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Simulations of the ripening of 3D, 2D and 1D objects
Materials Science and Engineering B88 (2002) 112– 117
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Simulations of the ripening of 3D, 2D and 1D objects C. Bonafos *, B. Colombeau, M. Carrada, A. Altibelli, A. Claverie CEMES/CNRS BP4347 F-31055 Toulouse Cedex 4, France
Abstract This paper presents simulations aimed at predicting the kinetic evolution during annealing of nanoparticles and extended defects having different geometry. This versatile model describes the capture and emission of single atoms by clusters. Within this approach, nanoparticles and defects only differ through their formation energies and capture cross-sections. This model has been applied to three particular cases relevant to semiconductor processing (i) spherical Si nanocrystals embedded in a SiO2 matrix; (ii) plate-shaped dislocation loops; and (iii) {311} planar defects in Si. Transmission electron microscopy (TEM) observations have been carried out on each system to measure the evolution of the size-histograms, mean radius and precipitate density during annealing. The simulation results well compared with the experimental data. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Nanoparticles; Extended defects; Ostwald ripening; Simulations
1. Introduction Particles embedded within a solid matrix lower their formation energy by transport of matter from the smaller to the larger members of the distribution. This competitive growth, called Ostwald ripening, leads to a decrease of the total number of particles and to an increase of the average particle size. Since more than a decade, different approaches for the predictive simulations of the precipitation of ion beam synthesised nanoparticles have been successfully developed [1]. Heinig et al. have developed a kinetic three-dimensional lattice Monte-Carlo approach for modelling the nucleation under irradiation and ripening of the nanoparticle population [2]. More complicated phenomena such as inverse Ostwald ripening of precipitates due to collisional mixing can be also predicted by using the lattice Monte-Carlo simulation [3]. The influence of the elastic stress on the ripening can also be addressed by thermodynamic model [4]. A non-local mean field thermodynamic model has recently been developed by Lampin et al. to describe the precipitation of dislocation loops in Si [5]. In this paper, we present simulations aimed at describing the Ostwald ripening during annealing of * Corresponding author. Tel.: +33-56-22-57911; fax: + 33-56-2257999. E-mail address: [email protected] (C. Bonafos).
nanoparticles and extended defects having different geometries. This versatile model describes the capture and emission of single atoms by clusters of every size. Within this approach, nanoparticles and defects only differ by their formation energies and capture efficiencies. This model has been applied to three particular cases relevant to semiconductor processing (i) spherical Si nanocrystals embedded in a SiO2 matrix; (ii) plateshaped dislocation loops in Si; and (iii) {311} elongated defects in Si. Transmission electron microscopy (TEM) observations have been carried out on each system to measure the evolution of the size-histograms, mean radius and precipitate density during annealing. The simulations are compared with the experimental data.
2. Modelling of the ripening This model was initially proposed by Cowern et al. [6] to describe the growth of interstitial clusters in Si. It is assumed that, at the beginning of annealing, all the clusters are of size 2. This assumption is experimentally confirmed by electron spin resonance (ESR) on irradiated Si [7] and will be discussed for the case of Si nanoparticles in SiO2 in Section 3. It is assumed that, since clusters of at least two atoms already exist in the matrix, the number of free Si atoms in the wafer is always much smaller than the number of Si atoms
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C. Bonafos et al. / Materials Science and Engineering B88 (2002) 112–117
contained within the clusters. Therefore, the system immediately reaches a quasi steady-state and the atoms lost by one cluster are captured by another cluster or by the free surface of the wafer. We also assume that the volume fraction occupied by the precipitates is small and thus that the distance between the precipitates is large enough so the precipitates only interact through the mean field between them. We describe the growth of ‘precipitates’ in a matrix by calculating the difference between the capture (Fn ) and emission (Rn ) rates of monomers to/from clusters of size n. The growth rate is classically written as the product of the capture crosssection (An ) of the particle by the net flux of atoms towards it. For a diffusion-limited (b= DiC *dS/dr) i growth, it can be written [8]:
)
dn A dS = Fn −Rn = Di C*A =Di C*i n (S( −Sn ) i n dt dR R = r Reff (1) where Di the coefficient of diffusion and C *i the solid solubility of the ‘impurity’ (equilibrium value) within the host matrix. An is the capture cross-section of the precipitate. Since Reff represents the radial extension of the diffusion field, An /Reff is characteristics of the capture efficiency of the precipitate. S( is the mean supersaturation of impurity atoms within the matrix and S(n) the supersaturation of atoms in equilibrium with a precipitate containing n atoms. S(n) is given by the Gibbs –Thomson equation and can be written, S(n)= exp[Ef(n)/kT]. Ef(n) is the formation energy (the derivative of the total energy dEtotal/dn) of a precipitate containing n atoms. The emission rate (Rn ) is thus an exponential function of the formation energy of the particle and is given by: Rn =Di C*i
An E (n) exp f Reff kT
(2)
The capture rate (Fn ) is a function of the environment of the precipitate and is proportional to S( , the mean supersaturation of impurity atoms between the precipitates: Fn = Di C*i
An S( Reff
(3)
Both the capture efficiency (An /Reff) and the formation energy (Ef) depend on the geometry of the precipitate. The model is based on a set of (n +1) coupled differential equations. The first n equations describe the fluxes of atoms from particles of size n to particles of size n + 1 and n− 1 [9]: dNn =Fn − 1Nn − 1 −FnNn +Rn + 1Nn + 1 −RnNn dt
(4)
Nn is the number of precipitates of size n. This equation drives the evolution of the defects in terms of size-distribution. The last equation describes the ‘free
113
component’ of the impurity atoms i.e. the concomitant evolution of the free atoms supersaturation in dynamical equilibrium with the precipitates [6]. These free atoms can be trapped by a precipitate or by the free surface of the sample where they ultimately annihilate (non-conservative ripening). The surface is thus characterised by a length Lsurf, mean free path for annihilation. The precipitates are located in a plane parallel to the surface at a depth of rp, the projected range of the implant.
S( = Di C*i
% inRnNn n=2
(5)
% (An /Reff)Nn + 1/(Lsurf + rp) n=2
The quantity i is the number of atoms released by the break-up of a cluster (i= 2 for n= 2, i=1 otherwise). This last equation describes the time-evolution of the supersaturation mean-field centred on the precipitates. This system is solved by means of the Runge– Kutta method. Now, for a precipitate with a given geometry, the purpose is to express the capture efficiency (An /Reff) and the formation energy (Ef).
3. Spherical Si nanoparticles embedded in a SiO2 matrix For this system, we also assume that at the beginning of annealing the whole initial supersaturation has already nucleated to form particles of size 2. This is an acceptable assumption as we implant large initial Si excess (a 10 at.%). In this case, the direct interaction between Si atoms among themselves abruptly increases with the formation of SiSi bonds and the formation of small Si clusters is very probable even in the as-implanted state [10]. The case of spherical objects is the easiest one in terms of geometrical description. Indeed, in this case the capture efficiency is simply An /Reff =4yr [11]. After high temperature annealing (\ 1000 °C), the SiO2 layer has relaxed and the formation energy of a precipitate is equal to its interfacial energy and thus is given by: Ef(n)3D =
2|6m 4y6 2m = 2| r 3n
1/3
with
n=
4yr 3 36m
(6) where, n is the number of atoms within a spherical particle of size r. 6m is here the atomic volume of Si and | the interfacial energy per surface unit. We have tested the model against TEM data obtained on populations of Si nanocrystals ion beam synthesised in SiO2 matrix. In these experiments, the initial Si excess implanted in SiO2 was varied from 10 to 30 at.%. The value of diffusivity of Si in SiO2 injected in the simulations has
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thus, the mean field approximation is no more valid (Eq. (1) does not hold). A correction of the model has been proposed [11]. It consists in assuming that the interactions between neighbouring precipitates take place at a distance l( r larger than the actual radius of the precipitate. Under this quite simple assumption, the capture (F *) n and emission (R *) n rates are now functions of the mean distance between the precipitates l( r and are given by [4]:
r r F*3nD = Fn 1+ 2 = Fn 1+ K l( r rmean Fig. 1. About 800 nm thick SiO2 layer implanted with Si at 150 keV with doses of 1017, 2× 1017 and 3 × 1017 cm − 2, corresponding to 10, 20 and 30 at.% of Si in SiO2 and annealed at 1100 °C. With symbols, the time-evolution of the mean radius measured by TEM is plotted (up-triangles, 30 at.%; circles, 20 at.%; and squares, 10 at.%) when annealing at 1100 °C. We have superimposed with plain-line the mean-field simulation. In dotted, dash-dotted and dashed lines we have represented the simulations taking into account interaction effects for the 10, 20 and 30 at.%.
been deduced from TEM measurements [11] (DiC *= i 4.5× 1013 exp(− 2.8/kT) cm − 1 s − 1, Di the diffusion coefficient and C *i the solubility of Si in SiO2). The effect of the surface as a sink has been tested and has been found to be negligible both because of this low diffusivity and of the high densities of precipitates. As a consequence, the term Lsurf +rp has been taken as infinite in the simulation (conservative ripening). As it can be seen in Fig. 1, for the lowest Si excess (10 at.%), the experimental evolution of the mean radius, for annealing time larger than 2 h, is well fitted by these mean-field simulations. The theoretical size-distribution obtained with this model for the 10 at.% sample after 16 h annealing at 1100 °C is shown on Fig. 2(a) and is in good agreement with the TEM stack-histogram. Nevertheless, when the initial Si concentration is very high, ( a10 at.%) the mean field simulation fail to describe the growth of the nanoparticles. The diffusion fields surrounding the precipitates can overlap and
r r R*3nD = Rn 1+ 2 = Rn 1+ K l( r rmean
and
(7)
with K=6 1/3/e 8Y(), Rn and Fn the capture and emission rates in the mean field approximation. rmean is the mean radius, the volume fraction of precipitates and Y()= Y(1/3,8) is the incomplete gamma function. Results of this new set of simulations are shown on Fig. 1 or the 20 and 30 at.% (respectively, in dash-dotted and dashed lines). Now, these simulations are in good agreement with the TEM results, for annealing times larger than 2 h. The theoretical size-distribution obtained with the ‘interaction model’ for the 20 at.% sample after 8 h annealing at 1100 °C is shown in Fig. 2(b) and is in good agreement with the correspondent TEM stack-histogram. It is to be noted that the interaction model completely fails to fit the 10 at.% experimental data.
4. Plate-shaped dislocation loops in Si The faulted dislocation loops are plate-shaped (2D) Si precipitates. They are usually formed by preamorphisation of Si by heavy ions (such as Ge) and subsequent annealing at high thermal budget (\ 900 °C) [12]. As a sink, the dislocation line surrounding the stacking fault can be represented by a torus having a core radius rc (usually taken equal to b, the Burgers vector of the
Fig. 2. About 800 nm thick SiO2 layer implanted with Si at 150 keV with doses of 1017, 2 × 1017 and 3 ×1017 cm − 2, corresponding to 10, 20 and 30 at.% of Si in SiO2 and annealed at 1100 °C. Theoretical size-distributions superimposed to the TEM histogram for (a) the 10 at.%, (1100 °C 16 h) with a mean-field model and (b) for the 20 at.% (1100 °C, 8 h) and interaction model.
C. Bonafos et al. / Materials Science and Engineering B88 (2002) 112–117
115
Fig. 3. (a) (001) Si preamorphised with Ge at 150 keV with a dose of 2 × 1015 cm − 2 and annealed at 1000 °C. Comparison between our simulations (solid and dash lines) and the time-evolution of the experimental mean radius and density; (b), theoretical size-distribution at t= 50 s superimposed to the TEM stack-histogram.
dislocation) and a loop of radius r. The capture efficiency (An /Reff) of these defects can be written An / Reff)2D =4yreff with reff =y(r 2 −r 2c )1/2/ln(8r/rc): yr/ ln(8r/b) for rrc [12]. Its formation energy is now the sum of the interfacial energy and of the fault energy and is given by: Ef(n)2D =
|6m 6mk y6m =| + br b bn
1/2
+
6mk b
measured by TEM. This value is not valid for short time annealing, where other extended defects precede and coexist with the dislocation loops. The number of atoms stored within the loops (not shown) is found to be constant and the ripening is, in this case, conservative.
(8) 5. Elongated {113} defects in Si
with n=(yr 2b/6m)the number of atoms within a diskshaped precipitate of size r. 6m is the atomic volume of Si, k is the fault energy by surface unit and | is the interfacial energy by surface unit, | =vb2 ln(8r/b)/ 4y(1 − 6), with v and 6, respectively, the shear modulus and the Poisson coefficient of Si. The asymptotic limit of the formation energy for very small loops (n] 2) has been taken as 1.3 eV, the value of the formation energy of di-interstitial clusters [13]. We have run our simulation to describe the growth of populations of Frank dislocation loops found after preamorphisation with 2×1015 cm − 2, 150 keV Ge ions and annealing at 1000 °C, under Ar atmosphere. The Si diffusivity inserted in the model (Di C*= 2× i 1025 exp(− 4.56/kT) cm − 1 s − 1) has been again extracted from TEM measurements [12]. The depth position of the defects has been measured on TEM images and is about 175 nm. The recombination length of Si interstitials at the surface (Lsurf) has been taken larger than 1 mm (conservative ripening) [13]. In Fig. 3(a), the simulated time-evolution of the mean radius and of the density are superimposed on the experimental data obtained by TEM. A good agreement is obtained between theory and experiment for the longer annealing time (t\50 s). The size-distribution obtained from the simulations after 50 s annealing is superimposed on Fig. 3(b) on the experimental stack-histogram and shows a good agreement. Nevertheless, a discrepancy is observed in Fig. 3 for short annealing times. This is probably due to the value of the diffusivity, which has been extracted from loop growth kinetics, as
These extended defects are usually formed after low dose Si implants and annealing at temperatures between 700 and 900 °C. They are called {113} defects (or ‘rod-like’ defects), are planar (1D) Si precipitates and are known to dissolve after some minutes of annealing when close enough to a surface. In order to access to the formation energies of the {113}’s, we have calculated the total energy of these defects based on their crystallographical characteristics i.e. taking into account the two edge dislocations plus the two mixed dislocations plus the stacking fault energy which altogether define this defect [8]. They have been assumed to be planar, rectangular and of constant width (4 nm). They contain 20 atoms nm − 1. This energy curve gently tends towards its asymptotical limit at 0.65 eV [14]. Cowern et al. [6] have experimentally shown that, for very small {113} defects (nB 10 atoms), the formation energies oscillates with stable configurations for 4 and 8 atoms and tends, for larger sizes, towards values expected for small {113} defects. Fig. 4(a) shows the overall variations of the formation energy of these defects as a function of their size expressed in atoms. Since little is known of the structure of the clusters, we have reasonably assumed that these very small {113} defects are spherical with a capture area An = 4yr 2. Approaching the capture area of a {113} defect is far more difficult. Gencer and Dunham [15] have proposed that capture only occurs through the edges of these elongated defects. We prefer to assume that the capture area offered by this type of
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Fig. 4. Variation of (a) the formation energies and (b) the capture efficiency of clusters and {113}’s as a function of the number of Si atoms they contain.
Fig. 5. Comparision between our simulations (solid lines) and experimental data (circles) on {113} defects dissolution [16]. (a) Average defect length. (b) Defect density.
defect still increases as they become longer. Thus, our capture cross-section is the sum of three terms, (i) the two cylinders at the width sides; (ii) the two cylinders along the length sides; and (iii) the four hemispheres at the corners. The overall variation of the capture efficiency An /Reff is shown on Fig. 4(b). The exact amplitude and position of the abrupt jump corresponding to the transition from spherical to elongated precipitates have little impact on the results. All simulations were run with the DiC *i =2 ×1023 exp( − 4.52/kT) cm − 1 s − 1 [6]. We have tested the model to describe the famous dissolution of the {113}’s experimentally observed by Eaglesham after 5×1013 cm − 2, 40 keV Si implantation [16]. Fig. 5 shows the comparison between our simulation and a compilation of their experimental results. An excellent fit is obtained by adjusting only (Lsurf + rp) at 80 nm, i.e. taking the surface as almost a perfect sink (annealing under N2 atmosphere). Clearly, these simulations show that dissolution occurs or not depending only on the distance and sink efficiency of the surface. The growth of the defects i.e. the size increase and density decrease they experience during this non-conservative Ostwald ripening, perfectly matches the experimental observations. This evolution strongly depends on the size dependence of the formation energy of the defects and cannot be reproduced without assuming at least one very stable cluster in the 2– 10 atoms range.
6. Conclusion In this paper we have presented simulations aimed at predicting the ripening of populations of precipitates embedded within a host matrix. This versatile model can be applied to objects presenting different geometry, to conservative or no conservative ripening and can take into account interaction effects between neighbouring precipitates when the mean field approximation is no more valid. This model has been applied to three particular cases relevant to semiconductor processing, from spherical Si nanoparticles embedded in SiO2 to 1 and 2D extended defects in Si. In each case, the theoretical evolution of the mean radius, density and size-distribution is in good agreement with the experimental values measured by TEM. References [1] P. Voorhees, M.E. Glicksman, Acta Mater. 3 (1984) 2001. [2] M. Stroebel, K.H. Heinig, W. Moeller, A. Meldrum, D.S. Zhou, C.W. White, R.A. Zhur, Nucl. Inst. Methods B 147 (1999) 343. [3] M. Stroebel, K.H. Heinig, W. Moeller, Mat. Res. Soc. Symp. Proc. 647 (2001) 02.3.1. [4] C.H. Su, P.W. Voorhees, Acta Mater. 44 (1996) 2001. [5] E. Lampin, V. Senez, A. Claverie, J. Appl. Phys. 85 (1999) 8137. [6] N.E.B. Cowern, G. Mannino, P.A. Stolk, F. Roozeboom, H.G.A. Huizing, J.G.M. van Berkum, W.B. de Boer, F. Cristiano, A. Claverie, M. Jaraiz, Phys. Rev. Lett. 82 (1999) 4460.
C. Bonafos et al. / Materials Science and Engineering B88 (2002) 112–117 [7] J.W. Corbet, J.P. Karins, T.Y. Tan, Nucl. Inst. Methods B 457 (1981) 182. [8] A. Claverie, B. Colombeau, F. Cristiano, C. Bonafos, MRS2001 Spring meeting, in press. [9] L. Kampmann, M. Kahlweit, Ber Bunsenges Physik. Chem. 72 (1970) 456. [10] A.F. Leier, L.N. Safranov, G.A. Kachurin, Semiconductors 33 (1999) 380. [11] C. Bonafos, B. Colombeau, A. Altibelli, M. Carrada, G. Ben Assayag, B. Garrido, M. Lopez, A. Perez-Rodriguez, J.R.
[12] [13] [14] [15] [16]
117
Morante, A. Claverie, Nucl. Inst. Meth. Phys. Res. B 178 (2001) 17. C. Bonafos, D. Mathiot, A. Claverie, J. Appl. Phys. 83 (1998) 3008. M. Omri, The´ se de doctorat, Toulouse (1999). B. Colombeau, F. Cristiano, C. Bonafos, A. Claverie, The´ se de doctorat, Toulouse (2001). A.H. Gencer, S.T. Dunham, J. Appl. Phys. 81 (1997) 631. D.J. Eaglesham, P.A. Stolk, H.J. Gossmann, J.M. Poate, Appl. Phys. Lett. 65 (1994) 2305.
www.elsevier.com/locate/mseb
Simulations of the ripening of 3D, 2D and 1D objects C. Bonafos *, B. Colombeau, M. Carrada, A. Altibelli, A. Claverie CEMES/CNRS BP4347 F-31055 Toulouse Cedex 4, France
Abstract This paper presents simulations aimed at predicting the kinetic evolution during annealing of nanoparticles and extended defects having different geometry. This versatile model describes the capture and emission of single atoms by clusters. Within this approach, nanoparticles and defects only differ through their formation energies and capture cross-sections. This model has been applied to three particular cases relevant to semiconductor processing (i) spherical Si nanocrystals embedded in a SiO2 matrix; (ii) plate-shaped dislocation loops; and (iii) {311} planar defects in Si. Transmission electron microscopy (TEM) observations have been carried out on each system to measure the evolution of the size-histograms, mean radius and precipitate density during annealing. The simulation results well compared with the experimental data. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Nanoparticles; Extended defects; Ostwald ripening; Simulations
1. Introduction Particles embedded within a solid matrix lower their formation energy by transport of matter from the smaller to the larger members of the distribution. This competitive growth, called Ostwald ripening, leads to a decrease of the total number of particles and to an increase of the average particle size. Since more than a decade, different approaches for the predictive simulations of the precipitation of ion beam synthesised nanoparticles have been successfully developed [1]. Heinig et al. have developed a kinetic three-dimensional lattice Monte-Carlo approach for modelling the nucleation under irradiation and ripening of the nanoparticle population [2]. More complicated phenomena such as inverse Ostwald ripening of precipitates due to collisional mixing can be also predicted by using the lattice Monte-Carlo simulation [3]. The influence of the elastic stress on the ripening can also be addressed by thermodynamic model [4]. A non-local mean field thermodynamic model has recently been developed by Lampin et al. to describe the precipitation of dislocation loops in Si [5]. In this paper, we present simulations aimed at describing the Ostwald ripening during annealing of * Corresponding author. Tel.: +33-56-22-57911; fax: + 33-56-2257999. E-mail address: [email protected] (C. Bonafos).
nanoparticles and extended defects having different geometries. This versatile model describes the capture and emission of single atoms by clusters of every size. Within this approach, nanoparticles and defects only differ by their formation energies and capture efficiencies. This model has been applied to three particular cases relevant to semiconductor processing (i) spherical Si nanocrystals embedded in a SiO2 matrix; (ii) plateshaped dislocation loops in Si; and (iii) {311} elongated defects in Si. Transmission electron microscopy (TEM) observations have been carried out on each system to measure the evolution of the size-histograms, mean radius and precipitate density during annealing. The simulations are compared with the experimental data.
2. Modelling of the ripening This model was initially proposed by Cowern et al. [6] to describe the growth of interstitial clusters in Si. It is assumed that, at the beginning of annealing, all the clusters are of size 2. This assumption is experimentally confirmed by electron spin resonance (ESR) on irradiated Si [7] and will be discussed for the case of Si nanoparticles in SiO2 in Section 3. It is assumed that, since clusters of at least two atoms already exist in the matrix, the number of free Si atoms in the wafer is always much smaller than the number of Si atoms
0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 0 8 6 1 - 3
C. Bonafos et al. / Materials Science and Engineering B88 (2002) 112–117
contained within the clusters. Therefore, the system immediately reaches a quasi steady-state and the atoms lost by one cluster are captured by another cluster or by the free surface of the wafer. We also assume that the volume fraction occupied by the precipitates is small and thus that the distance between the precipitates is large enough so the precipitates only interact through the mean field between them. We describe the growth of ‘precipitates’ in a matrix by calculating the difference between the capture (Fn ) and emission (Rn ) rates of monomers to/from clusters of size n. The growth rate is classically written as the product of the capture crosssection (An ) of the particle by the net flux of atoms towards it. For a diffusion-limited (b= DiC *dS/dr) i growth, it can be written [8]:
)
dn A dS = Fn −Rn = Di C*A =Di C*i n (S( −Sn ) i n dt dR R = r Reff (1) where Di the coefficient of diffusion and C *i the solid solubility of the ‘impurity’ (equilibrium value) within the host matrix. An is the capture cross-section of the precipitate. Since Reff represents the radial extension of the diffusion field, An /Reff is characteristics of the capture efficiency of the precipitate. S( is the mean supersaturation of impurity atoms within the matrix and S(n) the supersaturation of atoms in equilibrium with a precipitate containing n atoms. S(n) is given by the Gibbs –Thomson equation and can be written, S(n)= exp[Ef(n)/kT]. Ef(n) is the formation energy (the derivative of the total energy dEtotal/dn) of a precipitate containing n atoms. The emission rate (Rn ) is thus an exponential function of the formation energy of the particle and is given by: Rn =Di C*i
An E (n) exp f Reff kT
(2)
The capture rate (Fn ) is a function of the environment of the precipitate and is proportional to S( , the mean supersaturation of impurity atoms between the precipitates: Fn = Di C*i
An S( Reff
(3)
Both the capture efficiency (An /Reff) and the formation energy (Ef) depend on the geometry of the precipitate. The model is based on a set of (n +1) coupled differential equations. The first n equations describe the fluxes of atoms from particles of size n to particles of size n + 1 and n− 1 [9]: dNn =Fn − 1Nn − 1 −FnNn +Rn + 1Nn + 1 −RnNn dt
(4)
Nn is the number of precipitates of size n. This equation drives the evolution of the defects in terms of size-distribution. The last equation describes the ‘free
113
component’ of the impurity atoms i.e. the concomitant evolution of the free atoms supersaturation in dynamical equilibrium with the precipitates [6]. These free atoms can be trapped by a precipitate or by the free surface of the sample where they ultimately annihilate (non-conservative ripening). The surface is thus characterised by a length Lsurf, mean free path for annihilation. The precipitates are located in a plane parallel to the surface at a depth of rp, the projected range of the implant.
S( = Di C*i
% inRnNn n=2
(5)
% (An /Reff)Nn + 1/(Lsurf + rp) n=2
The quantity i is the number of atoms released by the break-up of a cluster (i= 2 for n= 2, i=1 otherwise). This last equation describes the time-evolution of the supersaturation mean-field centred on the precipitates. This system is solved by means of the Runge– Kutta method. Now, for a precipitate with a given geometry, the purpose is to express the capture efficiency (An /Reff) and the formation energy (Ef).
3. Spherical Si nanoparticles embedded in a SiO2 matrix For this system, we also assume that at the beginning of annealing the whole initial supersaturation has already nucleated to form particles of size 2. This is an acceptable assumption as we implant large initial Si excess (a 10 at.%). In this case, the direct interaction between Si atoms among themselves abruptly increases with the formation of SiSi bonds and the formation of small Si clusters is very probable even in the as-implanted state [10]. The case of spherical objects is the easiest one in terms of geometrical description. Indeed, in this case the capture efficiency is simply An /Reff =4yr [11]. After high temperature annealing (\ 1000 °C), the SiO2 layer has relaxed and the formation energy of a precipitate is equal to its interfacial energy and thus is given by: Ef(n)3D =
2|6m 4y6 2m = 2| r 3n
1/3
with
n=
4yr 3 36m
(6) where, n is the number of atoms within a spherical particle of size r. 6m is here the atomic volume of Si and | the interfacial energy per surface unit. We have tested the model against TEM data obtained on populations of Si nanocrystals ion beam synthesised in SiO2 matrix. In these experiments, the initial Si excess implanted in SiO2 was varied from 10 to 30 at.%. The value of diffusivity of Si in SiO2 injected in the simulations has
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C. Bonafos et al. / Materials Science and Engineering B88 (2002) 112–117
thus, the mean field approximation is no more valid (Eq. (1) does not hold). A correction of the model has been proposed [11]. It consists in assuming that the interactions between neighbouring precipitates take place at a distance l( r larger than the actual radius of the precipitate. Under this quite simple assumption, the capture (F *) n and emission (R *) n rates are now functions of the mean distance between the precipitates l( r and are given by [4]:
r r F*3nD = Fn 1+ 2 = Fn 1+ K l( r rmean Fig. 1. About 800 nm thick SiO2 layer implanted with Si at 150 keV with doses of 1017, 2× 1017 and 3 × 1017 cm − 2, corresponding to 10, 20 and 30 at.% of Si in SiO2 and annealed at 1100 °C. With symbols, the time-evolution of the mean radius measured by TEM is plotted (up-triangles, 30 at.%; circles, 20 at.%; and squares, 10 at.%) when annealing at 1100 °C. We have superimposed with plain-line the mean-field simulation. In dotted, dash-dotted and dashed lines we have represented the simulations taking into account interaction effects for the 10, 20 and 30 at.%.
been deduced from TEM measurements [11] (DiC *= i 4.5× 1013 exp(− 2.8/kT) cm − 1 s − 1, Di the diffusion coefficient and C *i the solubility of Si in SiO2). The effect of the surface as a sink has been tested and has been found to be negligible both because of this low diffusivity and of the high densities of precipitates. As a consequence, the term Lsurf +rp has been taken as infinite in the simulation (conservative ripening). As it can be seen in Fig. 1, for the lowest Si excess (10 at.%), the experimental evolution of the mean radius, for annealing time larger than 2 h, is well fitted by these mean-field simulations. The theoretical size-distribution obtained with this model for the 10 at.% sample after 16 h annealing at 1100 °C is shown on Fig. 2(a) and is in good agreement with the TEM stack-histogram. Nevertheless, when the initial Si concentration is very high, ( a10 at.%) the mean field simulation fail to describe the growth of the nanoparticles. The diffusion fields surrounding the precipitates can overlap and
r r R*3nD = Rn 1+ 2 = Rn 1+ K l( r rmean
and
(7)
with K=6 1/3/e 8Y(), Rn and Fn the capture and emission rates in the mean field approximation. rmean is the mean radius, the volume fraction of precipitates and Y()= Y(1/3,8) is the incomplete gamma function. Results of this new set of simulations are shown on Fig. 1 or the 20 and 30 at.% (respectively, in dash-dotted and dashed lines). Now, these simulations are in good agreement with the TEM results, for annealing times larger than 2 h. The theoretical size-distribution obtained with the ‘interaction model’ for the 20 at.% sample after 8 h annealing at 1100 °C is shown in Fig. 2(b) and is in good agreement with the correspondent TEM stack-histogram. It is to be noted that the interaction model completely fails to fit the 10 at.% experimental data.
4. Plate-shaped dislocation loops in Si The faulted dislocation loops are plate-shaped (2D) Si precipitates. They are usually formed by preamorphisation of Si by heavy ions (such as Ge) and subsequent annealing at high thermal budget (\ 900 °C) [12]. As a sink, the dislocation line surrounding the stacking fault can be represented by a torus having a core radius rc (usually taken equal to b, the Burgers vector of the
Fig. 2. About 800 nm thick SiO2 layer implanted with Si at 150 keV with doses of 1017, 2 × 1017 and 3 ×1017 cm − 2, corresponding to 10, 20 and 30 at.% of Si in SiO2 and annealed at 1100 °C. Theoretical size-distributions superimposed to the TEM histogram for (a) the 10 at.%, (1100 °C 16 h) with a mean-field model and (b) for the 20 at.% (1100 °C, 8 h) and interaction model.
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Fig. 3. (a) (001) Si preamorphised with Ge at 150 keV with a dose of 2 × 1015 cm − 2 and annealed at 1000 °C. Comparison between our simulations (solid and dash lines) and the time-evolution of the experimental mean radius and density; (b), theoretical size-distribution at t= 50 s superimposed to the TEM stack-histogram.
dislocation) and a loop of radius r. The capture efficiency (An /Reff) of these defects can be written An / Reff)2D =4yreff with reff =y(r 2 −r 2c )1/2/ln(8r/rc): yr/ ln(8r/b) for rrc [12]. Its formation energy is now the sum of the interfacial energy and of the fault energy and is given by: Ef(n)2D =
|6m 6mk y6m =| + br b bn
1/2
+
6mk b
measured by TEM. This value is not valid for short time annealing, where other extended defects precede and coexist with the dislocation loops. The number of atoms stored within the loops (not shown) is found to be constant and the ripening is, in this case, conservative.
(8) 5. Elongated {113} defects in Si
with n=(yr 2b/6m)the number of atoms within a diskshaped precipitate of size r. 6m is the atomic volume of Si, k is the fault energy by surface unit and | is the interfacial energy by surface unit, | =vb2 ln(8r/b)/ 4y(1 − 6), with v and 6, respectively, the shear modulus and the Poisson coefficient of Si. The asymptotic limit of the formation energy for very small loops (n] 2) has been taken as 1.3 eV, the value of the formation energy of di-interstitial clusters [13]. We have run our simulation to describe the growth of populations of Frank dislocation loops found after preamorphisation with 2×1015 cm − 2, 150 keV Ge ions and annealing at 1000 °C, under Ar atmosphere. The Si diffusivity inserted in the model (Di C*= 2× i 1025 exp(− 4.56/kT) cm − 1 s − 1) has been again extracted from TEM measurements [12]. The depth position of the defects has been measured on TEM images and is about 175 nm. The recombination length of Si interstitials at the surface (Lsurf) has been taken larger than 1 mm (conservative ripening) [13]. In Fig. 3(a), the simulated time-evolution of the mean radius and of the density are superimposed on the experimental data obtained by TEM. A good agreement is obtained between theory and experiment for the longer annealing time (t\50 s). The size-distribution obtained from the simulations after 50 s annealing is superimposed on Fig. 3(b) on the experimental stack-histogram and shows a good agreement. Nevertheless, a discrepancy is observed in Fig. 3 for short annealing times. This is probably due to the value of the diffusivity, which has been extracted from loop growth kinetics, as
These extended defects are usually formed after low dose Si implants and annealing at temperatures between 700 and 900 °C. They are called {113} defects (or ‘rod-like’ defects), are planar (1D) Si precipitates and are known to dissolve after some minutes of annealing when close enough to a surface. In order to access to the formation energies of the {113}’s, we have calculated the total energy of these defects based on their crystallographical characteristics i.e. taking into account the two edge dislocations plus the two mixed dislocations plus the stacking fault energy which altogether define this defect [8]. They have been assumed to be planar, rectangular and of constant width (4 nm). They contain 20 atoms nm − 1. This energy curve gently tends towards its asymptotical limit at 0.65 eV [14]. Cowern et al. [6] have experimentally shown that, for very small {113} defects (nB 10 atoms), the formation energies oscillates with stable configurations for 4 and 8 atoms and tends, for larger sizes, towards values expected for small {113} defects. Fig. 4(a) shows the overall variations of the formation energy of these defects as a function of their size expressed in atoms. Since little is known of the structure of the clusters, we have reasonably assumed that these very small {113} defects are spherical with a capture area An = 4yr 2. Approaching the capture area of a {113} defect is far more difficult. Gencer and Dunham [15] have proposed that capture only occurs through the edges of these elongated defects. We prefer to assume that the capture area offered by this type of
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Fig. 4. Variation of (a) the formation energies and (b) the capture efficiency of clusters and {113}’s as a function of the number of Si atoms they contain.
Fig. 5. Comparision between our simulations (solid lines) and experimental data (circles) on {113} defects dissolution [16]. (a) Average defect length. (b) Defect density.
defect still increases as they become longer. Thus, our capture cross-section is the sum of three terms, (i) the two cylinders at the width sides; (ii) the two cylinders along the length sides; and (iii) the four hemispheres at the corners. The overall variation of the capture efficiency An /Reff is shown on Fig. 4(b). The exact amplitude and position of the abrupt jump corresponding to the transition from spherical to elongated precipitates have little impact on the results. All simulations were run with the DiC *i =2 ×1023 exp( − 4.52/kT) cm − 1 s − 1 [6]. We have tested the model to describe the famous dissolution of the {113}’s experimentally observed by Eaglesham after 5×1013 cm − 2, 40 keV Si implantation [16]. Fig. 5 shows the comparison between our simulation and a compilation of their experimental results. An excellent fit is obtained by adjusting only (Lsurf + rp) at 80 nm, i.e. taking the surface as almost a perfect sink (annealing under N2 atmosphere). Clearly, these simulations show that dissolution occurs or not depending only on the distance and sink efficiency of the surface. The growth of the defects i.e. the size increase and density decrease they experience during this non-conservative Ostwald ripening, perfectly matches the experimental observations. This evolution strongly depends on the size dependence of the formation energy of the defects and cannot be reproduced without assuming at least one very stable cluster in the 2– 10 atoms range.
6. Conclusion In this paper we have presented simulations aimed at predicting the ripening of populations of precipitates embedded within a host matrix. This versatile model can be applied to objects presenting different geometry, to conservative or no conservative ripening and can take into account interaction effects between neighbouring precipitates when the mean field approximation is no more valid. This model has been applied to three particular cases relevant to semiconductor processing, from spherical Si nanoparticles embedded in SiO2 to 1 and 2D extended defects in Si. In each case, the theoretical evolution of the mean radius, density and size-distribution is in good agreement with the experimental values measured by TEM. References [1] P. Voorhees, M.E. Glicksman, Acta Mater. 3 (1984) 2001. [2] M. Stroebel, K.H. Heinig, W. Moeller, A. Meldrum, D.S. Zhou, C.W. White, R.A. Zhur, Nucl. Inst. Methods B 147 (1999) 343. [3] M. Stroebel, K.H. Heinig, W. Moeller, Mat. Res. Soc. Symp. Proc. 647 (2001) 02.3.1. [4] C.H. Su, P.W. Voorhees, Acta Mater. 44 (1996) 2001. [5] E. Lampin, V. Senez, A. Claverie, J. Appl. Phys. 85 (1999) 8137. [6] N.E.B. Cowern, G. Mannino, P.A. Stolk, F. Roozeboom, H.G.A. Huizing, J.G.M. van Berkum, W.B. de Boer, F. Cristiano, A. Claverie, M. Jaraiz, Phys. Rev. Lett. 82 (1999) 4460.
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