  102  results in growth of the Stage II layer into the melt. 
     Referring next to  FIG. 1F , a shift in the non-uniform thickness of the Stage II layer  144  can be seen over the length of the mold as the mold temperature increases and additional remelting progresses. The small arrows in  FIGS. 1E and 1F  qualitatively indicate the relative solid layer growth rates at different locations along the interface between the Stage II solid layer  144  and the molten semiconducting material  120 . 
     As illustrated in  FIGS. 1A-1F , during submersion, a Stage I solid layer  142  forms over and optionally in direct contact with the exposed surface  102  of the mold  100 . In turn, a Stage II solid layer  144  forms over and in direct contact with the Stage I solid layer  142 . In embodiments, absent complete remelting of the solid layer  140 , the thickness of the Stage I solid layer remains substantially constant during submersion and withdrawal, while the thickness of the Stage II solid layer is dynamic and a function of heat transfer dynamics, which can be controlled, for example, by the local thickness of the mold. A dashed line in  FIGS. 1D-1K  marks the boundary between the Stage I and Stage II solid layers  142 ,  144 . 
     Additional aspects of the growth and remelting of the solid layer as a function of the submersion time of the mold are described in commonly-owned U.S. patent application Ser. Nos. 12/466,104 and 12/466,143, each filed May 14, 2009, the disclosures of which being hereby incorporated by reference. 
     The portion of the exocasting process when the mold  100  is being submersed into the molten semiconducting material  120  is described above and is shown schematically in cross-section in  FIGS. 1A-1F . In particular  FIG. 1F  shows the position of the mold and the formation of solid layer  140  when the mold is at its maximum extent of submersion and the velocity of the mold with respect to the molten semiconducting material  120  is zero. A further portion of the exocasting process (i.e., when the mold  100  is being withdrawn from the molten semiconducting material  120 ), including the formation of a Stage III solid layer  146  over a surface of the mold, is described next with particular reference to  FIGS. 1G-1L . 
     During withdrawal of the mold, because the exposed solid surface is solidified semiconducting material rather than the original mold material, the wetting dynamics between the solid surface and the melt are likely different from those encountered during submersion. Referring to  FIG. 1G , in the example of molten silicon solidifying over a silicon solid layer  140 , a dynamic, concave meniscus  134  forms at the solid-liquid-gas triple point. As a result of this dynamic meniscus  134 , during withdrawal of the mold from the molten semiconducting material  120 , an additional solid layer  146  (Stage III solid layer) forms over the previously-formed solid layers (Stage I and Stage II solid layers). The Stage III solid layer  146  is also referred to herein as the overlayer, and determines the minimum thickness of a solid layer obtained through exocasting. 
     Although the Stage II solid layer  144  that has formed over the Stage I solid layer  142  will continue to grow or remelt according to the local heat flux dynamics beneath the surface  122  of the molten semiconducting material  120 , the Stage III solid layer  146  forms above the equilibrium surface  122  of the molten semiconducting material  120  due to the wetting of the solid layer (e.g., exposed surface of the Stage II solid layer  144 ) by the molten semiconducting material  120 . During withdrawal, a Stage III solid layer growth front  136  is continuously fed by molten material from beneath the dynamic meniscus  134 . 
     In embodiments, a majority of the thickness of the solid layer  140  will be formed during Stage II (i.e., growth that is substantially perpendicular to the mold&#39;s external surface). Referring to  FIGS. 1G-1J , the dynamic meniscus  134 , the Stage II solid layer  144  and the Stage III layer  146  formed during withdrawal define a dynamic volume  128  or “dragged volume” of the melt that is located above the equilibrium surface  122  of the molten semiconducting material  120 . The dynamic volume  128 , which is approaching solidification as a result of the various heat transfer mechanisms, continuously feeds the Stage III solidification front  136  during withdrawal. 
     In embodiments, the withdrawal rate can be from about 0.5 to 50 cm/sec, e.g., 1, 2, 5, 10 or 20 cm/sec. Higher withdrawal rates may cause fluid drag that can induce perturbations into the dynamic meniscus, which can be transferred to the Stage III overlayer. In further embodiments, as with the submersion rate, the withdrawal may be changed (i.e., increased or decreased) during the act of withdrawal such that the mold is accelerated or decelerated. In one example, during withdrawal the mold velocity is increased from 0 cm/sec to about 3 cm/sec at 10 cm/sec 2  over 7.5 cm of submersed mold. 
     After mold  100  is removed from vessel  110  and sufficiently cooled, the solid layer  140  of semiconducting material may be removed or separated from the mold  100  using, for example, differential expansion and/or mechanical assistance. Alternatively, the solid layer  140  may remain on mold  100  as a supported article of semiconducting material. 
     Referring again to  FIG. 2 , because the solid layer thickness versus submersion time curve displays a thickness maximum at the transition time, a solid layer having a particular thickness (i.e., other than the maximum thickness) can be obtained using a submersion time that is less than or greater than the transition time. In the example of  FIG. 2 , a 200 micron solid layer could be produced using a submersion time of either ˜1.2 seconds or ˜5 seconds. 
     It will be appreciated that either submersion time would produce a ˜200 micron thick solid layer, but that the respective times offer process trade-offs. A process involving a 1.2 second submersion time can be completed more rapidly than a process involving a 5 second submersion time, which can become increasingly important upon scale-up. On the other hand, because the rate of thickness change (i.e., slope of the thickness versus submersion time curve) at about 1.2 seconds is much greater than the rate of thickness change at about 5 seconds, small fluctuations in the more rapid process will lead to greater variability in solid layer thickness. 
     Due to the local slope in the thickness versus submersion time curve, any variability in submersion time or other process parameters would lead to variability in the solid layer. An exocasting example that illustrates this principle is described with reference to  FIG. 3 , which is a plot of solid layer thickness versus time for a silicon solid layer formed over a silica mold having dimensions of 15 cm×15 cm×1.5 mm thick, and an initial mold temperature of 100° C. 
     The target thickness for the solid layer is 200 microns with as low a TTV as possible, e.g., less than 30 microns. As seen with reference to  FIG. 3 , the transition time is about 1.5 sec, which corresponds to a maximum solid layer thickness of about 350 microns. Submersion times within either the solidification regime (˜0.25 sec) or the remelt regime (˜7.2 sec) could be selected to solidify a solid layer having the target thickness of 200 microns. 
     For a mold of these dimensions, a submersion/withdrawal velocity of 20 cm/sec leads to an excess residence time of 1.5 sec between the leading and trailing edges of the mold. With an average submersion time of 7.2 sec, the leading edge of the mold experiences a local submersion time of 7.95 sec, while the trailing edge of the mold experiences a local submersion time of 6.45 sec. This variability in local submersion time leads to a sold layer thickness adjacent to the leading edge of 215 microns, and a solid layer thickness adjacent to the trailing edge of 182 microns. This thickness variability represents a maximum TTV of ˜33 microns. The residence time window between the leading and trailing edges is illustrated using the vertical dashed lines in  FIG. 3 . 
     Applicants have discovered that minimization of the total thickness variability can be achieved by choosing mold attributes, such as the composition, thickness profile and initial temperature of a suitable mold such that the solid layer thickness adjacent to the leading edge of the mold is substantially equal to the solid layer thickness adjacent to the trailing edge of the mold. By changing (e.g., increasing or decreasing) the thickness of the leading edge of the mold with respect to the trailing edge of the mold, it is possible to impact the heat transfer kinetics of the exocasting process and offset the residence time dispersion between the leading and trailing edges. 
     The impact on solid layer thickness of the mold thickness is illustrated with reference to  FIG. 4  and one particular example. In the example, the target thickness for a silicon solid layer is 200 microns. The mold has opposing external surfaces each having areal dimensions of 15 cm×15 cm and the initial mold temperature is 800° C. Assuming that the submersion velocity and the withdrawal velocity are each 7.5 cm/sec, the leading edge of the mold would be submerged 4 seconds longer than the trailing edge, where the leading edge submersion time is about 21 seconds and the trailing edge submersion time is about 17 seconds. 
       FIG. 4  shows a series of plots of solid layer thickness versus effective submersion time for molds having a different thickness. By associating the effective submersion time with the mold thickness, the time dispersion (and hence thickness variation) across the length of the mold can be compensated and the attendant thickness variation can be decreased. 
     Referring to  FIG. 4 , as seen at intersection A, when submersed for 21 seconds a mold having a thickness of about 4.5 mm will yield a solid layer having a thickness that is roughly equal to the target thickness of 200 microns. In a similar vein, as seen at intersection B, when submersed for 17 seconds a mold having a thickness of about 4 mm will yield a 200 micron thick solid layer. In the foregoing example, by configuring the mold to have a leading edge thickness of 4.5 mm and a trailing edge thickness of 4 mm, variation in the solid layer thickness due to the time dispersion can be offset and a solid layer having constant thickness (˜200 microns) can be formed. 
     For submersion times greater than the transition time, the respective curves in  FIG. 4  are approximately parallel and equally spaced for constant increments in mold thickness over the range of 3.5 to 7 mm. In this embodiment, this approximately linear relationship attributes to a roughly linear dependence of the resulting solid layer thickness on the mold thickness. Thus, a suitable mold design for achieving a constant thickness solid layer may take the shape of a trapezoidal prism with the lengths of the parallel sides corresponding to the leading and trailing edge thicknesses as determined from numerical simulations. 
     Example mold designs are illustrated in  FIG. 5 .  FIG. 5   a  shows a trapezoidal prism  400  having a main body  410  with a leading edge  404  and a trailing edge  406  where a thickness of the mold at the leading edge  404  is greater than a thickness at the trailing edge  406 . The length and width of the mold are each constant.  FIG. 5   b  shows an additional mold design in the shape of a truncated pyramid  500 . The truncated pyramid has a main body  510 , a leading edge  504 , and a trailing edge  506 . In the mold embodiment of  FIG. 5   b , both a thickness and a width of the mold at the leading edge  504  are greater than a thickness and a width at the trailing edge  506 . 
     In an embodiment, a method of forming a solid layer of semiconducting material includes submerging a mold into and withdrawing the mold from a molten semiconducting material where the mold has a leading edge thickness that is at least 1% greater (e.g., 1, 2, 4, 5, 10, 15 or 20% greater) than the trailing edge thickness. The thickness of the mold can vary continuously or discontinuously along a length of the mold. The thickness along a length of the mold may change monotonically or parabolically, such that the thickness may be defined by a suitable equation (e.g., linear or higher-order polynomial). 
     It will be understood that the shape of the thickness versus submersion time curves can be manipulated using various combinations of at least the mold material and the initial mold temperature. The choice of mold material can affect, for example, the thermal conductivity, density, and heat capacity of the mold. The effect of the mold thickness and the initial mold temperature on the maximum solid layer thickness is shown in  FIG. 6  for a silica mold. 
       FIG. 6  is a graph of maximum solid layer thickness versus initial mold temperature for various mold thicknesses. In  FIG. 6 , the dependence of the maximum solid layer thickness on initial mold temperature is shown for mold thickness values of 1, 2, 3 and 4 mm, as indicated. From graphs such as this, which can be developed for molds having a range of thermal properties (e.g., thermal conductivity, density, heat capacity, etc.) a skilled practitioner can determine the appropriate parameter space for a desired low TTV process. 
     The disclosed methods can be used to produce articles of semiconducting material having one or more desired attributes related to, for example, total thickness, thickness variability, impurity content and/or surface roughness. Such articles, such as, for example, silicon sheets, may be used to for electronic devices such as photovoltaic devices. By way of example, an as-formed silicon sheet may have areal dimensions of about 156 mm×156 mm, a thickness in a range of 100 μm to 400 μm, and a substantial number of grains larger than 1 mm. In embodiments, a total thickness of the solid layer is 150, 200, 250, 300, 350 or 400 μm. In further embodiments, a total thickness of the solid layer is less than 400 μm (e.g., less than 350, 300, 250, 200 or 150 μm). 
     One advantage of the disclosed method includes the ability to minimize the total thickness variability (TTV) due to residence time variability over an areal dimension of the solid layer. A further advantage is the ability to minimize the total thickness variability due to fluctuations of process parameters such as mold and melt temperatures. 
     Total thickness variability means the normalized maximum difference in thickness between the thickest point and the thinnest point within a sampling area of a solid layer. The total thickness variability, TTV, is equal to (t max −t min )/t target , where t max  and t min  are the maximum and minimum thicknesses within the sampling area and t target  is the target thickness. The sampling area may be defined as the whole or a portion of the solid layer. In an embodiment, the total thickness variability of a solid layer is less than 30% (e.g., less than 10% or less than 5%). In processes involving the formation of more than one solid layer, a thickness dispersion is defined as the standard deviation of the ratio of average solid layer thickness to target thickness. 
     Unless otherwise indicated, all numbers used in the specification and claims are to be understood as being modified in all instances by the term “about,” whether or not so stated. It should also be understood that the precise numerical values used in the specification and claims form additional embodiments. Efforts have been made to ensure the accuracy of the numerical values disclosed herein. Any measured numerical value, however, can inherently contain certain errors resulting from the standard deviation found in its associated measuring technique. 
     It is noted that, as used in this specification and the appended claims, the singular forms “a,” “an,” and “the,” include plural referents unless expressly and unequivocally limited to one referent, and vice versa. Thus, by way of example only, reference to “a solid layer” can refer to one or more layers, and reference to “a semiconducting material” can refer to one or more semiconducting materials. As used herein, the term “includes” and its grammatical variants are intended to be non-limiting, such that recitation of items in a list is not to the exclusion of other like items that can be substituted or added to the listed items. 
     It will be apparent to those skilled in the art that various modifications and variation can be made to the programs and methods of the present disclosure without departing from the scope its teachings. Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the teachings disclosed herein. It is intended that the embodiments described in the specification be considered as exemplary only.