Modeling of Threading Dislocation Density Reduction in Heteroepitaxial Layers

Transcription

1 A. E. Romanov et al.: Threading Disloation Density Redution in Layers (II) 33 phys. stat. sol. (b) 99, 33 (997) Subjet lassifiation: 6.72.C; Ln; S5.; S5.2; S7.; S7.2 Modeling of Threading Disloation Density Redution in Heteroepitaxial Layers II. Effetive Disloation Kinetis A. E. Romanov 2 (a), W. Pompe (a), G. Beltz (b), and J. S. Spek () (a) Arbeitsgruppe Mehanik Heterogener Festkorper, Max Plank Gesellshaft, Hallwahsstr. 3, D-0069 Dresden, Germany (b) Mehanial Engineering Department and () Materials Department, University of California, Santa Barbara, CA 9306, USA (Reeived July 5, 996) The ªeffetive kinetisº of threading disloations (TDs) in growing epitaxial films is developed on the basis of the rystallographi and geometrial onsiderations from Part I of the paper. A system of oupled first-order nonlinear differential equations for twenty-four families of TDs is derived and solved for several initial onditions for TD densities. Numerial solutions demonstrated two general types of asymptoti behavior with inreasing film thikness: linear derease or saturation of total TD density. This behavior agrees with experimental data on TD redution in mismathed homogeneous buffer layers.. Introdution In Part I of this paper [] a simple geometrial approah was developed to explain the experimentally observed =h saling behavior for the threading disloation (TD) density with the film thikness h for mismathed epitaxial buffer layers [2 to 4]. The approah was based on the idea of effetive motion of inlined TDs along the film surfae as a result of film growth and subsequent reations between TDs. These reations are annihilation, fusion, or sattering of TDs and are assoiated with a harateristi distane for initiating a reation: r A ;r F, and r S, respetively. Additionally, in Part I, the rystal geometry of TDs and the possible reations between them was developed for (00) epitaxial growth of (00) semiondutor films. It was argued that for this geometry twenty-four unique Burgers vetor/slip plane ombinations exist (whih are inluded in Table I. 3 : In this part of the paper, we treat the derivation and numerial solutions of the systems of oupled differential equations for ªkinetisº of twenty-four TD families that is appropriate for the (00) growth of f... semiondutor films. It will be demonstrated that there exist two fundamental types of solutions for the oupled series of equations. The first type orresponds to the =h saling dependene and desribes balaned TD Part I see phys. stat. sol. (b) 98, 599 (996). 2 Permanent address: A. F. Ioffe Physio-Tehnial Institute, Russian Aademy of Sienes, St. Petersburg, Russia. 3 We will use the Roman numeral ªIº to denote tables, figures, and equations from Part I of this paper. Furthermore, the often referened table in Part II will be designated as ªTable II.º so as to distinguish it from Table I..

2 34 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek distributions, that is, when the net Burgers vetor ontent of all TDs is zero. The seond type of solutions is appliable to non-balaned TD populations and predits saturation behavior for TD densities at large film thiknesses. The saturation behavior orresponds to the experimental data presented in Part I. 2. Derivation of the Governing Equations for TD Redution In the development of equations for TD kinetis, we onsider the example of TDs belonging to the family designated ªº in Table I. with density r, that is, for disloations with Burgers vetor b ˆ a=2 0Š and line diretion l approximately parallel to [23Š. First, we write the terms responsible for diminishing disloation density. In the general ase, all reations of the type -j for j ˆ! 24 (see (I.22)) an ontribute to the proess of r dereasing, dr dh ˆ P24 K j r r j ; j ˆ where r j is the density of TDs in the j-th family. K ij are the ªkinetiº oeffiients that desribe the ªrateº of reations between disloations from families i and j. These oeffiients are uniquely determined by the geometry of mutual motion of interating disloations and by harateristi distanes, the interation radii r I. The general expression for K ij was found in Part I (see (I.24)). The possible reations between TDs are annihilation, fusion, and sattering. Here we will take into aount only the first two reations, with harateristi radii r A and r F, respetively. Details of the formalism assoiated with sattering reations are presented in Appendix A. As derived in Part I and shown in Table I., TDs from family an only have annihilation reations with TDs from family 4. TDs from family an have fusion reations with TDs from families, 2, 5, 6, 9, 20, 23, and 24. Therefore, only reations with these families will ontribute to the proess of diminishing the density of TDs from family, i.e., dr dh ˆ K ;4r r 4 K ; r r K ; 2 r r 2 K ; 5 r r 5 K ; 6 r r 6 K ; 9 r r 9 K ; 20 r r 20 K ; 23 r r 23 K ; 24 r r 24 : 2 It follows from the analysis of Table I. that TDs from family ªº may be generated only as the produt of fusion reations between TDs from the following pairs of families: 3±7, 3±8, 4±7, 4±8. Therefore, for prodution of TDs from family as a result of fusion, one an write dr dh ˆ K 3; 7r 3 r 7 K 3; 8 r 3 r 8 K 4; 7 r 4 r 7 K 4; 8 r 4 r 8 : 3 Combining the generation of TDs as a result of fusion (3) with TD redution as a result of annihilation and fusion (2) we arrive at the governing differential equation for the density of TD family, r, in the absene of sattering reations dr dh ˆ K ;4r r 4 r K ; r K ; 2 r 2 K ; 5 r 5 K ; 6 r 6 K ; 9 r 9 K ; 20 r 20 K ; 23 r 23 K ; 24 r 24 K 3; 7 r 3 r 7 K 3; 8 r 3 r 8 K 4; 7 r 4 r 7 K 4; 8 r 4 r 8 : 4

3 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 35 Similar equations an be readily derived for all twenty-four disloation families. These twenty-four equations will inlude in the most general ase 252 oeffiients K ij. These oeffiients an be diretly alulated from (I.24), however, some physial arguments and assumptions permit us to restrit the number of independent oeffiients by taking into aount the symmetry of the problem. First of all, by definition, K ij ˆ K ji. Other relations between K ij follow from Fig. I.4b where the rystallography of TDs for the ase of (00) epitaxy is presented. For example, we onlude that K 2; ˆ K 6; 5 ˆ K 4; 9 ˆ K 5; 2 ˆ...; K ;4 ˆK 2;3 ˆK 9;2 ˆ K 0; ˆ...; K 2; 24 ˆ K 22; 23...; 5 and so on. For the sake of simpliity, we assume an idential value for the oeffiients K ij ˆ D for all fusion reations and that for the annihilation reation K 4 ˆ C. dr dh ˆ Cr r 4 Dr r r 2 r 5 r 6 r 9 r 20 r 23 r 24 D r 3 r 7 r 3 r 8 r 4 r 7 r 4 r 8 : 6 We an rewrite (6) by inluding the oeffiients r A and r F, dr dh ˆ r Ar r 4 r F dr r r 2 r 5 r 6 r 9 r 20 r 23 r 24 r F d r 3 r 7 r 3 r 8 r 4 r 7 r 4 r 8 ; 7 where and d are dimensionless parameters that depend on the relative disloation motion with hanging film thikness. For example, it follows from (I.24) that for annihilation reations p (in the approximation that disloations have h2i line diretions) ˆ 2 2 (for more realisti line diretions, suh as h23i; will have a smaller value). For fusion reations, we may also onsider situations in whih d. Finally, we an represent (7) in dimensionless form, d~r d h ~ ˆ ~r ~r 4 k~r ~r ~r 2 ~r 5 ~r 6 ~r 9 ~r 20 ~r 23 ~r 24 k ~r 3 ~r 7 ~r 3 ~r 8 ~r 4 ~r 7 ~r 4 ~r 8 ; 8 where ~ h ˆ h=r A ; ~r i ˆ r i r 2 A, and k ˆ r F=r A and now d ˆ. Using the same approximations and designations, we an write the equations for the rest of the TD densities (see Appendix B). The derived system of oupled ordinary nonlinear differential equations will serve as the basis for the analysis of TD kinetis in growing films under different initial onditions. 3. Numerial Solutions and Interpretation The numerial solutions of the system of oupled differential equations for TD densities, (B) to (B24), an be obtained using standard mathematial software. We will illustrate the possible behavior of TD densities by several examples of solutions for different initial onditions whih are listed in Table II.. Two groups of possible initial onditions should be treated separately. The parameter responsible for this subdivision is the net Burgers

4 36 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek Table Initial TD densities for numerial solutions to oupled differential equations. The first eight ases (B to B8) orrespond to zero net Burgers vetor ontent B ˆ 0. The unbalaned ases (N to N8) orrespond to a net Burgers vetor ontent B 6ˆ 0 ~r 0 ~r 0 2 ~r 0 3 ~r 0 4 ~r 0 5 ~r 0 6 ~r 0 7 ~r 0 8 ~r 0 9 ~r 0 0 ~r 0 ~r 0 2 ~r 0 3 ~r 0 4 ~r 0 5 ~r 0 6 ~r 0 7 ~r 0 8 ~r 0 9 ~r 0 20 ~r 0 2 ~r 0 22 ~r 0 23 ~r 0 24 B B B B B B B B N N N N N N N N

5 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 37 vetor ontent B of the TDs in the film, defined as B ˆ P24 i ˆ ~r 0 i b i ; where ~r 0 i are the normalized TD densities for the initial thikness h 0, and b i is the Burgers vetor of a TD from the i-th family. We note that the net Burgers vetor ontent B of the TDs is a onserved vetor that an be distributed amongst several speifi TD families. 3. Balaned TD densities (B = 0) 3.. Relation between growth struture and initial onditions The first group of initial onditions (ases B through B8 in Table II.) orrespond to ªbalanedº ases; i.e., the TDs in the film have no net Burgers vetor ontent B ˆ 0. This an happen when the families with opposite Burgers vetors b and b have equal initial densities (ases B through B7). Speial attention has to be given to ase B7 for whih TDs with the same Burgers vetor, but from different slip systems, have equal initial densities. The other speial example is presented for ase B8 where there is no equilibrium between some b and b pairs ~r 0 ˆ ~r0 2 6ˆ ~r0 3 ˆ ~r0 4 ; ~r0 3 ˆ ~r0 4 6ˆ ~r0 5 ˆ ~r0 6 ; and ~r 0 8 6ˆ ~r0 9 but the global ondition B ˆ 0 is maintained. Cases B and B2 in Table II. orrespond to the presene of populations of TDs with inlined Burgers vetors with respet to the film/substrate interfae and zero populations of TDs with Burgers vetors parallel to the film/substrate interfae. Suh a situation an result from surfae nuleated TDs during layer-by-layer film growth. In ase B, all TD populations have the same initial densities that orrespond to a uniform generation of disloations by surfae soures. In ase B2, populations 5! 8 have higher densities than other TDs. This may happen, for example, as a result of heterogeneous stresses, or an imperfet biaxial stress state. For ases B3 and B4, initially TDs with their Burgers vetor parallel to the film/substrate interfae are present. This disloation onfiguration may be a onsequene of oalesene of impinging islands. Case B5 is a ombination of ases B and B3; similarly, ase B6 is a ombination of ases B2 and B4. Regarding ase B7, we propose that this initial TD onfiguration may be formed due to an artifially-indued imbalane between slip systems. Finally, ase B8 represents a situation that does not orrespond to any partiular TD formation mehanism, but may be of interest from an aademi point of view Numerial solutions Examples of numerial solutions of the equations for TD kinetis in growing films for the balaned ases are presented in Fig. and 2 (for these figures, the values of the parameters ˆ 2 and k ˆ were used for the numerial solution of the system of equations (B) to (B24)). All families follow a =h dependene for large film thiknesses h and onverge to a single urve. However, at the initial stage of film growth the behavior of speifi TD densities may be different. For example, ase B (see Fig., upper panel) orresponds to an idealized situation where all TDs are generated by surfae nuleation of half-loops. For these initial onditions, the TD densities r through r 6 derease monotonially with inreasing film thikness. In ontrast, TD densities r 7 through r 24 9

6 38 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek Fig.. Threading disloation densities for balaned Burgers vetor ontent B ˆ 0 orresponding to surfae nuleated TDs (ase B from Table II.) (upper panel) and to TDs generated during island oalesene (ase B4 from Table II.) (lower panel). Left olumn: linear±linear sale; right olumn: log±log sale. The speifi TD designation is indiated in the figure and in Table I.; ˆ 2 and k ˆ show a rapid initial inrease and then also begin to derease with inreasing film thikness. The initial inrease in TD densities r 7 through r 24 is due to fusion reations between primary TDs (surfae nuleated TDs). With inreasing thikness, the densities r 7 through r 24 reah a maximum value at ~ h 5, and at a thikness of ~ h 00, the TD densities onverge for all families. An idealized ase of island growth, where the TDs are generated as a result of island oalesene, is shown in Fig., lower panel. In this balaned ase (B ˆ 0, families 7 through 20 have a smaller initial density than families 2 through 24. The differene in densities diminishes with inreasing thikness as a result of higher initial onentration of TDs in families 2 through 24 and thus a higher initial rate of annihilation. The mixed ase, in whih all types of TDs are initially present in equal densities (for example ase B5 shown in Fig. 2, upper panel), demonstrates the idential monotoni derease in density for all families. Families through 24 have been treated equivalently in (B) to (B24), and thus with equal initial onditions, the system maintains this equivalene in TD densities. However, if the symmetry of the initial onditions is bro-

7 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 39 Fig. 2. Threading disloation densities for balaned Burgers vetor ontent B ˆ 0 orresponding to TDs for ases B5 (upper panel) and B8 (lower panel) from Table II.. These TDs orrespond to those for a mixed growth mode. Left olumn: linear±linear sale; right olumn: log±log sale. The speifi TD designation is indiated in the figure and in Table I.; ˆ 2 and k ˆ ken, as in ase B8 (Fig. 2, lower panel), then there is a ªtransition thiknessº neessary for all of the families to onverge to the same value for inreasing h. Similar to ase B, some families show an inreasing initial TD density as a result of fusion reations Thikness dependene of total TD density For the balaned ases, the total TD density ~r T ˆ P24 ~r i always obeys the =h dependene at large film thiknesses. This an be easily seen from the log±log plot in Fig. 3a. i ˆ Note that in this plot, the ordinate is the logarithm of the relative TD density, i.e., ~r T =~r 0 T, where ~r0 T ˆ r0 T r2 A and r0 T is the initial total TD density, and thus the offset of the urves at large h is simply log ~r 0 T, whih will be disussed shortly. From the analysis of Fig. 3a, and in lose analogy with (I.0), we propose that the behavior of the total TD density r T in balaned ases may have the following dependene: r T ˆ ~r T r 2 ˆ A K T h ^h T

8 40 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek Fig. 3. Logarithmi plots of the total relative threading disloation densities as a funtion of normalized film thikness for balaned ases B ˆ 0 : a) Dependenies for different initial onditions given in Table II.; ˆ 2 and k ˆ. b) Dependenies for different values of the relative radius for fusion k; ˆ2 (ase B from Table II.). The k values are indiated in the figure. Solid lines represent solutions obtained with the help of (0). ~K T values are indiated in the figure. ) Dependenies for different values of k for ase B5 (+) and normalized ase B (solid lines) suh that the total initial TD density ~r 0 T ˆ 0:6 for both ases. The geometri parameter ˆ 2 was used for these solutions log ~r T ~r 0 ˆ log ~K T h ~ h ~ 0 Š; T 0 where K T is the global TD ªkineti reationº oeffiient (and ~K T ˆ K T r 0 T r A is its normalized value), ^ht ˆ = K T r 0 T h 0, and ~h 0 ˆ h 0 =r A. Clearly, (0) orretly predits the behavior of the total TD density for h ^h; the value of the oeffiient ~K T an be found from the extrapolation of the linear parts of the urves in Fig. 3a to the value at log h ~ ˆ 0. To onfirm the general dependene of (0) for any film thikness, we first need to examine other features of the solution of the general system of differential equations. or

9 Threading Disloation Density Redution in Heteroepitaxial Layers (II) Competition between annihilation and fusion reations We reiterate that the results presented in Fig. and 2, and orrespondingly Fig. 3a, were obtained for ˆ 2 and k ˆ in (B) to (B24). Reall here the results of Setion 2 (also see Appendix B), where the parameter was assoiated with the geometry of effetive motion of TDs in growing films and k haraterized the relative radius for fusion reations with respet to the annihilation radius. From the struture of the system of equations (B) to (B24), it is lear that a hange of the parameter will lead to a simple renormalization of the film thiknesses. That is, for larger, the same values of ~r i and ~r T will be ahieved for a smaller value of the film thikness h. This is beause a larger value of leads to larger disloation motion for the presribed hange in film thikness. The influene of the parameter k (the ratio of fusion radius to the annihilation radius) on the TD redution is shown in Fig. 3b for initial TD densities given by ase B5 in Table II.. For larger k, the TD densities fall more rapidly, indiating that larger k orresponds to a larger global ~K T. This behavior an be understood in the framework of annihilation and fusion reations. First, if we onsider ases where the annihilation is the only possible reation, k ˆ 0, then a TD from a speifi family only has the possibility of reating with TDs from one of the twenty-four families (Fig. 3b). However, in the ase in whih both fusion and annihilation are possible, then a TD from a speifi family has the possibility of reating with TDs from nine of the twenty-four families; one of these reations will orrespond to annihilation and eight to fusion. However, fusion between TDs from other families will lead to TD generation and for a speifi family, there will exist four sets of generation reations. For the ase where fusion and annihilation have equal interation radii, i.e., k ˆ, we expet the global ~K T to be five times larger than for the ase where k ˆ 0, as seen in Fig. 3b. Further, we note that the =h behavior an also be obtained when there is effetively only fusion, as seen in Fig. 3b for inreasing values of k. The validity of the general behavior (0) was heked for two very different values of k: For k ˆ 0:5 and 50 from the large thikness region ( slope in the log±log plots), ~K T was found to be equal to 0.5 and 0, respetively. Sine h ~ 0 is defined by the initial onditions h ~ 0 ˆ, the full predited behavior for log ~r T =~r 0 T an be simply heked by plotting (0) for the values of ~K T estimated above. The agreement for both k values is exellent, as shown in Fig. 3b. The solutions in Fig. 3a orrespond to unequal total initial densities ~r 0 T. To verify the offset of normalized TD density for a large normalized thikness in the plots of total TD density in Fig. 3a (ªoffset effetº), we have repeated the alulations for renormalized initial densities for ase B in whih the total density ~r 0 T ˆ 0:6 for this modified ase is equal to the total density for ase B5. This normalization of total density leads to oinident urves for ~r T for ases B and B5 for several values of k, as shown in Fig. 3. This result demonstrates both the offset effet is a onsequene of total initial density and that the initial distribution of TDs over the families does not hange the overall harater of the solution. 3.2 Non-balaned TD densities (B 6ˆ 0) 3.2. Basis for non-balaned TD densities In the seond group of initial onditions (ases N through N8 in Table II.), the initial TD densities are not balaned suh that the film has a net Burgers vetor ontent B 6ˆ 0. TD reations happen loally, and in many ases on a loal length sale, there

10 42 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek may be a loal net Burgers vetor ontent of TDs. However, globally, B ˆ 0 may or may not be ahieved. The flutuations in the net Burgers vetor ontent thus beome a very important topi for future investigations. Here, however, we treat the general ase in whih there is a net Burgers vetor ontent and do not onsider the origin of this behavior. Again, variations for the presene and absene of TDs with Burgers vetors parallel to the film/substrate interfae are onsidered. The results show many similar features to the solutions for the balaned ases. However, sine there is a net Burgers vetor ontent, at least one, and often a set of families will show saturation behavior Numerial solutions For the non-balaned situations, the initially presribed net Burgers vetor ontent is invariant with regard to film growth. An example of suh behavior is given in Fig. 4, upper panel where r 0 ˆ r0 2 > r0 i for i ˆ 3 to 6 and TD families 7 through 24 (with Burgers vetor parallel to the film/substrate interfae) are initially absent. As in the bal- Fig. 4. Threading disloation densities for non-balaned Burgers vetor ontent B 6ˆ 0 orresponding to surfae nuleated TDs (ase N from Table II.) (upper panel) and to TDs generated during island oalesene (ase N4 from Table II.) (lower panel). Left olumn: linear±linear sale; right olumn: log±log sale. The speifi TD designation is indiated in the figure and in Table I.; ˆ 2 and k ˆ

11 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 43 aned ase, the densities of primary disloations r through r 6 begin to derease with inreasing h due to both annihilation and fusion reations. The densities of seondary disloations r 7 through r 24 initially inrease, but also fall with inreasing film thikness. For larger values of h; ~ r 3 through r 6 asymptotially approah zero. However, r and r 2 asymptotially approah saturation values Dr ˆ r r i. The total density r T also saturates to a value 2 Dr beause of the initial net Burgers vetor ontent. The other non-balaned ases (Fig. 4, lower panel and Fig. 5) present different situations where the saturation behavior manifests itself in the ensemble of TDs in growing films. For example, ase N4 (Fig. 4, lower panel) orresponds to island growth, where only TDs with Burgers vetors parallel to the film/substrate interfae exist and have an initial imbalane. In this ase, the imbalane between pairs of TD families remains onstant, i.e., r 23 r 24 r 2 r 22 ˆ r 7 r 8 r 9 r 20 ˆ2Dr; where Dr ˆ r 0 23 r0 22 ˆ r0 7 r0 2. Fig. 5 orresponds to non-balaned ases in whih there are TDs that may appear by both layer-by-layer and island growth. We note that for ase N5 the TD behavior is similar Fig. 5. Threading disloation densities for non-balaned Burgers vetor ontent B 6ˆ 0 orresponding to TDs for ase N5 (upper panel) and ase N6 (lower panel) from Table II.. These TDs orrespond to those for a mixed growth mode. Left olumn: linear±linear sale; right olumn: log± log sale. The speifi TD designation is indiated in the figure and in Table I.; ˆ 2 and k ˆ 4 physia (b) 99/

12 44 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek to N. Interestingly, the solutions for the initial onditions N5 orrespond to twenty-four distint values of the TD families (no pairing of families). For this ase, B ˆ 0:02 a=2 0Š and for large h, the two families that saturate to a finite density are TD families 7 and 8, both with Burgers vetor b ˆ a=2 0Š eah with densities ~r 7 ˆ ~r 8 ˆ 0:0. The behavior of the solutions for ase N6 (see Fig. 5, lower panel) demonstrates saturation in whih a given exess of Burgers vetors is unequally distributed in the two families and 2. This demonstrates that the net Burgers vetor ontent is preserved yet redistributed into different families (line diretions). This behavior an be understood in the framework of the fusion reations between families with non-zero densities. We will use ase N6 with different values of ~r 0 ˆ ~r0 2 to demonstrate some of the general features of saturation behavior Thikness dependene of the total TD density The behavior of the total TD density for the non-balaned ases is shown in Fig. 6, where Fig. 6a summarizes all of the ases of initial onditions N through N8 and Fig. 6b demonstrates the saturation behavior for varying ratios jbj=~r 0 T. Notie in Fig. 6a that for a range of thiknesses, the TD density follows the =h saling behavior and then saturates. The magnitude of the saturation value approximately sales as jbj=~r 0 T. Detailed onsiderations of the absolute magnitude of the saturation value ~r T depends on how B is partitioned into individual omponents related to disloation families. In Fig. 6b, we demonstrate this dependene of the hange in saturation value with dereasing jbj=~r 0 T. Together with the derease in saturation level, the =h region inreases. Furthermore, the value of ~K T is approximately independent of jbj=~r 0 T. Fig. 6. Logarithmi plots of the total relative threading disloation densities as a funtion of normalized film thikness for non-balaned ases B 6ˆ 0. a) Dependenies for different initial onditions given in Table II., ases N through N8; ˆ 2 and k ˆ. b) Dependenies for different magnitudes of the net Burgers vetor ontent jbj (and orrespondingly Dr for the ase N6; k ˆ ; ˆ2. The value of ~r 0 ˆ ~r0 2 is indiated in the figure

13 Threading Disloation Density Redution in Heteroepitaxial Layers (II) Disussison and Conlusions The results reported in this paper demonstrate that for initially balaned B ˆ 0) ases, the total TD density follows the global redution law (0). The model presented in Part I treated only one type of TD density. It was shown here in Part II that the total redution is desribed by the oeffiient ~K T. This result appears to be valid at least for situations when the series of differential equations inludes only a single parameter for the geometry of TD motion (parameter and two parameters for TD reations: the annihilation radius r A and the fusion radius r F. We have also demonstrated that for balaned ases the total TD density r T h appears to be independent of the initial distribution of TDs between different subfamilies. We onlude that ~K T in (0) depends on the value of k and thus on the possibility of fusion reations within an ensemble of TDs. When fusion is not possible due to initial onditions, for example if only the TDs of the initially present families have Burgers vetors that lie in the plane of the film/substrate interfae (e.g., see ase B4), then the value of ~K T orresponds to that of k ˆ 0. In the general ase, however, varying the ratio of r F to r A (parameter k does not hange the global behavior for the balaned ases; rather, inreasing k only inreases the overall rate of redution by inreasing the value of ~K T. For non-balaned TD distributions B 6ˆ 0, the total TD density shows the same initial behavior as the balaned ases. However, at large film thikness, ~r T saturates to a onstant value. The saturation behavior is due to the net Burgers vetor ontent B whih is defined by (9). The main harateristi of B is that it remains onstant during reations in a TD ensemble, however, it an be redistributed amongst several TD families. We believe that saturation in real experiments [5] is a onsequene of lateral flutuations in the density of different TD families and thus flutuations in B. The treatment presented in Part II provides new insight into the details of TD redution. In a future paper we plan to inlude a omplete treatment of the geometri oeffiients K ij for the reations between speifi pairs of TDs. This modifiation will break the overall symmetry of the set of the twenty-four differential equations in suh a way that equations for families through 6 will be inequivalent to the equations for families 7 through 24. Therefore, the possibility of the global redution behavior other than that predited by (0) will be examined. Aknowledgements This work was supported in part by the UC MICRO program in onjuntion with Hughes Airraft orporation, by the AFOSR through ontrat F (Dr. Gerald Witt ontrat monitor), by the NSF MRL program grant No. DMR , and by the Max Plank Soiety. Additional support for GEB was provided by an Alexander-von-Humboldt Fellowship. Appendix A In this appendix, we onsider the addition of sattering reations to the series of differential equations for TD kinetis. Equation (), in priniple, inludes all possible reations between TDs from family with TDs of any other of the twenty-four families. Based on the analysis of Part I, we may only onlude that K ˆ K 3 ˆ 0 beause the TDs in these pairs have no relative motion with inreasing film thikness. Some of the possible reations in () orrespond to sattering (fusion and annihilation have been treated in the main body of the text). For the ase of sattering, it is important that 4*

14 46 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek there is the onservation of Burgers vetor during sattering reations, this means that TDs of family an appear only from sattered TDs from families or 2. Consider now an example of sattering. For the entry [(, 2)±(5, 6)] in Table I., disloations of family are produed by sattering reations. In differential form, the sattering may be represented as dr dh ˆ b ; 5 K ; 5r r 5 b ; 6 K ; 6r r 6 b 2; 5 K 2; 5r 2 r 5 b 2; 6 K 2; 6r 2 r 6 ; A where, again, b k ij desribes the probability that the reation TDs from the i-th and j-th families form a TD from the k-th family. If only one TD is formed, then this is a fusion reation, if two TDs result, then it is a sattering reation. It was argued in Part I that for sattering reations the following relation must be true: b k ij bm ij bi ij bj ij ˆ 2 ; A2 where, again families i and k have the same Burgers vetor and families j and m have the same Burgers vetor (but not the same b as for families i and k). This relation simply reflets the onservation of the Burgers vetors of the reating TDs, i.e., sattering reations do not diretly redue r T. If there is no sattering, then b k ij ˆ bm ij ˆ 0 and b i ij ˆ bj ij ˆ. If there is sattering with overall equal probability, then bk ij ˆ bm ij ˆ bi ij ˆ b j ij ˆ 2. The ase when disloations from the i-th and j-th families have the same Burgers vetor, but different line diretions (orresponding to diagonal entries in Table I.) should be treated separately. Here only two oeffiients have to be used: b i ij and bj ij. For suh ªdiagonalº reations, the ase with no sattering is indistinguishable from the ase with equal probability of sattering: for both ases b i ij ˆ bj ij ˆ. Now we an ombine all ontributions, in aordane with Table I., to the hange of disloation density of r. dr dh ˆ b ;2 K ;2r r 2 K ;4 r r 4 b ;5 K ;5r r 5 b 2;5 K 2;5r 2 r 5 b ;6 K ;6r r 6 b 2;6 K 2;6r 2 r 6 b ;7 K ;7r r 7 b 2;7 K 2;7r 2 r 7 b ;8 K ;8r r 8 b 2;8 K 2;8r 2 r 8 b ;9 K ;9r r 9 b 2;9 K 2;9r 2 r 9 b ;0 K ; 0r r 0 b 2; 0 K 2; 0r 2 r 0 K ; r r K ; 2 r r 2 b ;3 K ; 3r r 3 b 2; 3 K 2; 3r 2 r 3 b ;4 K ; 4r r 4 b 2; 4 K 2; 4r 2 r 4 K ; 5 r r 5 K ; 6 r r 6 b ;7 K ; 7r r 7 b 2; 7 K 2; 7r 2 r 7 b ;8 K ; 8r r 8 b 2; 8 K 2; 8r 2 r 8 K ; 9 r r 9 K ; 20 r r 20 b ;2 K ; 2r r 2 b 2; 2 K 2; 2r 2 r 2 b ;22 K ; 22r r 22 b 2; 22 K 2; 22r 2 r 22 K ; 23 r r 23 K ; 24 r r 24 K 3; 7 r 3 r 7 K 3; 8 r 3 r 8 K 4; 7 r 4 r 7 K 4; 8 r 4 r 8 : A3

15 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 47 The equations for densities of TDs from the other families will have a similar struture. Note that in the absene of sattering, (A3) transforms to (4). Appendix B This appendix provides the system of oupled first-order nonlinear ordinary differential equations for TD families in growing (00) epitaxial f... films. d~r d h ~ ˆ ~r ~r 4 k~r ~r ~r 2 ~r 5 ~r 6 ~r 9 ~r 20 ~r 23 ~r 24 k ~r 3 ~r 4 ~r 7 ~r 8 ; B d~r 2 d h ~ ˆ ~r 2 ~r 3 k~r 2 ~r ~r 2 ~r 5 ~r 6 ~r 9 ~r 20 ~r 23 ~r 24 k ~r 9 ~r 0 ~r 2 ~r 22 ; B2 d~r 3 d h ~ ˆ ~r 2 ~r 3 k~r 3 ~r 9 ~r 0 ~r 3 ~r 4 ~r 7 ~r 8 ~r 2 ~r 22 k ~r 5 ~r 6 ~r 9 ~r 20 ; B3 d~r 4 d h ~ ˆ ~r ~r 4 k~r 4 ~r 9 ~r 0 ~r 3 ~r 4 ~r 7 ~r 8 ~r 2 ~r 22 k ~r ~r 2 ~r 23 ~r 24 ; B4 d~r 5 d h ~ ˆ ~r 5 ~r 8 k~r 5 ~r ~r 2 ~r 5 ~r 6 ~r 7 ~r 8 ~r 2 ~r 22 k ~r 9 ~r 0 ~r 9 ~r 20 ; B5 d~r 6 d h ~ ˆ ~r 6 ~r 7 k~r 6 ~r ~r 2 ~r 5 ~r 6 ~r 7 ~r 8 ~r 2 ~r 22 k ~r 3 ~r 4 ~r 23 ~r 24 ; B6 d~r 7 d h ~ ˆ ~r 6 ~r 7 k~r 7 ~r 9 ~r 0 ~r 3 ~r 4 ~r 9 ~r 20 ~r 23 ~r 24 k ~r ~r 2 ~r 7 ~r 8 ; B7 d~r 8 d h ~ ˆ ~r 5 ~r 8 k~r 8 ~r 9 ~r 0 ~r 3 ~r 4 ~r 9 ~r 20 ~r 23 ~r 24 k ~r 5 ~r 6 ~r 2 ~r 22 ; B8 d~r 9 d h ~ ˆ ~r 9 ~r 2 k~r 9 ~r 3 ~r 4 ~r 7 ~r 8 ~r 9 ~r 20 ~r 2 ~r 22 k ~r ~r 2 ~r 23 ~r 24 ; B9 d~r 0 d h ~ ˆ ~r 0 ~r k~r 0 ~r 3 ~r 4 ~r 7 ~r 8 ~r 9 ~r 20 ~r 2 ~r 22 k ~r 5 ~r 6 ~r 7 ~r 8 ; B0 d~r d h ~ ˆ ~r 0 ~r k~r ~r ~r 2 ~r 5 ~r 6 ~r 7 ~r 8 ~r 23 ~r 24 k ~r 3 ~r 4 ~r 2 ~r 22 ; B

16 48 A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek d~r 2 d h ~ ˆ ~r 9 ~r 2 k~r 2 ~r ~r 2 ~r 5 ~r 6 ~r 7 ~r 8 ~r 23 ~r 24 k ~r 7 ~r 8 ~r 9 ~r 20 ; B2 d~r 3 d h ~ ˆ ~r 3 ~r 6 k~r 3 ~r 3 ~r 4 ~r 7 ~r 8 ~r 7 ~r 8 ~r 23 ~r 24 k ~r 5 ~r 6 ~r 2 ~r 22 ; B3 d~r 4 d h ~ ˆ ~r 4 ~r 5 k~r 4 ~r 3 ~r 4 ~r 7 ~r 8 ~r 7 ~r 8 ~r 23 ~r 24 k ~r ~r 2 ~r 9 ~r 20 ; B4 d~r 5 d h ~ ˆ ~r 4 ~r 5 k~r 5 ~r ~r 2 ~r 5 ~r 6 ~r 9 ~r 20 ~r 2 ~r 22 k ~r 7 ~r 8 ~r 23 ~r 24 ; B5 d~r 6 d h ~ ˆ ~r 3 ~r 6 k~r 6 ~r ~r 2 ~r 5 ~r 6 ~r 9 ~r 20 ~r 2 ~r 22 k ~r 3 ~r 4 ~r 7 ~r 8 ; B6 d~r 7 d h ~ ˆ ~r 7 ~r 20 k~r 7 ~r 3 ~r 4 ~r 5 ~r 6 ~r ~r 2 ~r 3 ~r 4 k ~r ~r 2 ~r 5 ~r 6 ; B7 d~r 8 d h ~ ˆ ~r 8 ~r 9 k~r 8 ~r 3 ~r 4 ~r 5 ~r 6 ~r ~r 2 ~r 3 ~r 4 k ~r 7 ~r 8 ~r 9 ~r 0 ; B8 d~r 9 d h ~ ˆ ~r 8 ~r 9 k~r 9 ~r ~r 2 ~r 7 ~r 8 ~r 9 ~r 0 ~r 5 ~r 6 k ~r 3 ~r 4 ~r 3 ~r 4 ; B9 d~r 20 d h ~ ˆ ~r 7 ~r 20 k~r 20 ~r ~r 2 ~r 7 ~r 8 ~r 9 ~r 0 ~r 5 ~r 6 k ~r 5 ~r 6 ~r ~r 2 ; B20 d~r 2 d h ~ ˆ ~r 2 ~r 24 k~r 2 ~r 3 ~r 4 ~r 5 ~r 6 ~r 9 ~r 0 ~r 5 ~r 6 k ~r ~r 2 ~r ~r 2 ; B2 d~r 22 d h ~ ˆ ~r 22 ~r 23 k~r 22 ~r 3 ~r 4 ~r 5 ~r 6 ~r 9 ~r 0 ~r 5 ~r 6 k ~r 7 ~r 8 ~r 3 ~r 4 ; B22 d~r 23 d h ~ ˆ ~r 22 ~r 23 k~r 23 ~r ~r 2 ~r 7 ~r 8 ~r ~r 2 ~r 3 ~r 4 k ~r 3 ~r 4 ~r 9 ~r 0 ; B23 d~r 24 d h ~ ˆ ~r 2 ~r 24 k~r 24 ~r ~r 2 ~r 7 ~r 8 ~r ~r 2 ~r 3 ~r 4 k ~r 5 ~r 6 ~r 5 ~r 6 : B24

17 Threading Disloation Density Redution in Heteroepitaxial Layers (II) 49 The equations in this system were derived in the same way as (8). Normalized TD densities ~r i ˆ r i r 2 A and thikness h ~ ˆ h=r A are used in this system of equations. The oeffiient k ˆ r F =r A denotes the relative radius for fusion reations. The parameter desribes the dependene of TD motion on their inlination with respet to the surfae normal. For perpendiular orientations, ˆ 0, for large inlination, tends to infinity. For the real ase of (00) epitaxy, 2. Referenes [] A. E. Romanov, W. Pompe, G. Beltz, and J. S. Spek, phys. stat. sol. (b) 98, 599 (996). [2] P. Sheldon, K. M. Jones, M. M. Al-Jassim, and B. G. Yaobi, J. appl. Phys. 63, 5609 (988). [3] M. Tahikawa and M. Yamaguhi, Appl. Phys. Letters 56, 484 (990). [4] R. Beanland, D. J. Dunstan, and P. J. Goodhew, Adv. Phys. 45, 87 (996). [5] J. S. Spek, M. A. Brewer, G. E. Beltz, A. E. Romanov, and W. Pompe, J. appl. Phys. 80, 3808 (996).