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Evolution of point defects during swirl formation in semiconductors
Mater& Letters North-Holland
HATENALS LETTERS
15 ( 1993) 347-352
Evolution of point defects during swirl formation in semiconductors I.V. Vemer ‘, N.N. Gerasimenko i, Y. Zhang and J.W. Corbett Ins~ii~~e.f~rthe Study ofDefects in Solids, Physrcs Departments ~l~i~ers~t~at Afhany-SUNY, Received
Albany, NY 12222, US.4
31 July 1992
It is shown that in a system of impurities and intrinsic point defects (vacancies and interstitials) that are generated during the growth of a crystal from the melt, there may arise instabilities in the distribution of the defects. A model is presented for the evolution ofthese instabilities which may describe the origin ofthe quasi-periodical distribution of micro-defects in silicon, called swirl defects. This model is an exampie of the chaos phenomena which occur widely in defect processes due to the nonlinearity of the equations describing the defect processes, in particular, an example of the instabilities which occur in the are-chaos regime, i.e. short of full chaos.
1. Introduction The equations describing defect interactions are often highly nonlinear, as we will illustrate below. Such nonlinear equations may give rise to the variety of mathematical behaviors which are encompassed in the term c/zaos. We wiil describe here how the instabilities encountered in this theory can be exploited to describe swirl defects encountered in semiconducting crystals. During the melt-growth of large, dislocation-free semiconductor crystals (such as silicon or germanium, whether grown by the moating-zone (FZ) or Czochralski technique (CZ) ), microdefects can form due to the agglomeration of point defects (PD), i.e. vacancies (V ). self-interstitials (I ) and chemical impurities (X). Due to their specific quasiperiodictype distribution these defects are generally referred to as “swirl defects” (SDS). Since 1965, when Plaskett [I] published the first paper on this subject, there have been a number of excellent experimental and theoretical works in which the problem of formation and properties of SDS were considered from many points of view [ 2-61. Four types of SDS have been discussed (A, B, C and D), but we will focus ’ Permanent
address:
Moscow Institute of Electronic Federation.
ogy, Moscow, 103498. Russian
Ekevier Science Publishers
B.V.
Technol-
on two types, A and B, which form successively during crystal growth [ l-31 and about which the most is known. The A-type SD consists of extrinsic (interstitial-type) dislocation loops, which argues that these defects form by the a~lomeration of interstitial atoms. The structure of the B-type SD is still unclear; they do not give a detectable contrast in highvoltage electron microscopy, and so they must consist of defects (self-interstitials, vacancies, and/or impurity atoms) arranged without a strong strain field. Experimental investigations have shown that the formation of the SDS depends on various growth parameters. For example, there are critical values of crystai growth rate ( &) at which the impurity-defect structure of semiconductor dramatically changes during growth [ 3,6,7]. In particular, for FZ growth at Y0’,>4 mm/min, dislocation loop (A-type SD) formation is suppressed, and at C’,> 5 mm/min Btype of SD disappear [ 7 1. When v0 is lower than 0.2 mm/min SD-free crystals were obtained [ 71. There also exist strong dependences on the SD distribution and SD concentration and on the type of impurity atoms in the crystal. In CZ silicon doped with donor impurities (P, As, Sb) in concentration [X] 1 lOI cmp3, complete suppression of A-type SD results, together with an increase in the density of B-type SDS. Doping with acceptors (B, Ga) above lOI cmb3 causes the opposite effect, i.e. no B-type SDS are 347
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15,number 5,6
MATERIALS LETTERS
formed [ 71. Carbon, oxygen and other impurities play an important role in the formation of SDS also; a more complete discussion in this area has been given in refs. [5,8,9]. It was also found that when dislocations are generated during growth, the SD concentration is rapidly reduced [ 71. Similar effects take place during changes in crystal diameter during growth [ 5 1. Recently, research concerned with some recombination properties of dislocation-free, floatzoned silicon due to growth-induced microdefects was made [ 16,171. It was shown that SDS are the carrier recombination centers in float-zone-grown Si crystals. Hydrogen doping can suppress SD formation but causes some new recombination centers. The above authors also showed that defect formation can be changed by growth conditions, and long minority charge-carrier lifetimes are achieved through the use of moderately high growth speeds and low thermal gradients during crystal growth, by which SDS are avoided. From these experiments it is possible to conclude that the interaction between intrinsic PDs and impurity atoms plays an important role in the process of SD formation in crystals in which the concentration of sinks (for example, dislocations) for PDs is small. Each type of intrinsic PD generated at high temperatures interacts with each type of impurity with different rates. The V- and I-generation rates in a given volume of crystal may be different, for instance, due to fluctuations in the temperature gradient and/or variations of other internal and external parameters (concentration and distribution of oxygen-silicon precipitates, the density of dislocations, the atmosphere surrounding the crystal, etc.). The generation rate also depends on the cooling rate which is increased with an increase in the growth rate VO(the zone-traveling rate) and decreased with an increase in the crystal diameter.
January 1993
PV
.&L---v;
(1)
PI
(2)
I;
g,-
;
r+v-+o
(3)
I+1 t---t 1.1) 1.1+1 +-’ 1.1.1 ) 1.1.1+1 +--+ I.I.I*I, ... .
(4)
Eqs. ( 1) and (2) schematically indicate the generation rates, gv and gr, for V and I, respectively; eq. (3) similarly indicates the annihilation process for vacancies and interstitials. The reactions encompassed in eq. (4) are the hierarchy of reactions for the agglomeration of interstitials, as if they were the only defect present. The two-directional arrows indicate the possibility of dissociation of the aggregates; we label the kinetic coefftcients for the forward ith reaction with pi and those for the back reaction with PI. This hierarchy of reactions was discussed early by Szilard in unpublished work on the molecular theory of nucleation, and later by others [ 1O- 13 1. In this theory the formation of the I, agglomerates proceeds with the forward and backward reactions in a quasi-equilibrium until the critical nucleus, I,, is reached, beyond which the forward reaction dominates and growth proceeds; the clusters with n
++ V.V)
v.v-+-v
t---t V.V,V)
v*v.v+v
++ v.v.v.v,
... )
(5)
and the equations describing the annihilation 2. Mathematical PDs
description of interactions between
II+vdI
;
I.I.I+v+I*I, V~VfI-+V
Let us consider a scheme of likely interactions between PDs and dopant atoms [X] which have the total concentration [ Xltot: 348
process:
v.v.v-l-I+V.V,
... ;
(6)
) ...
It is clear from our discussion
(7) of the SDS that im-
Volume
15,number 5,6
MATERIALS
purities can play a role in the formation of SDS, so we must also include the hierarchy of equations associated with impurity-mediated agglomeration: x+1
++ x.1,
x,1+1 x.1
+’
x.1.1,
+v+x
+-
x.v+v
(8)
, ;
x~I~I+v+x.I, x+v
;
x,v.v,
... :
(10)
.
(11)
We can simplify the discussion if we introduce a vector C; the components, C,, are our various reactants, e.g.. V. I. 1.1, etc., the order of which need not concern us now. If we only consider spatial variations in one dimension (r). one may introduce kinetic equations for the above reactions of the form
ac ar =F(C)+D.
that certain defect complexes have a greater stability than ones adjacent in the hierarchy, so our assumption of hierarchical homogeneity is an explicit assumption and we realize that it does not have general validity; since we are only illustrating the nature of these systems here, this question need not concern us further here.
3. Application of kinetic description to the problem of swirl defects
x.v+1I+x, x~v~v+IL+x~v,
1993
(9)
x.v, t-*
January
LETTERS
z.
(12)
The F(C) are nonlinear functions describing the interactions in above equations; these functions may be written using the usual quasi-chemical method for the description of reactions between PDs and impurities (as we will show for the explicit case that we consider here in the appendix); the D is a vector the components of which are the diffusion coefficients of the reactants, some diffusion coefficients being zero, reflecting immobile reactants; and the scalar product in the last term of eq. ( 12) denotes the product of D, with C,; we have followed the usual, simplifying assumption of spatially constant D,. The several coupled infinite hierarchies of equations encompassed in eq. ( 13 ) are no more solvable than the single hierarchy for conventional nucleation and growth. We argue, however, that we need not solve the full equations to find the essential features of the problem. We will assume that the early reactants determine the subsequent reactions and growth, much as is in folk-saying: “As the twig is bent, so grows the tree”. This implies an essential homogeneity of the parameters characterizing the agglomeration so that there are no bottlenecks in the process; we recognize
To illustrate the nature of the solution of these equations we will truncate the equations to include only seven reactants (see appendix): dC z=F(C,,C,
,..., C,)+D$,
(13)
where C,= [I], Cz= [VI, C,= [Xl, C,= [X.1], C,= [X.1.1], C,= [X.V], and C,= [X.V.V], and we have added equations for the vacancies and interstitials reacting with their sinks, S, and Si: v+s”+s”, 1+s,+s,.
(14)
(In specific situations one could include specific types of impurities and explicitly include the interactions for al1 possible (PD + impurity) combinations. For example, boron plays a role in the formation of swirl defects in silicon, and one could Bi + V-B,; consider the following reactions: B,+I~Bi; B,+V+B;V; B;V+I+B,, etc. Here Bi and B, are interstitial and substitutional atoms of boron, respectively, and a number of the defects in the hierarchies are experimentally identified. ) The analytical analysis of the strongly nonlinear kinetic equations describing these seven defects is still a very difficult problem. One possible way to reduce the complexity of the problem would be to decrease the dimension of the problem by assuming a quasisteady-state for the concentration of C, to C, (the impurity variables) so that they follow virtually inertly the concentrations of C, = [V ] and [ I ] ; this assumption is equivalent to setting x,/at=: .. E ac,/ dt z 0. We choose instead a second way by exploiting the “small parameters” in the equations for impurity variables. For example, we can assume that at high 349
Volume 15,number 5,6
MATERIAL LETTERS
January 1993
temperatures (Tz 1000°C), the rate of decay of complexes is much higher than the rate of losses to sinks in a dislocation-free crystal. Also we will include in our consideration only the diffusion coefficients for the faster variables, i.e. for the vacancies and self-interstitials. With these assumptions one can obtain (see appendix) the two kinetic equations for XN [I] and y- [V] in which the kinetic coefficients of our scheme are included as parameters:
the various cases and for the specific variables and instabilities. In particular for our scheme of interactions (A. l)(A. 10) in the appendix, for Sz L*, p, >> ,u7, ,& RZ0, ,M~B,u~ and ~~a,&,, from (A.22), (A.23) and (A.24) we have the following conditions for an instability of the saddle type (for a point system):
(15)
(18)
(16)
In this case for full system with diffusion terms when D, B Dv (i.e. ,uoB ,& ) for the saddle point the evolution of the spatial structure has an oscillatory character [ 151 and in our system there arises and evolves a periodic spatial distribution of defects.
The behavior of the steady-state positive solutions of these equations (x=, y,,) depends on the type of eigenvalues of the Jacobian matrix at the steady-state points. For wave-like perturbations about the steadystate values (AXE, and Ay,,zexp( u@*) exp[i(rrk/ L)r’], where L/k is the wave length), we may get the conditions for the existence of nontrivial solutions (i.e. other than x,,, ySS=O) for each mode k from the solutions of the following determinant: det{M, - (nk/L)*D,,
-
w,LI~}
=
W’.
(17)
In this equation, wk= Wik? ii+&, A, is the Kronecker symbol, D, is the diffusion matrix and Mij= (ah!,,,/ litax, .dxah. There is a very large mathematical erature concerning the solutions of this system. In particular in the linear approach it has been shown that there are six distinct kinds of solutions, depending on the parameters [ 15 1. From possible variants in this Letter, we consider the situation when the critical point in the solution for x8,, ySS(I,,, V,,) is a saddle-point, i.e. det{Mijj
4. Summary In summa~ we can draw some conclusions which follow from the present model: (a) Nonlinear interactions between intrinsic point defects and impurity atoms may be the reason for the instability in the impurity-defect subsystem of semiconductor crystals during growth. (b) Formation of the spatially quasiperiodic distribution of microdefects called swirl defects is a consequence of the evolution of this instability. (c) In this model the various system parameters play a sensitive role in the evolution of the instabilities (swirls, just as occurs expe~mentally ). The main controllable critical parameters are the generation rates of the PD (g,, g2), the quantity and type of impurity atoms (R, /I* ) and the concentration of sinks, e.g. dislocations. A more detailed investigation of the behavior of such systems is being carried out, and we must consider the other possible types of critical points which may exist in our system of kinetic equations. We must also consider cylindrical polar coordinates with corresponding initial and boundary conditions to emulate the cylindrical geometry of many crystal-growth systems; in this case it would also be possible to take into consideration a new controllable parameter, the speed of rotation of the crystal during growth, be-
Volume
15.number 5,6
MATERIALS
cause this parameter may also contribute to the mechanism of the formation of the spatial distribution of swirl defects. Results of these considerations will be published separately. It is possible also to conclude that the method of stability analysis for interactions between self-PD and impurity atoms can be applied for the study of evolution of radiation defects during irradiation of semiconductors (for example, for ion implantation). For many practical cases (for instance, silicon with boron irradiated by hydrogen, etc.) interactions between PDs are the same as those considered above (see eqs. (A.1 )-(A.lO)). For the case of irradiation, parameters g, and g2 are the generation temperature (rate) of radiation defects and they depend on external parameters from an external source such as the particle beam (density of ion current, type of ion. etc. ) For different combinations of external and internal parameters non-monotonic distributions of defects were observed in irradiated semiconductors. These types of behavior of a defect-impurity subsystem of irradiated semiconductors may be the result of the evolution of instabilities of different kinds.
January
LETTERS
I993
w
v+x.v --
x.v.v )
(A.6)
86 P7
1+x.v
x,
-
P8 1+x.v.v
(A.7)
x.v ,
A
(A.8)
PO s, .
I + s, F
(A.9)
4 v+sv
-
sv )
(A.10)
or if we define C,=[I], c5= [X.1.1], c,= [X,1], c,= [X.V.V],
[VI, c,= [XI, c,= [X,V], and
c2=
/II c, +c3 --81
C,)
(A.l*)
c 5,
(A.2*)
w Cl +c‘l +-Irz P3 c;+c,-
c,,
(A.3*)
C;,
(A.4*)
w c;+c,
-
(A.S*) Acknowledgement Pb
We thank Professor Andrew J. Yencha for helpful and effective discussion of this paper.
c;+c, --CT,
(A.6*)
86
/17
c, +c,---- cs, /18
Appendix
c,+c,-
The reactions specifically considered illustrative example are the following:
here in the
1+x.1
v+x.1
x.1 (
---+ BI e--f+ /12
x.1.1,
/‘3 ---X.
.-P
(A.8*)
M (A.9*)
C,+~,-.S, c,+s,-
S”.
(A.lO*)
(A.1) The specific equations (A.21
(A.3)
v+x.1.1--7, v+v
cg,
0x
,I1 1+x
(A.7*)
X.1, K
ac
,,=F(C,,C,
,...,
in the vector expansion
c;)+*.~,
(A.ll)
are the following: aC,/at=8,-~,CIC,+/jlC4
(A.4) -P2CIC4+82Cs-j43C,C,
X.V
(A.5)
-P~C,C~-P&
tD,a2C,/ar2
(A.12) 351
Volume 15, number
5,6
MATERIALS
LETTERS
January
1993
with C=gr
-x--filfi
+ 1 fy+by2),
, F,=gz
-a.v+,U.fi
,
(A.24)
where
These equations may be simplified by introducing the following dimensionless variables:
References
with Got,, the total concentration and conservation
of the impurity
of the impurity, implies that (A.20)
z3+z4+z5+2(j+27=1.
We can substitute these variables into eqs. (A. 12)(A.l8),andtheleftsideofeqs. (A.14)-(A.181 (the equations for the impurity variables) put in the form
az,
ex-=..’
(A.21)
with e now taken to be a “small parameter” and we need only be concerned with the two equations for x and y:
ax $
=F,(x,y)+
52 d4,, ap2
ay=FJx,y)+D~ ‘M*,
T&
352
(A.22)
(A.23)
[ 1] T.S. Plaskett, Trans. AIME 233 ( 1965) 809. [2] S.M. Hu, J. Vacuum Sci. Technol. 14 (1977) 17. [ 31 A.J.R. de Kock, in: Handbook on semiconductors, Vol. 3, ed. S.P. Keller (North-Holland, Amsterdam, 1980) pp. 269284. [4]A.J.R. de Kock, in: Defects in semiconductors, eds. J. Narayan and T.Y. Tan (Noah-Holland, Amsterdam, 198 1) pp. 309-3 16. [5] T. Abe, H. Harada and J.-I. Chikawa, in: Defects in semiconductors II, eds. S. Mahajan and J.W. Corbett (Materials Research Society, Pittsburgh, 1983) pp. l-18. [ 61A.J.R. de Kock, Philips Res. Rept. Suppl. No. 1 ( 1973). [ 71 H. Foil, U. Gosele and B.O. Kolbesen, in: Semiconductor silicon, eds. H.R. Huff and E. Sirtl (The Electrochemical Society, Princeton, 1977) p. 565. [ 81 A.J.R. de Kock and W.M. van de Wijgert, J. Crystal Growth 49 (1980) 718. [9] A.J.R. de Kock, Philips Tech. Rev. 34 (1974) 244. [IO] L. Farkas, Z. Phys. Chem. 12.5 (1927) 236. [ 1 I ] J.W. Corbett, H.L. Frisch, D. Peak and M.St. Peters, in: Computational methods for large molecules and localized states in solids, eds. F. Herman, A.D. McLean and R.K. Nesbet (Plenum Press, New York, 1973) pp. 67-78. [ 121 D. Peak and J.W. Corbett, J. Stat. Phys. 17 ( 1977) 97. [ 13 ] D. Peak and J.W. Corbett, Radiat. Eff. 36 ( 1978) 197. [ 141 A.M. Turing, Phil. Trans. Roy. Sot. B 237 ( 1952) 37. [ 151 V.A. Vasil’ev, Yu.M. Romanovskii and V.G. Yakhno, Autowave processes (Nauka, Moscow, 1987 ) (in Russian). [ 161 T.F. Ciszek, T.H. Wang, T. Schuler and A. Rohatgy, J. Electrochem. Sot. I36 ( 1989) 230. [ 171 T.H. Wang, T.F. Ciszek and T. Schuler, J. Crystal Growth 109 (1991) 155.
HATENALS LETTERS
15 ( 1993) 347-352
Evolution of point defects during swirl formation in semiconductors I.V. Vemer ‘, N.N. Gerasimenko i, Y. Zhang and J.W. Corbett Ins~ii~~e.f~rthe Study ofDefects in Solids, Physrcs Departments ~l~i~ers~t~at Afhany-SUNY, Received
Albany, NY 12222, US.4
31 July 1992
It is shown that in a system of impurities and intrinsic point defects (vacancies and interstitials) that are generated during the growth of a crystal from the melt, there may arise instabilities in the distribution of the defects. A model is presented for the evolution ofthese instabilities which may describe the origin ofthe quasi-periodical distribution of micro-defects in silicon, called swirl defects. This model is an exampie of the chaos phenomena which occur widely in defect processes due to the nonlinearity of the equations describing the defect processes, in particular, an example of the instabilities which occur in the are-chaos regime, i.e. short of full chaos.
1. Introduction The equations describing defect interactions are often highly nonlinear, as we will illustrate below. Such nonlinear equations may give rise to the variety of mathematical behaviors which are encompassed in the term c/zaos. We wiil describe here how the instabilities encountered in this theory can be exploited to describe swirl defects encountered in semiconducting crystals. During the melt-growth of large, dislocation-free semiconductor crystals (such as silicon or germanium, whether grown by the moating-zone (FZ) or Czochralski technique (CZ) ), microdefects can form due to the agglomeration of point defects (PD), i.e. vacancies (V ). self-interstitials (I ) and chemical impurities (X). Due to their specific quasiperiodictype distribution these defects are generally referred to as “swirl defects” (SDS). Since 1965, when Plaskett [I] published the first paper on this subject, there have been a number of excellent experimental and theoretical works in which the problem of formation and properties of SDS were considered from many points of view [ 2-61. Four types of SDS have been discussed (A, B, C and D), but we will focus ’ Permanent
address:
Moscow Institute of Electronic Federation.
ogy, Moscow, 103498. Russian
Ekevier Science Publishers
B.V.
Technol-
on two types, A and B, which form successively during crystal growth [ l-31 and about which the most is known. The A-type SD consists of extrinsic (interstitial-type) dislocation loops, which argues that these defects form by the a~lomeration of interstitial atoms. The structure of the B-type SD is still unclear; they do not give a detectable contrast in highvoltage electron microscopy, and so they must consist of defects (self-interstitials, vacancies, and/or impurity atoms) arranged without a strong strain field. Experimental investigations have shown that the formation of the SDS depends on various growth parameters. For example, there are critical values of crystai growth rate ( &) at which the impurity-defect structure of semiconductor dramatically changes during growth [ 3,6,7]. In particular, for FZ growth at Y0’,>4 mm/min, dislocation loop (A-type SD) formation is suppressed, and at C’,> 5 mm/min Btype of SD disappear [ 7 1. When v0 is lower than 0.2 mm/min SD-free crystals were obtained [ 71. There also exist strong dependences on the SD distribution and SD concentration and on the type of impurity atoms in the crystal. In CZ silicon doped with donor impurities (P, As, Sb) in concentration [X] 1 lOI cmp3, complete suppression of A-type SD results, together with an increase in the density of B-type SDS. Doping with acceptors (B, Ga) above lOI cmb3 causes the opposite effect, i.e. no B-type SDS are 347
Volume
15,number 5,6
MATERIALS LETTERS
formed [ 71. Carbon, oxygen and other impurities play an important role in the formation of SDS also; a more complete discussion in this area has been given in refs. [5,8,9]. It was also found that when dislocations are generated during growth, the SD concentration is rapidly reduced [ 71. Similar effects take place during changes in crystal diameter during growth [ 5 1. Recently, research concerned with some recombination properties of dislocation-free, floatzoned silicon due to growth-induced microdefects was made [ 16,171. It was shown that SDS are the carrier recombination centers in float-zone-grown Si crystals. Hydrogen doping can suppress SD formation but causes some new recombination centers. The above authors also showed that defect formation can be changed by growth conditions, and long minority charge-carrier lifetimes are achieved through the use of moderately high growth speeds and low thermal gradients during crystal growth, by which SDS are avoided. From these experiments it is possible to conclude that the interaction between intrinsic PDs and impurity atoms plays an important role in the process of SD formation in crystals in which the concentration of sinks (for example, dislocations) for PDs is small. Each type of intrinsic PD generated at high temperatures interacts with each type of impurity with different rates. The V- and I-generation rates in a given volume of crystal may be different, for instance, due to fluctuations in the temperature gradient and/or variations of other internal and external parameters (concentration and distribution of oxygen-silicon precipitates, the density of dislocations, the atmosphere surrounding the crystal, etc.). The generation rate also depends on the cooling rate which is increased with an increase in the growth rate VO(the zone-traveling rate) and decreased with an increase in the crystal diameter.
January 1993
PV
.&L---v;
(1)
PI
(2)
I;
g,-
;
r+v-+o
(3)
I+1 t---t 1.1) 1.1+1 +-’ 1.1.1 ) 1.1.1+1 +--+ I.I.I*I, ... .
(4)
Eqs. ( 1) and (2) schematically indicate the generation rates, gv and gr, for V and I, respectively; eq. (3) similarly indicates the annihilation process for vacancies and interstitials. The reactions encompassed in eq. (4) are the hierarchy of reactions for the agglomeration of interstitials, as if they were the only defect present. The two-directional arrows indicate the possibility of dissociation of the aggregates; we label the kinetic coefftcients for the forward ith reaction with pi and those for the back reaction with PI. This hierarchy of reactions was discussed early by Szilard in unpublished work on the molecular theory of nucleation, and later by others [ 1O- 13 1. In this theory the formation of the I, agglomerates proceeds with the forward and backward reactions in a quasi-equilibrium until the critical nucleus, I,, is reached, beyond which the forward reaction dominates and growth proceeds; the clusters with n
++ V.V)
v.v-+-v
t---t V.V,V)
v*v.v+v
++ v.v.v.v,
... )
(5)
and the equations describing the annihilation 2. Mathematical PDs
description of interactions between
II+vdI
;
I.I.I+v+I*I, V~VfI-+V
Let us consider a scheme of likely interactions between PDs and dopant atoms [X] which have the total concentration [ Xltot: 348
process:
v.v.v-l-I+V.V,
... ;
(6)
) ...
It is clear from our discussion
(7) of the SDS that im-
Volume
15,number 5,6
MATERIALS
purities can play a role in the formation of SDS, so we must also include the hierarchy of equations associated with impurity-mediated agglomeration: x+1
++ x.1,
x,1+1 x.1
+’
x.1.1,
+v+x
+-
x.v+v
(8)
, ;
x~I~I+v+x.I, x+v
;
x,v.v,
... :
(10)
.
(11)
We can simplify the discussion if we introduce a vector C; the components, C,, are our various reactants, e.g.. V. I. 1.1, etc., the order of which need not concern us now. If we only consider spatial variations in one dimension (r). one may introduce kinetic equations for the above reactions of the form
ac ar =F(C)+D.
that certain defect complexes have a greater stability than ones adjacent in the hierarchy, so our assumption of hierarchical homogeneity is an explicit assumption and we realize that it does not have general validity; since we are only illustrating the nature of these systems here, this question need not concern us further here.
3. Application of kinetic description to the problem of swirl defects
x.v+1I+x, x~v~v+IL+x~v,
1993
(9)
x.v, t-*
January
LETTERS
z.
(12)
The F(C) are nonlinear functions describing the interactions in above equations; these functions may be written using the usual quasi-chemical method for the description of reactions between PDs and impurities (as we will show for the explicit case that we consider here in the appendix); the D is a vector the components of which are the diffusion coefficients of the reactants, some diffusion coefficients being zero, reflecting immobile reactants; and the scalar product in the last term of eq. ( 12) denotes the product of D, with C,; we have followed the usual, simplifying assumption of spatially constant D,. The several coupled infinite hierarchies of equations encompassed in eq. ( 13 ) are no more solvable than the single hierarchy for conventional nucleation and growth. We argue, however, that we need not solve the full equations to find the essential features of the problem. We will assume that the early reactants determine the subsequent reactions and growth, much as is in folk-saying: “As the twig is bent, so grows the tree”. This implies an essential homogeneity of the parameters characterizing the agglomeration so that there are no bottlenecks in the process; we recognize
To illustrate the nature of the solution of these equations we will truncate the equations to include only seven reactants (see appendix): dC z=F(C,,C,
,..., C,)+D$,
(13)
where C,= [I], Cz= [VI, C,= [Xl, C,= [X.1], C,= [X.1.1], C,= [X.V], and C,= [X.V.V], and we have added equations for the vacancies and interstitials reacting with their sinks, S, and Si: v+s”+s”, 1+s,+s,.
(14)
(In specific situations one could include specific types of impurities and explicitly include the interactions for al1 possible (PD + impurity) combinations. For example, boron plays a role in the formation of swirl defects in silicon, and one could Bi + V-B,; consider the following reactions: B,+I~Bi; B,+V+B;V; B;V+I+B,, etc. Here Bi and B, are interstitial and substitutional atoms of boron, respectively, and a number of the defects in the hierarchies are experimentally identified. ) The analytical analysis of the strongly nonlinear kinetic equations describing these seven defects is still a very difficult problem. One possible way to reduce the complexity of the problem would be to decrease the dimension of the problem by assuming a quasisteady-state for the concentration of C, to C, (the impurity variables) so that they follow virtually inertly the concentrations of C, = [V ] and [ I ] ; this assumption is equivalent to setting x,/at=: .. E ac,/ dt z 0. We choose instead a second way by exploiting the “small parameters” in the equations for impurity variables. For example, we can assume that at high 349
Volume 15,number 5,6
MATERIAL LETTERS
January 1993
temperatures (Tz 1000°C), the rate of decay of complexes is much higher than the rate of losses to sinks in a dislocation-free crystal. Also we will include in our consideration only the diffusion coefficients for the faster variables, i.e. for the vacancies and self-interstitials. With these assumptions one can obtain (see appendix) the two kinetic equations for XN [I] and y- [V] in which the kinetic coefficients of our scheme are included as parameters:
the various cases and for the specific variables and instabilities. In particular for our scheme of interactions (A. l)(A. 10) in the appendix, for Sz L*, p, >> ,u7, ,& RZ0, ,M~B,u~ and ~~a,&,, from (A.22), (A.23) and (A.24) we have the following conditions for an instability of the saddle type (for a point system):
(15)
(18)
(16)
In this case for full system with diffusion terms when D, B Dv (i.e. ,uoB ,& ) for the saddle point the evolution of the spatial structure has an oscillatory character [ 151 and in our system there arises and evolves a periodic spatial distribution of defects.
The behavior of the steady-state positive solutions of these equations (x=, y,,) depends on the type of eigenvalues of the Jacobian matrix at the steady-state points. For wave-like perturbations about the steadystate values (AXE, and Ay,,zexp( u@*) exp[i(rrk/ L)r’], where L/k is the wave length), we may get the conditions for the existence of nontrivial solutions (i.e. other than x,,, ySS=O) for each mode k from the solutions of the following determinant: det{M, - (nk/L)*D,,
-
w,LI~}
=
W’.
(17)
In this equation, wk= Wik? ii+&, A, is the Kronecker symbol, D, is the diffusion matrix and Mij= (ah!,,,/ litax, .dxah. There is a very large mathematical erature concerning the solutions of this system. In particular in the linear approach it has been shown that there are six distinct kinds of solutions, depending on the parameters [ 15 1. From possible variants in this Letter, we consider the situation when the critical point in the solution for x8,, ySS(I,,, V,,) is a saddle-point, i.e. det{Mijj
4. Summary In summa~ we can draw some conclusions which follow from the present model: (a) Nonlinear interactions between intrinsic point defects and impurity atoms may be the reason for the instability in the impurity-defect subsystem of semiconductor crystals during growth. (b) Formation of the spatially quasiperiodic distribution of microdefects called swirl defects is a consequence of the evolution of this instability. (c) In this model the various system parameters play a sensitive role in the evolution of the instabilities (swirls, just as occurs expe~mentally ). The main controllable critical parameters are the generation rates of the PD (g,, g2), the quantity and type of impurity atoms (R, /I* ) and the concentration of sinks, e.g. dislocations. A more detailed investigation of the behavior of such systems is being carried out, and we must consider the other possible types of critical points which may exist in our system of kinetic equations. We must also consider cylindrical polar coordinates with corresponding initial and boundary conditions to emulate the cylindrical geometry of many crystal-growth systems; in this case it would also be possible to take into consideration a new controllable parameter, the speed of rotation of the crystal during growth, be-
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15.number 5,6
MATERIALS
cause this parameter may also contribute to the mechanism of the formation of the spatial distribution of swirl defects. Results of these considerations will be published separately. It is possible also to conclude that the method of stability analysis for interactions between self-PD and impurity atoms can be applied for the study of evolution of radiation defects during irradiation of semiconductors (for example, for ion implantation). For many practical cases (for instance, silicon with boron irradiated by hydrogen, etc.) interactions between PDs are the same as those considered above (see eqs. (A.1 )-(A.lO)). For the case of irradiation, parameters g, and g2 are the generation temperature (rate) of radiation defects and they depend on external parameters from an external source such as the particle beam (density of ion current, type of ion. etc. ) For different combinations of external and internal parameters non-monotonic distributions of defects were observed in irradiated semiconductors. These types of behavior of a defect-impurity subsystem of irradiated semiconductors may be the result of the evolution of instabilities of different kinds.
January
LETTERS
I993
w
v+x.v --
x.v.v )
(A.6)
86 P7
1+x.v
x,
-
P8 1+x.v.v
(A.7)
x.v ,
A
(A.8)
PO s, .
I + s, F
(A.9)
4 v+sv
-
sv )
(A.10)
or if we define C,=[I], c5= [X.1.1], c,= [X,1], c,= [X.V.V],
[VI, c,= [XI, c,= [X,V], and
c2=
/II c, +c3 --81
C,)
(A.l*)
c 5,
(A.2*)
w Cl +c‘l +-Irz P3 c;+c,-
c,,
(A.3*)
C;,
(A.4*)
w c;+c,
-
(A.S*) Acknowledgement Pb
We thank Professor Andrew J. Yencha for helpful and effective discussion of this paper.
c;+c, --CT,
(A.6*)
86
/17
c, +c,---- cs, /18
Appendix
c,+c,-
The reactions specifically considered illustrative example are the following:
here in the
1+x.1
v+x.1
x.1 (
---+ BI e--f+ /12
x.1.1,
/‘3 ---X.
.-P
(A.8*)
M (A.9*)
C,+~,-.S, c,+s,-
S”.
(A.lO*)
(A.1) The specific equations (A.21
(A.3)
v+x.1.1--7, v+v
cg,
0x
,I1 1+x
(A.7*)
X.1, K
ac
,,=F(C,,C,
,...,
in the vector expansion
c;)+*.~,
(A.ll)
are the following: aC,/at=8,-~,CIC,+/jlC4
(A.4) -P2CIC4+82Cs-j43C,C,
X.V
(A.5)
-P~C,C~-P&
tD,a2C,/ar2
(A.12) 351
Volume 15, number
5,6
MATERIALS
LETTERS
January
1993
with C=gr
-x--filfi
+ 1 fy+by2),
, F,=gz
-a.v+,U.fi
,
(A.24)
where
These equations may be simplified by introducing the following dimensionless variables:
References
with Got,, the total concentration and conservation
of the impurity
of the impurity, implies that (A.20)
z3+z4+z5+2(j+27=1.
We can substitute these variables into eqs. (A. 12)(A.l8),andtheleftsideofeqs. (A.14)-(A.181 (the equations for the impurity variables) put in the form
az,
ex-=..’
(A.21)
with e now taken to be a “small parameter” and we need only be concerned with the two equations for x and y:
ax $
=F,(x,y)+
52 d4,, ap2
ay=FJx,y)+D~ ‘M*,
T&
352
(A.22)
(A.23)
[ 1] T.S. Plaskett, Trans. AIME 233 ( 1965) 809. [2] S.M. Hu, J. Vacuum Sci. Technol. 14 (1977) 17. [ 31 A.J.R. de Kock, in: Handbook on semiconductors, Vol. 3, ed. S.P. Keller (North-Holland, Amsterdam, 1980) pp. 269284. [4]A.J.R. de Kock, in: Defects in semiconductors, eds. J. Narayan and T.Y. Tan (Noah-Holland, Amsterdam, 198 1) pp. 309-3 16. [5] T. Abe, H. Harada and J.-I. Chikawa, in: Defects in semiconductors II, eds. S. Mahajan and J.W. Corbett (Materials Research Society, Pittsburgh, 1983) pp. l-18. [ 61A.J.R. de Kock, Philips Res. Rept. Suppl. No. 1 ( 1973). [ 71 H. Foil, U. Gosele and B.O. Kolbesen, in: Semiconductor silicon, eds. H.R. Huff and E. Sirtl (The Electrochemical Society, Princeton, 1977) p. 565. [ 81 A.J.R. de Kock and W.M. van de Wijgert, J. Crystal Growth 49 (1980) 718. [9] A.J.R. de Kock, Philips Tech. Rev. 34 (1974) 244. [IO] L. Farkas, Z. Phys. Chem. 12.5 (1927) 236. [ 1 I ] J.W. Corbett, H.L. Frisch, D. Peak and M.St. Peters, in: Computational methods for large molecules and localized states in solids, eds. F. Herman, A.D. McLean and R.K. Nesbet (Plenum Press, New York, 1973) pp. 67-78. [ 121 D. Peak and J.W. Corbett, J. Stat. Phys. 17 ( 1977) 97. [ 13 ] D. Peak and J.W. Corbett, Radiat. Eff. 36 ( 1978) 197. [ 141 A.M. Turing, Phil. Trans. Roy. Sot. B 237 ( 1952) 37. [ 151 V.A. Vasil’ev, Yu.M. Romanovskii and V.G. Yakhno, Autowave processes (Nauka, Moscow, 1987 ) (in Russian). [ 161 T.F. Ciszek, T.H. Wang, T. Schuler and A. Rohatgy, J. Electrochem. Sot. I36 ( 1989) 230. [ 171 T.H. Wang, T.F. Ciszek and T. Schuler, J. Crystal Growth 109 (1991) 155.