Solute trapping of group 111, and V elements IV, stepwise growth mechanism Division in silicon by an aperiodic Riccardo Reitano,@ Patrick M. Smith,b) and Michael J. Aziz of Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138 (Received 24 February 1994; accepted for publication 25 April 1994) With rapid solidification following pulsed laser melting, we have measured the dependence on interface orientation of the amount of solute trapping of several group III, IV, and V elements (As, Ga, Ge, In, Sb, Sn) in Si. The aperiodic stepwise growth model of Goldman and Aziz accurately fits both the velocity and orientation dependence of solute trapping of all of these solutes except Ge. The success of the model implies a ledge structure for the crystal/melt interface and a step-flow mechanism for growth from the melt. In addition, we have observed an empirical inverse correlation between the two free parameters (-“diffusive speeds”) in this model and the equilibrium solute partition coefficient of a system. This correlation may be used to estimate values of these free parameters for other systems in which solute trapping has not or cannot be measured. The possible microscopic origin of such a correlation is discussed. 1. INTRODUCTION The description of solidification phenomena over a range of interface velocities several orders of magnitude wide (lo-*-lo’ m/s) requires knowledge of the equilibrium solidification properties as well as how the material behaves when the departure from equilibrium becomes significant. For low solidification velocities, the description of solidification is based on the assumption that the interface is in local equilibrium; i.e., the solid and the liquid immediately adjacent to the interface can be considered to be in equilibrium with each other. In this case the concentration of solute in the liquid at the interface can be obtained using transport theory in the liquid, while the concentration in the growing crystal can be estimated from the equilibrium phase diagram. At high velocities, such as those obtained by pulsed laser melting, there is much less time for equilibration between the crystal and the melt across the interface, and the assumption of local equilibrium is no longer valid; departure from equilibrium is expected. One readily apparent consequence of the deviations from local equilibrium in rapid solidification is the formation of a highly supersaturated solid;‘72 in the case of pulsed laser melting of doped silicon, supersaturations by factors of up to 10’ have been reported3 following growth at rates of a few meters per second. Because such supersaturation implies that the chemical potential of the solute actually rises upon crystallization,” the process is called “solute trapping.” A distinct but closely related phenomenon is the suppression of solute/solvent partitioning at the rapidly moving crystal/melt interface. This suppressed partitioning is the mechanism whereby such supersaturations are attained; as a consequence the term “solute trapping” is also used synonymously with suppressed partitioning. The partition coefficient k, the ratio of solute compositions in the solid and in the liquid at the “Permanent address:Dipartimento di Fisica, Universiti di Catania, Corso Italia 57, I-95129 Catania. “Present address:Lawrence Livermore National Laboratory,Livermore, CA 94550. interface, undergoes a transition from its equilibrium value k, to unity as the growth rate increases: This has been shown to occur for many impurities in silicon.4 The trapping process can be understood from a kinetic point of view in terms of the restricted mobility of the solute as it diffuses through the interface. When the interface velocity u approaches the maximum speed of diffusion (the “diffusive speed” uD, given by the ratio of the solute diffusivity at the interface to the atomic jump distance), the solute atoms cannot diffuse away rapidly enough to escape from the advancing interface, resulting in solute trapping. Some insight into these phenomena has been gained with a theoretical description of the interface and an atomistic analysis of the solidification process. Several solute trapping models qualitatively give the correct dependence of k on the solidification velocity, but it is difficult to discriminate between the models because of the large uncertainties in the available experimental data. For example, the experimental data for bismuth’ and some theoretical curves are shown in Fig. 1. Below we briefly describe the models shown in Fig. 1. Baker’s model6 is a continuum model for dilute solutions in which the solute is treated as if diffusing in a continuum along a steep energy gradient at an interface of width 8. The one-dimensional diffusion equation is solved in steady-state with the diffusive flux assumed proportional to the product of the local concentration and chemical potential gradient. The standard free energy for the solute is assumed to vary linearly with position, from a value of E,y in the solid, to Ej in the middle of the interface, to El in the liquid. The diffusivity is assumed to be independent of position, except that it takes a discontinuous jump from D, , its value in the bulk of the liquid, to D,, its value in the bulk of the solid at the center of the interface. The ratio of E, to El is fixed by the equilibrium partition coefficient, but Ei is permitted to take any value. This model predicts the possibility of a nonmonotonic k(u) curve. For example, if there is strong adsorption of solute at the interface,7 then k can undergo a transition from k, to a value greater than unity and then back to unity. This will occur if local equilibrium between the interface Q 1994 American Institute of Physics 1518 J. Appl. Phys. 76 (3), 1 August 1994 0021-8979/94/76(3)/1518/12/$6.00 Downloaded 22 Jul 2003 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp Bi in Si(OO1) .$ 2. 0.4 / z s 6- 0.3 0 - -I‘-...’ Data from Aziz et al. Continuous Growth Model Stepwise Growth Model Two-Level Baker Model Aperiodic Stepwise Growth. Model .z 0.2 .e -t: fz 0.1 IO0 IO’ Interface .. speed (m/s) and aperiodic stepwise growth model (ASGM)16--are similar to Jackson’s in that they use nonlinear rate equations for solute-solvent redistribution across a sharp interface. The key difference is that interface motion is treated differently from solute-solvent redistribution, as a separate reaction with a higher mobility. Crudely speaking, if the solute does nothing it ehds up trapped in the solid as solvent atoms crystallize around it. In the Jackson model, if a solute atom does nothing it remains in the liquid. The growth rate in the Aziz models can exceed the diffusive speed by a large factor, and virtually complete solute trapping can occur. The CGM treats the case when the’interface is atomically rough enough that growth and redistribution occur simultaneously as strictly steady-state processes, even on the microscopic scale of the,crystal lattice. It predicts a velocitydependence of the partition coefficienf given by .~ k(u)= FIG. 1. Interface velocity dependence the partition coefficient for bisof muth in silicon (001). Data points are from Ref. 5; the liries areffiom various models as indicated. uh,+k, u/u&l m-i ’ 0) level and the solid is lost at much lower speeds than local equilibrium between the interface level and the liquid. Whereas this behavior is physically plausible, it has never been observed experimentally. To simplify the model and reduce the number of free parameters, therefore, we have removed the energy level in the middle of the interface, resulting in the two-level Baker model.’ We continue to require the diffusivity to remain constant throughout the interfacial region, but its value is treated as a free parameter when co-mparisons are made with experimental data. In Jackson’s mode19*” chemical rate equations are written for the hopping of each species across an atomically sharp interface. The interface velocity is determined from the sum of the individual fluxes across the interface; the partition coefficient is determined from their ratio. The rate equations are not linearized. A particular form for the coupling between the fluxes of individual species is assumed. The model yields the same expression for k(u) as does the continuous growth model of Aziz discussed below, but two important differences exist. (i) In the Jackson model, the maximum possible growth rate is of the order of uD, so k(u) is truncated at some u,,, and complete solute trapping is never approached. This is in conflict with the observation of growth rates well in excess of uD in metallic systems,” and with numerous observations of virtually complete solute trapping, e;g., of As in Si.” (ii) Because solute trapping depends on one species. being less mobile than the other at the interface, if A is significantly trapped in I3 at some velocity, ‘then B cannot be significantly trapped in A at any velocity. Because this result is also contrary to observation, Jackson’s modelI is not considered further below. The model fails to account for a solute atom “detaching” from a high-energy site on the solid side of the interface being by far the most likely atom to be the next to rejoin that site. This is due to stearic constraints on its diffusive escape imposed by the presence of a dense liquid.’ The models of Aziz and co-workers-the continuous growth model (CGM),‘.t4 stepwise growth model (SGM),8P15 J. Appl. Phys., Vol. 76, No. 3, 1 August 1994 The SGM treats the case in which an atomically smooth, sharp interface- advances by the periodic lateral passage of monolayer steps. The passage of a step results in the incorporation of a single liquid monolayer, including any solute atoms in the layer. Solute diffuses back into the liquid during the period before the passage of the next step, at which point any remaining solute is assumed to be permanently trapped into the solid. The predicted velocity-dependence of k for this mechanism is ” k(u)=k,+(l-k,)exp(LuD/u). (2) The ASGM treats the same case as the SGM, except that the passage of steps is assumed to. occur randomly, rather than periodically, in time. For a lattice in which the direction of step motion is normal to the direction of interface motion [expected to occur for the (111) interface in Si and fee metals] the ASGM predicts, with otherwise identical boundary conditions to those in the SGM, a k(u) relation identicalI to that of the CGM, Eq. (1). Hence the ASGM for the (111) interface cannot be distinguished from the CGM by a measurement of k(u) alone. Unlike ‘the CGM, however, the ASGM can be extended to predict the orientationdependence of k, as described below, with the addition of a second free parameter. ” The two-level Baker model, the CGM, and the SGM each have only a single unknown parameter, uD , which enters into the expression for k(u) only as the ratio u/uD. If ttD is treated as-a fitting- parameter and the models are plotted (see Fig, 1) on a log-velocity scale, then the only effect of a change in uD is a rigid shift of the curve to the left or the right, without any change inshape. Figure 1 shows that the CGM fits the,k(u) data17for Si(Bi) quite well, whereas the SGM and two-level Baker mbdel k rise too steeply with u to fit the data. The ASGM fits the’data almost as well as the .. CGM. In one of the first solute trapping measurements, Baeri et al. l8 observed a lower value for k for the (001) orientation than for the (111) at’the same interface velocity in Bi- implanted silicon. An explanation based on a reduced atomic mobility resulting from greater undercooling on the (111). Reitano, Smith, and Aziz 1519 Downloaded 22 Jul 2003 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp than on the (100) was proposed,” but successful fitting requires substantially more interfacial undercooling than has been measured.” The first systematic study of the orientation dependence of the partition coefficient was done by Aziz and White” on Bi-implanted Si; samples were cut at 5” increments from (110) through (111) to (OOl), so that the systematic variation of k with interface orientation could be measured. The partition coefficient at constant interface velocity was found to be sharply peaked at (111) and to decrease monotonically with increasing interface inclination from (111) [see Fig. 6(a)]. The aperiodic stepwise growth model16 predicts this orientation dependence. It assumes that at any orientation the interface is broken into (111) terraces of single double-layer height (3.13 A) and random width and that solidification proceeds by the lateral passage of these steps at random intervals. Some (not all, as in the SGM) of the solute atoms in the monolayer of liquid adjacent to the interface are trapped as a step passes. Because steps are expected to be atomically rough, the lateral trapping is assumed to obey CGM kinetics as a function of ledge speed. Some solute then diffuses back into the liquid through the terrace before the next layer isadded, permanently covering all remaining solute. The model has two free parameters, diffusive speeds at the ledge and at the terrace (z& and ZIP, respectively). Escape through the terrace is expected to be slower than at the step edge because the solute atoms on the terrace are more highly coordinated with the crystal. The sharp peak at (111) in k(B) occurs because as (111) is approached, the steps become more widely separated and must move faster in order to maintain a constant u imposed by heat tlow. The model accounts very well for the observed velocity and orientation dependence of k of Bi in Si with only these two parameters; The ASGM expression. for k(u, 6) is given by k(v)= TABLE I. Ion implantationparameters. IOII species Gallium Indium Tin Arsenic Antimony Germanium Energy ReVI 180 180 100 100 100 150 Dose (cm-‘) 1.0x 10’” LOX 10’5 3.0X10’S 1.0x10’6 2.5x10’5 3.0x 10’5 Substrate temperature(K) 300 300 77 77 77 77 Amorphous thickness(nm) 200 150 130 140 100 170 to make an “educated guess” for diffusive speeds in alloy systems where they have not been or cannot be measured. The best that might be hoped for is the discovery of a correlation between the diffusive speeds that fit the partitioning behavior and some other readily-measurable physical property of the alloy system. In this work we present a systematic study of the orientation dependence of the partition coefficient for several group III, IV, and V elements in silicon. The objective of the work is to determine whether the ASGM is of general validity and, if so, to gain some insight into the relation between & u;, and other material properties. II. EXPERIMENT k,+Pt(Pr+kJl(Pr+ Pt+l 1) ’ (3) where P,=u/[zJ~ cos(e)] and /?I=vl[uh sin(e)]; 0 is the angle of inclination from (111). Reasonable numerical values of ZI; and r& are obtained from a fit of the orientation dependence Si-Bi data (see Table III). The model has also been shown to reproduce fairly well the velocity dependence of the partition coefficient for Bir6 in (001) Si (see Fig. 1) and for Ast’ in polycrystalline Si. One problem with the ASGM is that it is doubtful that a physical experiment could eyer be designed to obtain an independent measurement of the quantities treated as free parameters in the model: the speeds of atomic diffusion across a crystal/melt interface. Molecular dynamics simulations have had some success in this regard. Calculations by Cook and Clancy” for a rough-interface (Lennard-Jones) system have confirmed the underlying hypothesis of the continuous growth model, namely that the solute diffusion coefficient at the interface is related to the growth rate at the center of the transition from local equilibrium to complete solute trapping. Even .with molecular dynamics, however, it would be extraordinarily difficult to determine separately the terrace and ledge diffusivities in a faceting system. It is also unclear how to predict diffusive speeds from first principles, or how even 1520 J. Appl. Phys., Vol. 76, No. 3, 1 August 1994 Float-zoned Si wafers were cut at several inclination angles from (111) toward (001) and (110). Six different impurities (In, Sn, Sb, Ga, Ge, and As) were ion-implanted at energies and doses detailed in Table I. Rutherford backscattering spectrometry (RBS) was used to measure the implanted depth profile and the thickness of the amorphous layer resulting from implantation. The samples were then irradiated with a pulsed XeCl excimer laser [wavelength 308 nm, pulse duration of 30 ns full. width half maximum (FWHM)] at an energy high enough to melt through the entire amorphous layer and allow liquid phase epitaxial crystallization. The laser beam was passed through a homogenizer which produced a spatial uniformity of 22% over a 2-mm-sq beam spot. Time-resolved reflectivity (TRR) measurements were performed during each shot in order to measure the melt duration; a fast digitizing oscilloscope was used to measure the intensity of a low-power argon ion laser beam reflected from the surface of the sample with a time resolution of approximately 1 ns. The laser pulse energy was calibrated by comparing the melt du‘rations measured on a pure silicon single crystal sample with the results of heat-flow simulations.23 In the case of As-, Ge-, and Sb-implanted samples, the back surface was heated with a high power CO2 laser for several seconds; this reduced the interface velocity during solidification by more than an order of magnitude, as will be discussed later in detail. A schematic drawing of the experimental configuration is shown in Fig. 2. The laser irradiation parameters are listed in Table II. The diffused impurity profile after irradiation was measured by grazing-exit RBS to enhance depth resolution. Numerical solutions to the diffusion equation were used to determine values of the partition coefficient by comparing the Reitano, Smith, and Aziz Downloaded 22 Jul 2003 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp Excimer Laser Trigger Line I-7 FIG. 2. Schematicdrawing of the experimentalsetup. experimental impurity profile with the results of the simulations. Details on the procedure are presented below. ill. RESULTS In analyzing our data we used computer simulations of pulsed-laser induced melting and resolidification and of impurity diffusion in the liquid. In the next two subsections we will describe briefly the information extracted from these simulations. A. Laser irradiations and heat-flow simulations The irradiation conditions were chosen depending on the partitioning behavior of each impurity. In order to determine a value for k we required a detectable amount of surface impurity accumulation. It has been observed, for example, that about 40% of the In atoms segregate to the surface for an interface velocity of 5 m/~,~ while until now no one has measured the partition coefficients for As in Si because of the extremely small amount of surface accumulation due to its high k value.12 For three of the six impurities (Ga, In, and Sn) irradiation of the samples at room temperature with a pulse energy density of 1.5 J/cm’ (1.2 J/cm2 for In) produced a measurable surface accumulation of the impurity. The energy density of each shot was checked by comparing the measured melt duration with that calculated by computer simulations. The simulation was performed with a numerical solution of the TABLE II. Laser irradiation parameters. Ion species Gallium Indium En Arsenic Antimony Germanium Fluence (J/cm’) 1.2 1.2 1.5 1.0 1.0 1.0 Substrate temperature(K) 300 300 300 1300~1500 1350-1420 1270-1450 Solidificatbn velocity (m/s) 4.3 4.5 4.2 0.1-0.4 0.17-0.28 0.13-0.61 1-D heat diffusion equation using the well-established optical and thermal parameters of crystallive, amorphous, and liquid Si. It has been shown that these parameters are known with sufficient accuracy to give quite good agreement between experiment and calculations for room-temperature substrates.” Agreement between the melt durations predicted by the simulations and the ones measured experimentallj was of the order of a few percent, while the reproducibility of the laser output was about 52%. The melt depth versus time profile calculated by the simulations was used as input to the diffusion simulations and was used to determine thesolidification velocity. The average velocity was slightly different for the three impurities listed above due to differences in the thicknesses of the amorphous layers produced during implantation of the impurities prior to irradiation. The thickne?s of the amorphous layer was determined to be independent of orientation by RBS and ion channeling; it has been checked foi all the substrate orientations for indium implantation and only for (111) and (001) for G? and Sn implantations. For the other three impurities (Ge, As, and Sb) regrowth velocities of a few meters per second produced immeasurably small surface accumulation on room-temperature substrates. However, by heating the substrate with a cw-mode CO, laser (X=10.6 pm) for several seconds prior to the pulsed laser melting, we were able to obtain solidification at much slower speeds; heating times of 8 to 10 s were used to allow heat to diffuse uniformly throughout the sample. The vniformity of CO, heating at the moment of pulsed excimer laser melting was checked by measuring the melt duration simultaneously in the center of the sample and other points within the 2 mmX2 mm area of the sample. Using the simulated dependence of melt duration on substrate temperature described below, we determined that throughout this area the maximum temperature variation was approximately 23%. The excimer pulse was synchronized with the end of the COz heating time, and the actual substrate temperature at which the sample was shot was estimated using heat flow simulations that determined the dependence of the melt duration, for the measured excimer pulse energy density, on substrate temperature. Melt durations scattered throughout the range 800-3200 ns were obtained, which correspond to a temperature range of 1300-1500 K for a 1 J/cm2 pulse. The corresponding calculated average velocities were in the range 0.1-0.6 m/s. This large shot-to-shot variation in the substrate temperature was found to be ‘unavoidable; we believe it to be due in part to variations in the output of the CO, laser and to differences from sample to’sample in the thermal contact between the sample and the sample holder. Because k is -a strong function .of both orientation and solidification velocity, for a study of the orientation dependence of k it was necessary to scale to a single interface velocity the partition coefficients obtained at different velocities, as discussed below. Finally, several samples were heated solely with the CO, laser (without any excimer laser pulse) and air-cooled, to determine whether the’ initial impurity depth profile prior to pulsed laser melting was altered by the CO2 heating technique. Reifano, Smith, and Azlz 1521 J. Appl. Phys.;Vol. 76, No. 3, 1 August 1994 Downloaded 22 Jul 2003 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 6000 5000 4000 3000 r 2 2000 1000 0 3.0 2.5 2.0 0 500 10001500200025003000 b I- ) G2*5 ~ / -Ei2*o - . 1.5 : 1.0 3.0 initial profi ‘le 2ji!l-0 1000 2000 3000 4000 Time I cl ; I I 300 (ns) 3-or; d) ns LIS t-f Depth (A> ns i i.5 i I s2-5 2.0 4-J 23 1.5 . g 1.0 0 c-l 0.5 600 1000 2000 3000 4000 0.0 Depth (a) s Depth 3-or; f) 4 23 g 0 (a) .~ ns -I i ! -If I f I 0 1200 ns L;S CL’ v: ;1 I 2.5 2.0 i -I 2100 L!S +--f. v ! I ‘_ . 1.5 -I 1.0 -I 0.5 0.0 -I 1000 2000 3000 4000 0 I ’ 1000 I 2000 I 3000 4000 3% 9). iT2*5 -g -: L;S I - -14-l c--c 2.0 1.5 1.0 . . Depth 2700 (8) ns * I : :_ : 0 0 0.5 I A! 0 I 0.0 , 00 Depth (a) 0 s 1000 2000 3000 4000 Depth (A) FIG. 3. (a) Calculatedmelt depth us time profile for a 1 J/cm2 pulse at a substratktemperatureof 1500 K, (b) starting impurity profile for the diffusion simulation; (c)-(h) calculatedimpurity profiles at severaltimes after the excimer pulse as indicated. (93. Diffhon simulations The heat-flow simulations predict the interface position or.melt depth (ignoring the volume decrease upon melting), as a function of time. The result of a typical simulation is 1522 J. Appl. Phys., Vol. 76, No. 3, 1 August 1994 shown in Fig. 3(a) for a 1 J/cm2 pulse at a substrate temperature of 1500 K. The maximum melt depth is about 5000 A, the melt duration is 3200 ns, and the average solidification velocity is. 0.1 m/s. Melt-depth profiles such as this are calReitano,~Smith, and Aziz Downloaded 22 Jul 2003 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 0 Experirnek~l our data. The sensitivity of the best-fit value of k to the value of D, used is not excessive, but uncertainties in DI do contribute to the total uncertainty ‘in k. Comparison will be made to literature values for D, where the latter exist. ‘C. Fitting procedure We used two different criteria to determine the “best-fit” values of k and D, . The first was the standard least-squares fitting method. With D, fixed at a reasonable value, the partition coefficient was varied in steps of 0.01, and the standard deviation (cr) between measured and simulated depth profiles was calculated point by point; a parabola was fit to the o-(k) points, and the minimum of the parabola. was chosen as the best-fit k for that D,. The liquid diffusivity was then varied and the procedure repeated again; the best-fit values of DI and k were those corresponding to the absolute minimum of cr. An alternative method is, we believe, less susceptible to systematic errorsT6 A systematic error such as a potentially inaccurate melt depth simulation at large depths might skew the results determined solely by a least-squares fit. The alternative method is based on the steady state limit, in which the area under the surface peak is given by A=C, i ;-I ;, 1 (4) ‘i; 1.5 E2 -ii $ 1.0 z 0 0’ profile 0.5 nnnth 1x1 ; FIG. 4. Initial and diffused (circles) arsenic profiles for a (001) implanted sample. Dashed and solid lines are the calculated profiles for D,=3X104 cm% and k=0.41 and 0.47, respectively; the former gives too large a surface peak while for k=0.47 the surfacepeak is too small. Our best-fit value in this caseis k=0.44. culated for each shot, and, together with the measured initial impurity depth profile, are used as input in the diffusion simulations. The program solves the one-dimensional diffusion equation, assuming the impurity diffuses only in the liquid. The boundary condition at the solid-liquid interface is that the ratio of solute concentration’in the solid and liquid across the interface be k. This is the same simulation program used in the past to determine partition coefficients.‘7 While we neglected the velocity dependence of the partition coefficient as the velocity slows down during solidification in any particular sample, this is not as crude an approxima: tion as it may at first appear. In fact the velocity varies only slightly as the interface moves through the implanted region, especially for the high-temperature irradiations; the differ-. ence between calculations with constant k and those allowing a velocity dependence is quite small. We believe this&is a negligible source of error in our analysis; good agreement between the simulated and measured impurity depth profiles, even in the tail region, confirms this. In Figs. 3(b)-3(h) the solute profile at several different times is shown, and the progressive accumulation of solute ahead of the interface is evident. When the interface returns to the free surface, the segregated solute forms a sharp surface peak; this profile must be convolved with the RBS detector resolution function in order to allow comparison with the experimental profile. In Fig. 4 the As profile for a (001) sample before’and after irradiation ‘is shown as an example; the simulated profiles were calculated using D,=3X10w4 cm’/s, k=0.41 (dashed line) or k=0.47 (solid line), and the melt depth protie shown in Fig. 3(a); in this case the best-fit (see below) value of k is 0.44. 3 -‘_ Because we fit an entire curve and not just a single point, the fitting procedure uniquely determines not only a best-fit value of k but also a best-fit value of the liquid diffusion coefficient D, if the latter is allowed to vary. Because the value of D, used influences the best-fit value of k somewhat, we used the value of DI that gives the best agreement with J. Appl. Phys., Vol. 76, No. 3, 1 August 1994 where C, is the concentration of the growing solid. (Note that Fig. 3 .represents a worst case for the steady-state hypothesis, as this is the slowest velocity used in the entire study, corresbonding to the largest D,lv width, 30 run, of the diffusive boundary-layer.) Keeping D, fixed, we compared the ratio A/C, after solidification for the data and for the fully aonstea