Solid phase crystallization under continuous heating: kinetic and microstructure scaling laws

Transcription

1 Solid phase crystallization under continuous heating: kinetic and icrostructure scaling laws J. Farjas * and. Roura RM, Departent of hysics, University of irona, Capus Montilivi, dif. II, 77 irona, Catalonia, Spain *Corresponding author: jordi.farjas@udg.es Abstract he kinetics and icrostructure of solid phase crystallization under continuous heating conditions and rando distribution of nuclei are analyzed. An Arrhenius teperature dependence is assued for both nucleation and growth rates. Under these circustances, the syste has a scaling law such that the behavior of the scaled syste is independent of the heating rate. Hence, the kinetics and icrostructure obtained at different heating rates only differ in tie and length scaling factors. Concerning the kinetics, it is shown that the tended volue evolves with tie according to α = + [ p( κ Ct' )] where t ' is the diensionless tie. his scaled solution not only represents a significant siplification of the syste description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on the kinetic paraeters. Concerning the icrostructure, the istence of a length scaling factor has allowed the grain size distribution to be nuerically calculated as a function of the kinetic paraeters. ACS: 8..Aj, 8..Jt, 5.7.Fh.. Introduction Crystallization of aorphous aterials and other solid state transforations usually involve rando nucleation and growth. Under this assuption, the phase

2 transforation is described by the Kologorov-Johnson-Mehl-Avrai theory (KJMA) [-6]. he transfored fraction, α, is related with the tended transfored fraction, α α, through the so-called KJMA relation: [ ( )] α( t) = p α t. () would be the transfored fraction if grains grew through each other and overlapped without utual interference, i.e.: = t I( u) v α ( t) ( u, t) du, () where I is the nucleation rate per unit volue and v (u,t) is the tended volue transfored at tie t by a single nucleus created at tie u v t ( ( z dz) ( u, t) = σ ). (3) In q. (3), σ is a shape factor (e.g., σ =4π /3 for spherical grains), is the growth rate and depends on the growth echanis [7] (e.g., =3 for three diensional, 3D, growth). u For the particular case of isotheral transforations, where growth and nucleation rates are constant in tie, qs. ()-(3) have an analytical solution: σi + + α ( t) = t. (4) Unfortunately, owing to the dependence of and I on teperature, general act solutions do not ist for non-isotheral conditions. Accordingly, a nuber of published works have developed different theoretical and nuerical approaches to analyze non-isotheral phase transforations within the fraework of KJMA theory [8-3]. Recently, a quasi-act solution of the KJMA theory was obtained under continuous heating conditions [3]. A useful approach to investigate the kinetics and grain orphology consists of finding a scaling law such that the syste behavior is universal. his ethod has been successfully used for the isotheral case [3]. In this case the tie, τ, and length, λ, scaling factors are [33]:

3 ( I ) /( + ) /( + ) τ = and λ =. (5) I When tie is scaled in q. (4), one gets a universal solution (independent of I and ): where, + α ( t') = ( κt'), (6) and t' t / τ is the diensionless tie. σ + κ +, (7) In this paper we will show that a siilar scaling law applies for transforations at a constant heating rate (Sec. ). For a given ratio between the activation energies of I and, there ists an approxiate scaled solution independent of the heating rate. Accordingly, the kinetics and icrostructure for any heating rate can be obtained fro this scaled solution siply by ultiplying the diensionless tie and length values by the corresponding scaling factors. In Sec. 3 we obtain the scaled solution for the transforation kinetics, α (t'), which represents a significant siplification when copared to the quasi-act solution recently published [3]. Apart fro the transforation kinetics it would be very useful to know the resulting aterial s icrostructure because any of the aterial s physical properties are icrostructure-dependent. Surprisingly, work related to the icrostructure obtained under continuous heating conditions is very scarce. As far as we know, only Crespo et al. [34] have addressed this proble for a particular case. In Section 4, and thanks to the siplicity of the scaled solution, an analytical pression is obtained for the average grain size. Additionally, we nuerically analyze the dependence of the grain size distribution on the ratio between the nucleation and growth activation energies. Finally, in Section 5 the liits of therally activated nucleation are analyzed. It will be shown that when the activation energies of nucleation and growth are significantly different, the odel of pre-isting nuclei is ore adequate. A scaled act solution for preisting nuclei is also included. 3

4 . he scaling law In ost practical situations where continuous nucleation takes place, it is possible to assue an Arrhenius teperature dependence for both I and [9-,,5,35]: I = I p( / k ) and = p( / k ), (8) B where and are the respective activation energies for nucleation and growth, k B is the Boltzann constant and is the teperature. Under this assuption, qs. ()-(3) have a quasi-act solution [3]: + α = p kc p, (9) βkb kb ( + )! p( k κ I,, C, p( x) + x Π( + u i ) i= where ( ) and β is the constant heating rate; + B u) du β d / dt. ote that, according to qs. () and (9), α = k C p. Moreover, α (t) in q. (9) is the act solution of the nonisotheral KJMA rate equation βkb kb [3]: where k( ) dα = ( + ) C k( ) ( α) dt / k B ke. / + [ Ln( α) ], () he tie, τ, and length, λ, scaling factors we propose here are inspired by the isotheral case, q. (5). Since I and depend on tie through teperature for constant heating, we define the scaling factors using the values of I and for a particular teperature. A logical choice is the well-defined peak teperature,, i.e., the teperature at which the transforation rate is axiu: 4

5 τ λ I ( ) + I = ( I ) + = = = I + + p kb p ( + ) kb, () where, is given by q. (A) (see Appendix A). Under the approxiation that the crystallization takes place in a relatively narrow teperature range,, () the diensionless growth and nucleation rates becoe (see Appendix A): ' = p κ Ct' and I' = p κct', (3) where t' t / τ is the diensionless tie. ote that the diensionless growth and nucleation rates do not depend on the heating rate, they only depend on the geoetrical factor σ and the ratio / through the constants κ, / and C, respectively. Hence, for a given ratio / the transforation kinetics and grain orphology for different β differ only by the tie and length scaling factors τ and λ, respectively. herefore kinetics and icrostructure can be obtained fro the scaled syste siply by ultiplying the diensionless tie and length by τ and λ, respectively. quation (9) is obtained under the assuptions that the critical nuclei size, the transforation rate at the initial teperature,, and the incubation tie for nucleation are negligible. he first assuption relies on the fact that the average grain size is usually uch larger than the critical nuclei size. hus, this approxiation only affects the very early stages of crystallization. Concerning the second assuption, it is based on the fact that, in well designed perients, is low enough to ensure that the periental results do not depend on. Finally, the istence of a finite incubation tie would odify q. (9). However, as the incubation tie is linked to the 5

6 crystallization kinetics, in any cases an approxiate relation equivalent to (A) is pected and the scaling law is still valid. For instance, we have verified the validity of the scaling law for the case of crystallization of a-si where the activation energy of the incubation tie is siilar to that of crystallization [36]. 3. Scaled approxiate solution for the transforation kinetics In this section, we will find a scaled pression for α (t) (i.e., independent of β ) which virtually coincides with the quasi-act solution. Let us rewrite the nonisotheral KJMA equation [q. ()] for α : dα = ( + ) k( ) Cα +, (4) dt and show that, under the approxiation of q.(), it is scalable with tie. With q. () k() becoes: κ t k ( ) p κc τ τ. (5) Once k() is substituted in q. (4), a scaled equation results: dα κct' = ( + ) κ Ce α + dt', (6) where t' t / τ is the diensionless tie. Integration of q. (6) delivers the scaled solution for α : + [ p( κct' )] α ( t' ) =, (7) after iposing that α = at the peak teperature [3] (i.e., at t =). Finally the scaled solution for the transfored fraction is obtained after cobining qs. () and (7): + ( [ p( ')] ) α ( t' ) = p κct. (8) Alternatively, q. (7) can be obtained after integration of qs. ()-(3) once the diensionless rate constants and I [q. (3)] are substituted there. 6

7 o verify the validity of the tie scaling and the proposed scaled solution for α (t'), we have calculated α (t) and the transforation rate, d α / dt, fro the crystallization of aorphous silicon at two tree heating rates of β =.5 and K/in which can be considered the lower and upper liits for ost perients. hen, tie and transforation rate are scaled for these particular heating rates and copared to the scaled solution α (t') given by q. (8). and its derivative d α ' / dt. he results are plotted in Fig.. he calculation of α (t) has been done using a nuerical ethod which delivers the act solution of qs. ()-(3) [3]. All calculations described in this section have been done for isotropic 3D crystallization ( = 3, σ = 4 / 3π ) of aorphous silicon (I and, detailed in able I). he coincidence for both heating rates and the scaled solution is cellent. he discrepancies in the transfored fraction for both heating rates are lower than 3-4 despite the large shift in peak teperatures fro 57.6 to ºC and the very different tie-scaling factors of.7 5 and 3. s for β =.5 and K/in, respectively. It is worth noting that, although the two tie scales differ by ore than five orders of agnitude, the scaling law is still valid. In fact, the usefulness of the approxiation ade in q. () is based on the ponential dependence of the growth and nucleation rates on teperature, i.e., the Arrhenius dependence. his strong dependence on teperature liits the crystallization process to a narrow teperature range even when the heating rate is as low as.5 K/in. he accuracy of the scaled solution can be tested for a wide range of / values through the full width at half axiu (FWHM) of the transforation rate peak. Fro the scaled solution, q. (8), the calculation of the FWHM, straightforward and results in (see Appendix B): ΔtHM = τ ( + )κ C Δ thm, is. (9) 7

8 In Fig. we see that this value departs only slightly fro the act one when is very different fro, the discrepancy being higher for <<. Let us highlight the foral siplicity of the scaled solution [q. (8)] when copared with the quasi-act solution [q. (9)]. his siplicity has been reached without any significant loss of accuracy in the range of transfored fractions of practical interest (say,. < α <. 99 ). In contrast with the isotheral case [q. (6)] we see that the scaled solution for continuous heating is not universal (independent of and I) but depends on the particular value of the ratio between the activation energies and (through the paraeter C). his dependence has iportant consequences for the icrostructure developent, which will be analyzed in the nt section. 4. rain size orphology In this section we will verify the length scaling law proposed in Sec. [q. ()] and analyze the dependence of the final icrostructure on the kinetic paraeters. o characterize the final icrostructure we have calculated the grain size distribution, where the grain size of an individual grain, i, is defined as: 3v 3 i ri, () 4 π and v i is the actual grain volue. he nuerical algorith used for the calculation of the grain size distribution is described in [3]. o verify the length scaling law we have calculated the grain size distributions fro the crystallization of aorphous silicon at two tree heating rates of β =.5 and K/in. hen the grain size distributions are scaled by dividing the grain size by the length scaling factor λ [q. ()]. he results are plotted in Fig. 3. he length 8

9 scaling factors are.3 and.7 icrons for β =.5 and K/in, respectively. he discrepancies between both distributions are within the nuerical accuracy of the algorith. hus, once scaled, the grain size distributions do not depend on the particular heating rate and erge in one single distribution. Indeed, as plained in Sec. the scaled grain size distribution only depends on the / ratio. o characterize the grain size distributions fro the nuerical siulations, we have calculated the average grain size, < r >, the ean grain radius, r and its standard deviation, σ r, defined as: < 3 r > 3 r i, r r i i= i σ r i, () and ( r r ) i where is the final nuber of grains. he average grain size after 3D crystallization as a function of / is reported in Fig. he grain size distribution for a series of / ratios below or above unity are plotted in Figs. 4(a) and 4(b), respectively, and, finally, the values of r and σ r are detailed in Fig Analytical solution for the average grain size Before looking at the grain size distributions in detail, let us take advantage of the scaled solution for α [q. (8)] which allows us to find an analytical pression for the average grain size. According to Ref. [3], < r > can also be calculated fro the nuber of grains fored after coplete crystallization: where can be obtained fro α (t) : r V σ < >=, () ( ( u) ) = V I( u) α du. (3) In Appendix B, the scaled solution has been substituted in q. (3) and an analytical pression for < r > has been obtained: 9

10 < r > κ = Γ λ C ( ) +, (4) where Γ is the gaa function [37]. It has been plotted in Fig. where it can be copared with the act values obtained for the two tree heating rates of.5 and K/in. It can be concluded that q. (4) gives the average grain size with very good accuracy unless / <<, where discrepancy is below %. 4. Dependence of the grain size distribution on / Let us now focus our attention on the grain-size distributions of Fig. 4. When / =, the distribution coincides with the isotheral distribution obtained in ref. [3]. his is as pected because and I have the sae teperature dependence and their ratio is constant. Consequently, the teperature has an effect on the rate at which the transforation proceeds but not on the icrostructure. In this particular case, the grain size distribution is independent of the theral history. For, the grainsize distribution departs progressively fro the isotheral one, and when the ratio / is far fro unity, the distributions have characteristic shapes which can be readily understood. When / <, during the first stages of the transforation, nucleation doinates over growth. Consequently, the nuclei density is higher when copared to the isotheral case. hus, when / diinishes, the average grain is reduced (Fig. 3). Concerning the bell-shaped grain-size distribution for / << [Fig. 4(a)], it can be plained by the fact that ost nuclei are fored at a teperature range where they are not allowed to grow significantly. his eans that they grow together at higher teperatures leading to a narrow distribution of grain sizes. In Fig. 5 we see that, indeed, the standard deviation diinishes drastically for / <<.

11 In contrast, when / > during the first stages of crystallization, growth doinates and the nucleation rate increases progressively as crystallization proceeds. Since the tie left for growing is lower for the nuclei that appear later, the density of sall grains will be higher than for larger grains [as shown in Fig. 4(b)]. In fact, fro Fig. 4(b), one can infer that the slow initial nucleation results in the foration of a sall quantity of large grains. Moreover, this initial low nucleation rate results in a reduction of the transforation rate which is anifested in Fig. as a onotonous increase of t HM Δ with /. In addition, nucleation takes place during a longer tie interval and, consequently, the grain size distribution contains larger grains as Fig. 5). / increases (see Finally, it is worth noting that when / increases, the delayed nucleation results in an increased nuber of phanto nuclei []. Soe authors have claied that, in clear disagreeent with Avrai s assuption, they ust be cluded in the calculation of α. For the siulations carried out in this work, the ratio between phanto nuclei and real grains increases fro. when / =. to.98 when / =. In all these siulations, the agreeent between the transfored fractions calculated fro the icrostructure and fro the nuerical solution of Avrai s odel [q. ()] is cellent. 5. Liits of therally activated nucleation Fro a foral point of view, the analysis given in Sec. -4 for continuous nucleation can be applied for any arbitrary value of the ratio /. In the following, we will argue that, when this ratio is far fro unity, the aterial will follow the kinetics of pre-isting nuclei (described in Appendix C), when nucleation is not therally activated but a constant density of nuclei, n, already ists before they grow. When <<, nucleation takes place early and, eventually, its rate ay vanish before the onset of particle growth (site saturated nucleation [,38]). Consequently the nuclei

12 grow as if they were preistent to the growth stage. On the other hand, when >>, hoogeneous nucleation is less viable. In ost practical situations, when >>, nucleation is catalyzed by inclusions and the container walls [35,39,4], i.e., it is virtually ipossible to prevent heterogeneous nucleation. In this case, again, one can also apply the odel of preisting nuclei provided that nuclei are randoly distributed [4,4]. In the case of heterogeneous nucleation, the latter condition can be jeopardized by a particular distribution of the ternal nucleation sites. However, in several practical situations and in the case of heterogeneous nucleation localized at the container walls it is possible to assue that nucleation sites are randoly distributed. hus, the proble can be solved by assuing an initial surface density of preisting nuclei [43-45]. A universal scaled solution can also be obtained for the case of pre-isting nuclei (see Appendix C). Calculations, like those done for continuous nucleation in Sec. 4, show that the kinetics is scalable with a siilar accuracy ( Δα < 4. between.5 and K/in). 6. Conclusions In this paper, we have shown that, when tie and length are properly scaled, the description of solid state crystallization under annealing at a constant heating rate becoes very siple. For a given aterial, the tie dependencies of the transfored fraction obtained at different heating rates erge into one single scaled solution. he accuracy of this siplified kinetics has been tested against act nuerical solutions of the KJMA equations. Within the range of α values of interest (. < α <. 99 ), the agreeent is cellent. Apart fro a geoetrical paraeter, this scaled solution depends only on the particular growth and nucleation rates through one single paraeter: the ratio of activation energies, /.

13 In addition to the crystallization kinetics, it has been shown that the grain size distribution can be scaled with a characteristic length. Again, for a given / ratio, the scaled distributions do not depend on the particular heating rate. Fro the scaled kinetic equation, it has been possible to obtain the analytical dependence of the average grain size on /. he grain size obtained after isotheral crystallization coincides with that obtained after continuous heating only when / =. Although sall deviations are predicted for iportant changes in the aterial s properties. <<, they are probably not high enough to induce he scaled distributions have been calculated for a series of / values ranging fro. to with a nuerical algorith which siulates the icrostructure developent. It has been shown that, for <<, the distribution of grain sizes is quite narrow around the average value whereas, for diinishes onotonically as the radius increases. >>, the density of grains For the sake of copleteness, the kinetics and grain size distribution have been calculated for the case of preisting nuclei. It has been shown that it is also possible to find appropriate tie and length scaling factors. Finally, our analysis relies on the fact that the transforation is therally activated and, consequently, that it takes place in a narrow teperature range. Indeed, any real transforations are therally activated, thus we believe that our approach can by applied to a large nuber of transforations. Acknowledgents his work has been supported by the Spanish rograa acional de Materiales under contract nuber MA

14 4 Appendix A. Diensional scaling law for the case of continuous nucleation he tie, τ, and length, λ, scaling factors are defined in q. () where the peak teperature,, is given by: = dt d α. (A) Substitution of qs. (9) and () in q. (A) leads to the value of as the solution of an algebraic equation: B k B e C k k = β (A) he scaled syste is universal (independent of β ) provided that the diensionless growth and nucleation rates do not depend on β. Actually, the diensionless growth and nucleation rates are: = = B B k k e I I e ' and ' τ λ λ τ. (A3) Unfortunately, the result does depend on β through the relationship between and t. o suppress this dependence we will suppose that the teperature range where the crystallization takes place is relatively narrow:... + =. (A4) Furtherore, selecting a tie scale requires selecting a scale factor as well as a tie origin. his origin ust correspond to an equivalent state for any diensional syste (any particular value of β ). Here again the natural choice is the tie at which the transforation rate is axiu: t t β β = + =. (A5) hen, substitution of qs. (A5) and (A) into q. (A4) gives: B k B t C k e C k k t B τ κ β β = =. (A6) hus, the diensionless growth and nucleation rates becoe:

15 where t' t / τ is the diensionless tie. ' = p κ Ct' and I ' = p κct', (A7) Appendix B. Analytical calculation of < r > and Δ thm for the scaled syste he total nuber of grains is given by q. (3). Cobining qs. (3), (8) and (3), one gets: λ where (8): [ ( [ ] ) + p κ Ct' ] + = t e ' e dt' = κ C ( + ) κ C + s ( + ) e s ds = Γ ( + ) κ C ( + ) (B) t' s e. Finally, < r > is obtained fro substituting q. (B) into q. (): < r > κ = Γ λ C ( ) + For the calculation of dα t') = ( + ) κc p dt' and the transforation at the axiu is: consequently, where. (B) Δ thm we first calculate the transforation rate fro q. + [ ( p[ κct' ]) ]( p[ κct' ]) ( + dα dt' ( t') t' = Δt τ = ( + ) κce HM = t ' ' t, (B3), (B4), (B5) dα( t') dt' t ' dα( t') dα( t') = = = ( + ) κce dt' dt' t ' substituting q. (B6) into (B3) one gets t' =, (B6) e = e x ( x), x + ( p[ Ct' ]) κ. (B7) 5

16 quation (B7) has two solutions: x = and x = By substituting these solutions and q. (B) one obtains: ΔtHM x = ln = τ ( + ) κc x ( + ) κc. (B8) Appendix C. Universal scaled solution for the case of pre-isting nuclei When nucleation is copleted prior to crystal growth, the kinetics of the transforation is sipler because it is clusively governed by the growth rate: where ( ) k' α = k p k B kb β ', (C) σ n and n is the pre-isting nuclei density. hen, the corresponding peak teperature is given by: β k' k B = e k B. (C) On the other hand, according to the scaling law for the isotheral case [3,46], the tie and length scaling factors are defined as: / ( n ) = ( n ) / ' e and ' n τ = λ =, (C3) = and the diensionless growth rate and nucleation density are: τ ' ' = e λ' k B k B 3 and n' Iλ' = /. (C4) Supposing again that the teperature range where the crystallization takes places is relatively narrow, one gets [ '] and a uch sipler pression results for α : ' = p σ t, (C5) ( p[ t' ]) α = σ. (C6) 6

17 For pre-isting nuclei, the grain-size distribution f(r) coincides with the distribution obtained under isotheral conditions. In [3] it has been shown that, for 3D, it can be fitted to a aussian distribution (the square correlation coefficient is.9998): where μ =.693λ ' and σ =.89λ '. ( r μ ) σ f ( r) = e, (C7) πσ For the case of pre-isting nuclei < r > is obtained directly fro q. (): < r > = λ ' σ. (C8) For the calculation of Δ t HM we follow the sae procedure developed in Appendix B for continuous nucleation. First we calculate the transforation rate fro qs. () and (C6): ( p[ σ t' ]) ( p[ t' ]) dα( t') dt' he transforation rate at the axiu is (t =): and = ( ) σ p σ dα( t') dt' dα dt' ( t') t' = = σ e dα( t') dα( t') = = = σ e t ' dt' ' ' t dt t' = substituting q. (C) into (C9) one gets ( p[ t' ]). (C9), (C), (C) e x = e ( x), x σ. (C) quations (C) and (B7) are identical so they have the sae solutions. By substituting these solutions one obtains: Δt τ x ln HM = = ' σ x σ. (C3) 7

18 able I. xperiental paraeters of aorphous silicon nucleation and growth rates [36]. Activation energy, 5.3 ev ucleation reponential ter, I.7 44 s - -3 Activation energy, 3. ev rowth reponential ter,. 7 s - 8

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23 ..8 Continuous nucleation.4.. α τ dα/dt t/τ Figure. ransfored fraction and transforation rate versus tie for 3D crystallization of aorphous silicon under continuous heating. Heating rates:.5 K/in (squares) and K/in (triangles). ie and transforation rates have been scaled according to q. (). he solid line is the solution of the scaled syste q. (8). 3

24 .8..8 <r>/λ.6.6 Δt HM /τ.4.4. / Figure. Average grain size, < r >, and the FWHM of the transforation rate evolution, Δ thm. he solid and dashed lines have been calculated fro the analytical solution of the scaled syste, qs. (4) and (9) respectively. he discrete points are the result of a siulation of the crystallization process and can be considered act (squares:.5 K/in; triangles: K/in). 4

25 oralized nuber of grains.6.4. Continuous nucleation K/in.5 K/in r/λ Figure 3. Final grain radius distribution for 3D crystallization of aorphous silicon under continuous heating. Heating rates:.5 K/in (black bars) and K/in (grey bars). he radius has been scaled according to q. (). 5

26 5 (a). noralized nuber of grains / =..5. r/λ. / = (b) noralized nuber of grains r/λ Figure 4. Final grain radius distributions for several values of / resulting fro the siulation of the crystallization process at K/in. Owing to the length scaling law, these scaled distributions are alost independent of the heating rate. 6

27 r/λ.45.3 σ r /λ..4.. / Figure 5. Mean grain radius, r, and its standard deviation, calculated fro the grain size distributions of Fig. 4. σ r, versus / 7