Oscillatory crystal growth of alloys with the miscibility gap

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Journal of Crystal Growth 130 (1993) 101-107 North-Holland

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I' I,I,

,oo...Lo, C R Y S T A L GROWTH

Oscillatory crystal growth of alloys with the miscibility gap D.E. Temkin Institut fiir FestkiJrperforschung, Forschungszentrum Jiilich, D-W-5170 Jiilich, Germany and I.P. Bardin Institute for Ferrous Metals, Moscow 107005, Russian Federation * Received 2 February 1992; manuscript received in final form 10 September 1992

An interface motion during crystal growth of an alloy having the miscibility gap controlled by the interface kinetics is considered. It is shown that within a certain range of parameters, a continuous spectrum of oscillatory solution exists. The interface velocity is a periodic function of time for these solutions, and the composition of solid is a continuous periodic function of the distance.

1. I n t r o d u c t i o n

The major aim of the paper is to give a simple theory which predicts the possibility of a periodic growth of a mixed crystal in the region of immiscibility. There is much experimental evidence of compositional modulations in epitaxial layers of ternary and quaternary alloys AraB v [1-8]. These one-dimensional modulations have been observed in epilayers grown by vapor-phase epitaxy [1,6,7], by liquid-phase ep,taxy [2-4,8] and by molecular beam epitaxy [5]. Alternating layers with different composition, which were parallel to the growing interface, formed spontaneously instead of a homogeneous crystal. These modulations were not strongly periodic. The modulations were observed if growth was performed in the immiscible region. However, a rough estimate indicates that the diffusivity in Am B v compound semiconductors at ~tJt[.~lt~alt .--:,-,.-d

. . t. .V . w u,+!., ~ i

.L~,lttl.~.,Jtut.t.*Jt,~.,o . . . . . . t .... ~

IS

trw~

lnLv

fnr ~ny

significant composition modulation to be developed by bulk diffusion. This phenomenon has been interpreted as spinodal decomposition pc-

* Permanent address.

curring at the growing interface [4-6]. This mechanism, however, is not well understood yet. A theory of solidification of an alloy with a miscibility gap was developed previously [9,10]. It was shown that an oscillatory process, leading to formation of compositional modulations, inside the region of immiscibility, in a certain range of parameters develops. The composition modulations have the form of discontinuous oscillations with periodic jumps in composition. The characteristic length of the periodicity of these modulations is scaled with the diffusion length, l o , which is equal to the coefficient of diffusion in the liquid phase divided by the growth velocity. However, another microscopic characteristic length exists which may have influence on the periodicity of the modulations and which was ignored before [9,10]. This is a correlation length, Ic, which defines the width of a boundary between two domains of different composition [11]. We consider here a simple theory of growth from a completely stirred mother phase. In this case the characteristic periodicity of compositional modulations is scaled only by the correlation length lc and the formation of the modulations looks like a process of a spinodal decomposition at the growing interface. The main point of the theory is

0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

102

D.E. Temkin / Oscillatory crystal growth of alloys with miscibility gap

associated with the use of expressions for chemical potentials in a nonhomogeneous alloy which have been derived previously [12]. Using these expressions, two relations are obtained which describe the interface kinetics and which depend not only on composition, but also on the spatial derivatives of the composition of the growing crystal. The oscillations described here have the character of continuous modulations of the composition, contrary to the discontinuous modulations investigated previously [9,10]. These two types of oscillations with quite different periodicities may "interact". However, the theory of the "interaction", which should include both the diffusion in the liquid phase and the relations derived here for the interface kinetics, is not developed yet.

diffusion in the solid phase. Then these conditions take the form

V(1-C)=JA,

To illustrate the possibility of an oscillatory crystallization, we consider the following simple model. A two-component solid phase grows from a perfectly stirred fluid phase in which chemical potentials of both components,/x FA and IxFa, are given. The solid phase, which is a substitutional alloy of components A and B, is characterized by a phase diagram with unmixing. Thus, the free energy per atom of the solid, fs(C), considered as a function of its composition C (C is the mole fraction of B) has two minima and one maximum. We suppose that the interface between the solid and the fluid is flat and grows by the normal mechanism of growth. Let ~SA and /~sa be the chemical potentials of A and B in the solid at the interface. Within the scope of linear irreversible thermodynamics, we define the fluxes JA and Js of components A and B from fluid to solid phase at the interface as

(2)

where V = JA + JB is the interface velocity and C is the composition of the solid phase at the interface. The chemical potentials of a solid phase with a uniform composition C are

af~ /ZOA(C) =fs(C) -- (," aC'

(3a)

afs ~ ° B ( C ) = f s ( C ) + ( 1 - C) 0C"

(3b)

The free energy of a non-uniform solid solution contains a gradient term:

r= ~ 2. Model

VC=Ja,

f~(c)-,~ ~- ~

dx.

(4)

Here F is the free energy per unit area of the interface, 0 is the atomic volume, a is a positive coefficient and x is the space coordinate. Chemical potentials of the non-uniform solid at a given point depend not only on the composition at this point, but also on the space derivatives of the composi:ion, C ' - dC/dx and C "=- d2C/dx 2 [121 o ~SA = #SA(C) - ½a( C') 2 + aCC",

(5a)

/Xsa = Iz°s(C) - ½ a ( C ' ) Z - a ( 1 - C ) C "

(5b)

From eqs. (1)-(3) and (5), we obtain the following differential equation for spatial distribution of concentration, C(x), in the growing solid phase o/C" - ½o~(C')2[ C - r ( | - C)]./[ C 2 + r(1 - C) 2]

=fg(c) + { c ~ A - r ( 1 - c ) ~ -[c-

~( ~ - c ) ] £ ( c ) } / [ c ~

+ ~(1 -

c)~], #'K\

JA = WA(~A - IxsA),

JB = WB(~FB -- Ixs~),

(1) where 14;x and WB are kinetic coefficients with a dimension of velocity. (For simplicity, we neglect the cross-terms.) Now, we should write two conservation conditions at the interface. We ignore

and tke relation for growth velocity.

V/WB= [(1 - C)/.tFA + C ~ F s - f s ( C ) +½a(C")2]/[C2+ r ( 1 - C)2]. Here r = W j W A.

(7)

D.E. Temkin / Oscillatory crystal growth of alloys with miscibility gap

Thus, the constant B is proportional to the growth velocity V and must be positive or equal to zero, B >_0. As the function P(C) is known, the relation between space coordinate x (in the direction of the normal to the interface) and the composition C can be found by integration

Eqs. (6) and (7) describe the growth of a nonuniform and of a uniform solid phase. In the last case we have to put C ' = 0 and C " = 0. The order of the differential equation (6) can be lowered because the equation does not contain the variable x. Introducing a new unknown function p 2 ( c ) = ( d C / d x ) 2,

dC

x - x o= f P ( C ) '

(8)

(2/a)[BVI-C2+

(11)

where x 0 is a constant of integration.

we obtain a linear first order equation for p2. Its integration gives p2=

103

3. Qualitative consideration

r(1 - C ) 2 + f s ( C )

- ( 1 - C)/~FA -- C/ZFB],

Let us discuss in a qualitative way the behavior of the solutions given by eqs. (9)-(11) for different values of the constant of integration B (or V). We define two function fF(C) and ¢(C; B):

(9)

where B is an arbitrary constant. Substituting p2 from eq. (9) into eq. (7), we obtain

V / W e = BI[ C 2 + r( 1 - C) 2] '/~.

tp(C; B) - f s ( C ) + B I C 2 + r(1 - C ) 2.

(10)

(d)

(o)

Ill

zLI.I "

':-', i\j

LIJ UJ

o:: U-

i,

it

I

I

i~ 'J

Co CI CzCn |

I l

t I I I ~,,4 ~,~~ ~i~

(b)

,~ •~

-

f "'%

l,

\'!1

i

}

I

~

I

"NI

,

\ \,_,q \\-II fs~

C

I I I

It

,

\ ;I \~ "~

C(2) ..(~) o Lo

C

I

",,%

(e)

'

k',: C

%%. /

!

i I '

I_

[

Co ×

E

i i I i

I !

'

:

C

i

Ce

Fig. 1. Schematic drawing clarifying the appearance of a homogeneous and periodic distribution of concentration during growth of solid phase at a given temperature and given values of chemical potentials of both components in the mother phase (see text). In (c), the dependencies P - d c / d x on C and C on x are shown as well.

104

D.E. Temkin / Oscillatory crystal growth of alloys with miscibility gap

Here fF(C) is the mean chemical potential of the initial, fluid phase averaged with the composition of the growing solid phase at its interface. Eq. (9) can be written as

e 2= (2/a)[4,(C; B) - f F ( C ) ] . As p 2 ~ 0, the condition q~ > f F should be ful-

filled, for any physical solution. Thus, the periodic solution exists when the two curves, q~(C; B) and fF(C), have two points of intersection and q~> f F in the region between these points. On the other hand, as it can be shown, the points o f tangeney o f q~ and fF describe steady state growth of a homogeneous solid phase. Several different cases are possible: (a) At given temperature and chemical potentials t~ FA and t~ Fa, the condition fs(C) > f F ( C ) is fulfilled at any C (fig. la). No regime of growth is possible in this case. (b) The curves fs and fF intersect, as it shown in fig. lb. The curve q~(C; B) at some positive value of B will be tangent to the curve fv(C). The point of tangency (Bo, C 0) describes steadystate growth of a uniform solid phase. No periodic regime is possible. (c) The intersection of the curves fs and fv is like that shown in fig. lc. In this case, a periodic solution exists in a finite interval of B, 0 < B < B o. At B = 0 we have an equilibrium periodic distribution with composition C(x) oscillating in the interval C t < C(x)< C 2 (fig. lc). These equilibrium one-dimensional distributions have been analyzed previously [11] in connection with a process of decomposition of solid solutions. At the point B = B 0 the periodic solution disappears. At this point the period A goes to infinity, and the solution is like a "critical nucleus" [11] with a finite region of composition C a situated inside an infinite re3ion of the solid phase of composition C o (fig. Ic). At the same time, the point (Bo, C o) describes a steady-state growth of a uniform solid as in case (b). (d) The relative positions of the curves fs and fF are shown in fig. ld. In this case, periodic solution exists in the region B~o~) < B
composition (or of the growth velocity) equals zero. The disappearance of the periodic solution at the point B = B~02) is similar to that described in ease (c): the period is infinite and tile amplitude is finite. At the same time, the points (B(o1), C(o1)) and (B(o2), C(o2)) define regimes of growth of a homogeneous solid phase. (e) If the curve fF is situated far from fs (fig. le), the curve q~(C, Bo), which is tangent to fF, may have only one minimum instead of two minima and one maximum, which characterize the curve fs. In this case, at the point (B o, Co) , we have homogeneous growth, and no periodic growth at B > B o.

4. Growth in a symmetric system

To obtain some analytical results we consider, for simplicity, a symmetric system with equivalent components, having equal kinetic coefficients WA = W B - W and chemical potentials in the mother phase (ttFA = tZFB, in this case the mean chemical potential fF = ~FA and does not depend on C). For a symmetric system the critical point of decomposition in solid phase is situated at C = 1/2 and the free energy of the solid solution is an even function of r / - - C - 1/2. We suppose that the temperature is not far below the critical point and use power-law expansion for fs: fs(7/) =.to - tit/2 + br/4,

(12)

where t~ and b are positive and a/b << 1. At a given temperature, the compositions of two coexisting solid solutions are

rl = +tie- +(a/2) l/z << 1,

(13)

where a = ti/b. For the symmetric system, we obtain from eqs. ~,.i),,

~Jt~.~;

f.tllt~t

IkJ. g..,.~,.

P2(r~) -- ( 2 b / a ) [ t3/2 - a - ( a - / 3 ) r t 2 + r14], (14)

V/bW=/3(1 - 2rt2),

(!5)

where /3= 21/2B/b and A = ( f v - f o ) / b . Here we took into account that for a growing phase

D.E. Temkin / Oscillatory crystal growth of alloys with miscibility gap

r/= "" ~7~ << 1 and substituted (1 + 27/2) instead of

(1 + 47/2) t/2.

4.1. Growth of a homogeneous solid phase For homogeneous growth the composition r/(x) is constant and r / ' = r / " = 0. We can obtain two equations for an unknown composition and growth velocity from eqs. (6) and (7) (with C' = C" = 0 ) or from eqs. (14) and (15) taking the right-hand side of eq. (14) and its derivative with respect to 7/equal to zero. Thus we obtain: r/4 _ ( a - / 3 ) , 1 2 - a + / 3 / 2

= O,

(16a)

Only positive or zero values of W" and /3 have a physical meaning. The following solutions of eqs. (16) satisfy these conditions:

13= 1 + a - ( 1

when the mean chemical potential of the mother phase fF is greater than the free energy f0 of a growing phase, i.e., it is possible at A > 0. At the point A = a/2, two other solutions, given by eqs. (18) and by the lines MN, M ' N ' , and M"N' in fig. 2, bifurcate from the symmetrical solution. These new solutions disappear at the point of equilibria (points M, M', and M" in fig. 2) at which fF is equal to the free energy of the equilibrium solid phase fs( + r/e) = fo - a2/46. At this point A = - a 2 / 4 , the parameter/3 (and the growth velocity) is equal to zero and the compositions coincide with the equilibrium ones, r / =

(17)

A>__0;

4.2. Oscillatory growth The periodic solutions of eqs. (14) and (15) exist in the range of parameters/3 and A, which coincides with the shaded region OMNO in fig. 2a. Eq. (14) can be written in the form

+ 2 a - 4 A ) 1/2,

r/ = _+ ( ~~a - ~ /~3 ) 1/2 for

+ ( a / 2 ) 1/2.

+r/e=

(166)

47/3 -- 2( a - - / 3 ) r / = 0.

/3 = 2 A , 7 / = 0 for

105

2b

1 - ~ 1a 2 < A < ~ a ,

(18)

The dependencies /3 and r/ on A, given by eqs. (17) and (18), are shown schematically in fig. 2. Eqs. (17), and the corresponding lines ON and ON' in figs. 2a and 2b, respectively, describe growth of a solid phase with symmetric composition r / = 0. It is clear that this process is possible

where

rl l ,2 =

((a-fl)/2 ..T [(a --

)2/4 + A --

/211/2} '/2

(20)

(the minus sign corresponds to subscript 1). Performing the integration (11) with the initial condition x = 0 at r / = - r/l, we obtain n

x= f ,,:

dr/

e(,)

--r/1

= + fn

rl,

dr/

/2

-'7' [-~- (r/2 - r/2)( ~r/~- r/~'].)1] I

N' A M" ' / I

(b)

Fig. 2. Dependencies of fl on A (a) and r/on a (b) for growth of a homogeneous solid phase (schematic). The periodic solutions exist in the shaded region OMNO.

=+

a)1/2 1 - - F ( q , r/l/r/2), ~ r/2

(21)

where F(q, t) is an elliptic integral of the first kind, q = arcsin r//r/1, and r/1 and 7/2 are defined

106

D.E. Temkin / Oscillato~ co,stal growth of alloys with miscibility gap

by eq. (20). Eq. (21) determines x as a function of r/. The spatial periodicity A is given by

a = 2j_n,m' drlp(71)= 2 (2°t)

rl2

.

In another limiting case, when/3 increases and approaches/3 M, the period increases indefinitely and the amplitude rt~ approaches a finite value:

(22) A=2

Here K(k) is the complete elliptic integral. Thus, the periodic regime of growth exists for given values of parameters A a n d / 3 which correspond to the region OMNO in fig. 2. In this regime, a periodic distribution of concentration, which is defined by eq. (21) and is characterized by a periodicity A (eq. (22)) and by an amplitude rh (eq. (20A is formed. The solid phase grows with a periodically varying velocity given by eq. (15). For a given value of a , the poss~le values of /3 are determined by the conditions (see fig. 2a)

0
--a2/4
(23a)

0 < A < a/2,

(23b)

where /3m and /3 M are defined by eq. (I7) and (18), respectively:

tim = 2A,

tim =1 + a - ( 1

+2a-4A)

!/2

(24) Using the expansions of the elliptic integrals, we can consider different limiting cases. tn the vicinity of the line ON ~ith ~3 approaching /3m, we obtain from eqs. (20) and (22), that the period A remains finite and the amplitude r/z goes to zero:

A = ~ [ 2 a / b ( a - 2a)] '/2, rt, = [(/3 - / 3 ~ ) / 2 ( a -

(25a)

2A)]:,,2

-,(x) = - ~ cos 2 ,.-,x/h.

(96a~

and from eq. (15)

I !i

-

(27a)

a -/3M ) 1/2 r/l =

2

"

(27b)

These expressions are correct under the condition ( t i m - / 3 ) / ( a -/3M )2 << 1. The compositions r/l from eq. (27b) coincides with the composition of the homogeneous solid phase given by the second equation (18). As /3 approaches /3M, the growth velocity approaches the velocity of growth of a uniform solid phase of a composition + r h or -r/~. In this case, the structure of the solid phase consists of equally sized domains, with compositions + 77~ and - 77~, separated by antiphase boundaries with a width of the order of

I¢ = (a/ba) ,/2 = ( a / d ) ,/2,

(28)

and alternating with periodicity A. We have to point out, however, that A goes to infinity slowly, logarithmically. Therefore, in both limiting cases a scales with t c, and the characteristic scale of the periodic structure remains the microscopic length l¢. Only in the vicinity of the point N (fig. 2), where (1 - 2 A / a ) << 1, may the periodicity. A, corresponding to eq. (25a) be much greater than /c:

A/l,:-..(1 - 2A/a) -~/2 >> 1.

(29)

(25b)

These emttons are correct for (13 {3~)/(a 2A)2 << 1. In this case we obtain from eq. (2I)

V(x)= 2bwa 1 + 2rl~ ~

IX ] 1/2 4(a --/3M) In b(a_/3M) (/3M-/3) w2,

1 cos z ~ -

A , 26b )

Here ~ and rt~ are given by eqs. (25).

5. Conclusion A soiid soiufion, having a miscibiiity gap, can grow from the mother supersaturated phase homogeneously or with a periodic distribution of components. One or three regimes of the homogeneous gro'*xh exist, depending on the supersatvrafion: one regime at large supersaturations and three below some critical supersaturation. In the last case, a continuous spectrum of periodic

D.E. Temkin / Oscillator), crystal growth of alloys with miscibility gap

modes of growth exists besides the regimes of homogeneous growth. These modes are characterized by different values of the amplitude and the spatial periodicity of compositional oscillations in the growing phase and by different growth rates.

Acknowledgements This work was accomplished during my visit at the Forschungszentrum in Jiilich. I would like to express my gratitude for hospitality and financial support and to thank K. Kawasaki, H. MiillerKrumbhaar and V. Pokrovsky for fruitful discussions. References [1] K. Maksimov and E.N. Nagdaev, Soviet Phys.-Dokl. 24 (1979) 297.

107

[21 S.K. Maksimov, L.A. Bondarenko, V.V. Kuznetsov and A.S. Petrov, Soviet Phys.-Solid State 24 (1982) 355.

[31 P. Henoc, A. lzrael, M. Quillec and H. Launois. Appl. Phys. Letters 40 (1982) 963.

[41 H. Launois, M. Quillec, F. Glas and M.J. Treaey, in: Proc. 10th Intern. Symp. on GaAs and Related Compounds, Albuquerque, NM, 1981, Inst. Phys. Conf. Ser. 65, Ed. G.E. Stillman (Inst. Phys., Bristol, 1982) p. 537. [51 P.M. Petroff, A.Y. Cho, F.K. Reinhardt, A.C. Gossard and W. Wiegmann, Phys. Rev. Letters 48 (1982) 170. [61 M. Quillec, H. Launois and M.C. Joncour, J. Vacuum Sci. Technol. B 1 (1983) 238. [71 S.N.G. Chu, S. Nakahara, K.E. Strege and W.D. Johnston, Jr., J. Appi. Phys. 57 (1985) 4610. [81 A.G. Norman and G.R. Booker, J. Appl. Phys. 57 (1985) 4715. [91 M.B. Geilikman and D.E. Temkin, Soviet Phys.-JETP Letters 36 (!982) 292. [10l M.B. Geilikman and D.E. Temkin, J. Crystal Growth 67 (1984) 607. [111 A.G. Khachaturjan, Theory of Structure Transformations in Solids (Wiley, New York, 1983). [12l E.A. Brener and DE. Temkin, Soviet Phys.-Cryst. 30 (1985) !40