Interstep interaction in solution growth; (101) ADP face

CRYSTAL GROWTH

Journal of Crystal Growth 121 (1992) 643—655 North-Holland

o~*~o~

Interstep interaction in solution growth; (101) ADP face P.G. Vekilov ‘,Yu.G. Kuznetsov and A.A. Chernov Institute of Crystallography, Russian Academy of Sciences, Leninskii Prospekt 59, 117333 Moscow, Russia Received 22 August 1990; manuscript received in final form 20 February 1992

Michelson interferometry combined with X-ray topography was applied to study growth morphology and kinetics of the (101) ADP face in a flowing solution. Step velocity was measured as a function of different parameters. The data reveal a step—step interaction at average distances of the order of 1 j~m.This interaction is interpreted in the terms of surface diffusion. Parameters of step motion and adsorption on the surface are determined. Azimuthal growth anisotropy is related to crystal structure. A formula giving the azimuthally anisotropic hillock slope versus supersaturation on a complex dislocation source plus experimental dependencies of the hillock slope on the supersaturation for various dislocation sources enabled us to determine the 2 and the Burgers vector of the growth sources. “Quantization” of the measured free surface energy of the step riser a = 29 erg/cm hillock slopes in the whole supersaturation region supposes identical structure of the dislocation sources of equal strength.

1. Introduction Two pathways exist for a particle to enter the crystal from the surrounding mother solution. The first, diffusion through the solution followed by direct incorporation into the step from solution was proposed in ref. [1], and developed to its contemporary form in refs. [2—4]. The mechanism has been supported by experiments due to Kaischev, Budevski and other Bulgarian scientists for the electrocrystallization of silver only [5—71. Many experimental works on conventional solution growth appeared lately, where no step diffusion field overlap was observed, making the authors believe that surface diffusion is insignificant [8—121. The second mechanism, diffusion through the solution, adsorption on the crystal surface, twodimensional diffusion on the terraces towards the steps and incorporation into the step was discussed by Bennema [13—16].The best analytical description of this mechanism was given by Gilmer, Ghez and Cabrera [17,18]. Important

Permanent address: Zhk. Svoboda, BI. 13, Vh. B, Ap. 34, Sofia 1229, Bulgaria. 0022-0248/92/$05.00 © 1992

—

theoretical contributions were made in refs. [19,201. Numerous measured growth rate versus supersaturation curves were confronted mainly with the BCF surface diffusion formula for vapour growth (see ref. [21] and references therein). These curves are relatively simple and it is not too difficult to fit experiment and theory, making use of several free parameters. So these fittings are not convincing. Thus the validity of the surface diffusion mechanism for solution crystal growth in general is still an open question. In our earlier experiments [22], we found that the average velocity of steps forming the growth hillock increases with the supersaturation slower than the linear dependence demonstrating a step—step interaction. Analogous dependencies were recorded by Japanese colleagues [12], but the bend is not dwelt upon. So the aim of the present study was to investigate this phenomenon. We also try to analytically relate the slope of an anisotropic growth hillock around a complex dislocation source to the supersaturation, and hence to determine the Burgers vector of the dislocation source leading to growth and the free surface energy of the step riser on (101) ADP face.

Elsevier Science Publishers B.V. All rights reserved

644

P.G. Vekilov eta!.

/ Interstep interaction in solution growth; (101) ADP face

2. Experimental procedure

We suppose linear step kinetics: v, b,u. The back-influence of the already generated steps on the step source activity is neglected. We suppose also that the hillock anisotropy is due to anisotropy of the kinetic coefficient b, the effective free surface energy of the step rise a being isotropic. The retardation of the step rotation due to elastic stress fields around the dislocation source and to the Gibbs—Thomson effect is accounted for by the numerical coefficient 4, the dimensionless frequency of spiral rotation (see appendix B). Let the complex dislocation source consist of n dislocations (fig. 2, n 3). Let h0 be the smallest =

The experiments were carried out on an in-situ Michelson interferometer [221. In addition, a quick change of supersaturation facility was elaborated, making use of two interswitchable thermostats [23]. The solution replacement in the growth chamber and temperature relaxation took about 10 s. Preparation for the experiments and measurement were identical to those described earlier [22]. The saturation temperature of the solution in most experiments was about 35°C, pH 4.4. The solution flow rate was 50—60 cm/s if not specified. Supersaturation was calculated as de=

scribed in appendix A.

=

crystallographically possible step height. On (101) ADP face (space group 142d), h0 ~(101) [27,28]. Let the U01) component of the Burgers vector of the jth dislocation be equal (see appendix C for the meaning of the symbols): =

3. Dislocation hillock slopes 3.1. Theory Let us derive a formula to describe the dependence of the hillock slope on the relative supersaturation for the system with asymmetric triangular anisotropy and complex dislocation source (fig. 1) following the logic of refs. [24,25].

b~>=m~ ~(101), Em~=m, ~ b b mh j 1 ~‘

=

=

I

=

...

n.

0’

The linear dimensions of the dislocation source in the directions of step movement will be L1, L2 and L3.

1jiI~ I 1 _________________________________________

b

Fig. 1. (a) Outlook of the growth hillock on (101) ADP face. (b) scheme of the vicinal sectors on the face.

PG. l/eki!ov eta!.

/ Interstep

interaction in solution growth; (101) ADP face

645

tance between the steps on growth sector I

~(uniting 1 4:~1/~L Lbe (3)) will B+L* v1 (1),1 (2) and

1

L1_______ C2 L3 C3

1T+—+—, L*=_+_+_. )L2 L3 B=~— Hence the b2 b1hillock b3f2~I~12a slope on b11 sector b2 I isb3

p1 =h/11

‘,/~V~\\

hk 4 (2%/.~Qa

=

b1

Fig. 2. Scheme of formation of a growth hillock on a complex dislocation source containing three dislocations giving birth to three growth steps (C1, C2 and C3: points of dislocation outcrop: other notations in appendix C).

(i+1~

+

L2+l~ + L3+lc)

V1

W1

(1)

V3

V2

b1

The step velocity in the case of linear kinetics may be either (1) inversely proportional to step height in case of surface diffusion [17] or (2) independent of step height (unless this height is much larger than D/f3) in case of direct incorporation from the solution [31. (1) Height dependent step velocily. In this case steps generated by different dislocations have different velocities and, consequently, on their movement round the dislocation source form steps of one and the same total height h shortly after supersaturation is imposed. Let there be k secondary steps with height h, each of them containing q rn/k steps of elementary height h0. Then =

qh0

=

(rn/k)h0,

kh

=

rnh0

=

B+L*

For a single elementary dislocation L* h0:

h1.

(3)

So under stationary conditions the mean dis-

=

0,

=

p~=4h0kTu/b12~I~flaB. In the symmetric case b1 b2 h0 2~ 3 Q~

=

.

=

b3, b,B

=

(6) 3:

h0

=

i~’

(7)

4

(2)

=

kTr

=

where l~is the critical size of the two-dimensional triangular nucleus

h

L* (5)

biI

T=

+

=

f2Viuia

Consequently, each step turns around the group of dislocations during the period

B kTr

which coincides with the standard relationship [25,26]. For small o-, 2~/~flaB/kTu>> L* and

~1

=

w1b 1kTcr 2~/iQab1B rnp1. =

(8)

If v~is the velocity of an elementary step of height h0, then v1 (k/rn)v~ and R =pv (k/m)v?mp~ kR°. As we seen in this case, at low supersaturations, the normal growth rate should be proportional to the number of secondary steps and not to the total Burgers vector of the dislocation source. Normal growth rates measured on dislocation sources whose strength =

=

=

646

PG. Veki!ov eta!.

/ Interstep interaction in solution growth; (101) ADPface

differs by several times could differ by tens of percents only. (2) Height-independent step Velocities. In this

R

case steps originating on each dislocation move without interaction. Hence

~1

1,

=

=

(b 1/n11)

V1

P 1V1,

=

(9)

=b1/n11 =~/l~, h =(1/n) Eh1.

(10)

As we see in this case, the normal growth rate is proportional to the total Burgers vector of the dislocation source.

(V1T)/n,

R= Eh1/T=(1/T)Eh1=b±/T.

4

Uniting these two relationships we get

b

a

II

//

ifi 1

6,102

o

2

21345

.3 0

1

2

3456

7

4

d

3

C

0

2

2

1

o 0

1 ~ffl

0

1

1

2

3

6102 4

6,1~2

.

0

1

2

3

4

5

6

Fig. 3. Dependence of the slope of the growth hillock p on the supersaturation a-. (a), (b) and (c) in the three vicinal sectors on the crystals with (a) triple, (b) double and (c) unitary Burgers vector of the dislocation source (see text). (d) The same dependence for sector I of all the studied crystals.

PG. Vekilot et al.

/ Interstep interaction in

For the interstep distance we now have V

n

=

I2~/~fla I B + L* 4n l~, b1

1T =

The slope is b1 nl~

=

=

b

I.

kTu

(11)

B+L*~

)

p1

L*

=

+

for crystal B, and even more for A, mean stronger dislocation sources. Fig. 4 shows the i/p versus 1/u plot for crystal A. Points for low fall out of the straight line. This discrepancy might be due to a stronger dislocation source existing at lower supersaturations. Increasing a- decreases the radius of the critical nucleus (p~ t2a/kTu) and the distance between some of the dislocations in the complex source no longer fits the condition d <9.Sp~[ii, and the initial growth source falls apart to give the one that exists at higher supersaturations. The least squares method analysis of the three =

which is identical to eq. (5) — the slope of the hillock is independent of whether the step velocity decreases with step height or not. In reciprocal coordinates we have b1

647

is linear, this hillock is, probably, generated by a single dislocation (see eq. (6)). The greater slopes

1 b 1w~ 2~/~Qa

1(

solution growth; (101) ADPface

b~ 2v~fla 1 B— b14 kT a-

straight lines in fig. 4 gave: i/Pi

=a+c(1/u),

(13)

117.17

=

+

6.7(1/u),

l/p2= 150.15+9.2(1/u), I/p3 244.44 + 20.8(1/u). =

where b1

_____

a=

~L*, b1w~

c =

b14

2V~f2a B. kT

(14)

The kinetic coefficients of the steps for this crystal were (see below) b1 0.09 cm/s, b2 0.13 cm/s and b3=0.25 cm/s, which gives B=24.7 s/cm. Knowing 3, we get thatfrom T=eq. 308914) K and (2 1.059 x 10— 22 cm b 3/erg. 1/a=5.1 X iO~ cm This value coincides with the one obtained for the same dislocation source from the threshold undersaturation for etch-pit formation during dissolution [23]. For crystal B the same procedure yields b 3/erg. 1/a 3.4 X 10~ cm Applying the single dislocation formula (6) to the straight lines in fig. 2c, we get for crystal C: =

=

=

3.2. Experimental results and discussion Fig. 3 shows the measured p(u) curves for the three sectors of the growth hillock for crystals A, B and C. Since the p(o-) dependence for crystal C

i/~103 3

=

b

o

3/erg. 1/a 1.75 x 108 cm Comparing the obtained b 1 /a values gives =

0I

(bI/a)A:(bI/a)B:(bI/a)C~3:2:1.

Now we suppose that on crystal C we have b1 5.33 x 108 cm, i.e. the shortest possible K101~ ADP translation [27,28]. On crystal B, b1 10.66 x 108 cm, ai~don crystal A, h1 15.99 x 10—8 cm. All three (b 1/a) ratios give for the free =

1115,102 0

0,5

jO

i.~

2~0

Fig. 4. Plot of the reverse slope i/p versus the reverse supersaturation 1/a- for crystal A.

=

=

648

P.G. Vekilov et a!.

/ Interstep interaction in solution growth; (101) ADP face At higher supersaturations, the hillock slope is determined by the linear dimensions of the dislo-

Table i Kinetic coefficients b b

1 (cm/s)

a-.

b2 (cm/s)

b3 (cm/s)

A

0.09

0.13

0.25

B

0.13

0.18

0.35

surface energy of the step riser on (101) ADP face: 2. a 29 erg/cm This a value is approximately twice as large as the one reported earlier [22,29,30]. This discrepancy comes from the assumptions made ad hoc in these works that the Burgers vector and the step heights are elementary ones, instead of, probably, =

cation source, L,. So we may conclude that the “quantization” of the slope for all supersaturations is due to identical structure of the dislocation sources. Dislocation sources of one and the same strength have the same structure.

4. Step velocity Fig. 5 shows the measured velocity of steps versus supersaturation curves for crystals A and C, i.e. on a triple and on a single dislocation source. At supersaturations of up to 2%, they are strictly linear on all three sectors, the kinetic coefficients b 1 (V1 b,u) being given in table 1. However, at a-> 2% there exists a strong bending towards lower V values. Neither impurities nor 2D nucleation on terraces should be responsible for this since these phenomena would lead to the opposite — to non-linear acceleration of step motion at higher supersaturation [9,21]. We shall discuss the following three possibilities for the V(u) non-linearity: (1) the effect of temperature; (2) overlapping of bulk diffusion fields; (3) overlapping of surface diffusion fields. =

being twice as large. Crystals with elementary dislocations as growth sources only are a real rarity — in a series of about 30 experiments we had just two! As seen in fig. 3d, growth hillock slopes versus supersaturation dependencies on all the studied crystals fall apart into groups of coinciding curves, according to the dislocation source’s Burgers vector in the whole a- region. Equal Burgers vectors are responsible for this coincidence only for low

4cm~

i~ V,ld4cm/s

100 V, 1~

75

7 _ 75

a

b

U

Fig. 5. Dependence of the velocity of steps

L’

on the supersaturation a- for crystals with (a) triple and (b) single Burgers vector of the dislocation source.

PG. Vekilov et al.

/ Interstep interaction

(1) Temperature. The supersaturation if achieved by cooling the solution and thus the kinetic coefficient should decrease when a- increases. According to refs. [31,32] the activation energy for the dipyramid face growth is 51 kJ/mol. This value should be mainly attributed to the step velocity, since the surface energy decreases linearly with temperature, the temperature coefficient being ~ 0.1 erg/cm2~K [331.The highest supercooling in the region of non-linearity was 2°C,which gives bliilear/bfinal 1.11. The actual bending is 4 to 5 times larger. To confirm this estimate, the following experiment was staged: First, the V(o-) dependence was measured, in which the supersaturation was achieved by cooling the solution. Then the temperature was kept constant at the lowest value reached and water was added to lessen the supersaturation. The results are shown in fig. 6. For each of the three sectors, the V(u) data, obtained making use of these two methods of supersaturation, control practically coincide with one another. (2) Bulk diffusion. Our data were confronted with the bulk diffusion formula [2,31. It was possible to achieve the best coincidence (fig. 7) only for 6 as large as 10 x i0~ cm, which is improbable for flow rates of 50—60 cm/s [11,34]. It should be mentioned, however, that the model of refs. [2,3] assumes a completely stagnant bulk diffusion

in solution growth; (101) ADP face 0.005 —

—

—

—

—

—

V,cm/s

=

V,lO4cm/s

649

—

0 0

6

0.00

Fig. 7. Confrontation of the experimental points for v(o) with Chernov’s bulk diffusion model: ( ) simulation, ~ = 10 ~sm; (+) experimental points.

layer, which is actually not the case. We measured the R, p and V dependence on the flow rate for supersaturations a- 0.054 and a- 0.074. The flow rate changed from 50 to 80 cm/s, but all the measured quantities remained perfectly constant (fig. 8). A third contradiction is the greater step velocity for a single dislocation step source than for complex dislocation sources, which is incompatible with the bulk diffusion this model (figs. 5 and 9). (3) Surface diffusion. When discussing the validity of the surface diffusion model for solution crystal growth, two conceptual character difficulties appear: =

=

o-T~26.7~const

(a) In contrast to the vapour growth [35], the concentration of the crystallizing substance in the surrounding medium is high enough to provide sufficient flux towards the step riser, so that the piling up of particles on the terraces is not necessary.

50

25 6 102

0

_____________________________ 2

4

6

Fig. 6. Comparison of step velocity v versus supersaturation adependence in case supersaturation is changed by changing the temperature and by the addition of water: (.) sector I; (o) sector II; (x) sector III, a- changed by cooling; (U) a- changed by water addition.

dered the solute solution particles “coat” to and its own the crystal adhesion (comparable energy of (b) Since the face is covered with a partly orwith the dissolution heat) is high, the mean free path on the surface cannot be long enough to account for the measured growth rates. Let us consider those objections: In the case of transport-controlled step propagation (step kinetic coefficients are high enough),

650

/ Inierstep interaction

PG. Vekikn’ et al. -7

RlOcm/s 180

L I

1 I

is the crystal molecular density, and thus ne/Ceh Cs/Ce 5. Consequently, the volume flux will prevail. However, if we suppose D~to be of the

~

r

=

order of D, the picture will be reverse. The criterion (15) differs from the one proposed in ref. [1]by the dimensionless factor Ae/6, which is of the order 0.4—0.01, and neglecting it could lead to a wrong conclusion about the dominating growth mechanism in the case of transport control. Of course, one may include in eq. (15) the bulk-to-surface-layer density ratio, Ceh/fle,

160 150

_______________________________ 140

‘~‘

50

Y

60

70

80

I 2,0

I

1 8

t

t

into the effective A~value and rather arbitrary assumptions following eq. (15) would not be needed. However, in this case one loses the direct physical sense of the diffusion length Ae, and of its numerical value, which should be compared with an average interstep distance. Because of the density factor, it is wrong to compare A~and honly. Step propagation usually proceeds under ki-

+

16

50

60 4cm/s

120

v,lO

in solution growth; (101) ADP face

70

80

1 I

100

~

netics control. In this case [3], J~, f3~a-Ce and /3s0~~le/1L Their ratio is (16)

80

=

____________________________________

50

60

70

=

80 u,cm/s

Fig. 8. Dependence of (a) normal growth rate R, (b) slope of growth hillock p and (c) step velocity t’ on flow rate u at constant supersaturation: (•) a-i = 0.074 and (0) a2 = 0.0544.

the bulk flux per unit step length should be of the order

J

/3

~

The first term should be less than unity, since each ion has to throw away its hydration cover when entering the step, which is easier for the surface adsorbed ion than for the bulk ion. The

/

7cm,~ R,1O

and the surface flux

a

J 5=D5(n5n~)/A5.

150

With the notation C~— Ce a-ne, we get

=

a-Ce and n~

—

ne

=

100

(15) J~

2/s [31], 6 2 x i0~ cm At Dand 5.5 x 10~ cm cm [22], and supposing [34] A~ 5 x iO~ D 2/s, we get for the first term of the 5 iO~ cm above expression 1.3 x i03. However strong the adsorption, it is hardly possible that ne/Ceh can exceed iO~.Indeed, even for a full monomolecular adsorption layer (Oe 1), 11e C 5h, where C~ =

=

b

50

=

=

s.id2 0 Fig. 9. Dependence of normal growth rate R on supersaturalion a- for (a) triple dislocation source and (b) single disloca-

=

lion.

P.G. Veki!oL’ et a!.

100

/ Interstep interaction in

solution growth; (101) ADPface

651

~10”cm~~

V,1~cnvt 30

75

.

20

50

sector U 6~2.68%

10 0

V(6). p16) AVI6), p°O,86103

25

3

1

2

P ,io_ 3

Fig. 11. Dependence of the step velocity L’ on the hillock slope p at constant supersaturation a- = 0.0268 for vicinal sector II.

ov)61. p.1,20103

3,1o2 0

1

2

3

4

5

6

~

Fig. 10. Comparison of v(a-) dependence measured on a growth hillock with a change of the supersaturation slope and when the slope was kept constant.

In order to discover the nature of the influence of the slope on the step velocity, we may write V(a-)

second term also does not exceed unity at any positive adsorption heat, and so surface flux will prevail, So far the surface diffusion model is the only one complying with our results, although the increased movability of the adsorbed solute partides on the surface is still a mystery to us. To obtain evidence about the surface diffusion field overlap, we carried out the following experiments: Changing quickly the supersaturation (on the two-thermostat interswitchable apparatus [23]), we were able to measure the step velocity at different supersaturations, keeping the slope of the hillock constant for two values of p (fig. 10). The steeper straight line on fig. 10 corresponds to p 0.86 x iO~ and the more shallow line to p 1.2 x i0~. Thus, at p constant, V 15 a unear function of a-. The same technique was employed to measure the step velocity dependence on the hillock slope, keeping the supersaturation constant (fig. 11): the measured V decreases when p increases. The results obtained with the above consideration mean that bending in the V(u) curves is due to surface diffusion field overlap, which increases with diminishing step distance (1 h/p).

=ba-[1+f(p)]~.

The term in the brackets is a function of p only since V(a-): p constant is linear. In order to determine f(p), the plot of a-/V versus p could be useful. Fig. 12 shows =

a-/V

=

a~+ cp

1, a-> a- * (17) for all three sectors of the growth hillock, the coefficient c being equal on them. This is in a very good agreement with the formula of Gilmer,

v ‘cm 15

6?216

=

=

=

=

10 ~0

~

________________________________________ 1

2

Fig. 12. Plot of a-/LI versus p for crystal A (see text).

PG. VekiloL’ et al.

652

/ Interstep interaction in solution growth;

Ghez and Cabrera [17]. When the flow rate is unimportant, it takes the form

h —

V

1 —

(18)

0

00 -

<111>

0

A

IA. 1 I—~+—coth_L). A1 £2CeD A. 2 2A1

(19)

h

A

1, eq.

0

0 0 P

~

0<111) 0

1\

IA~1

0<101>

0 0

0

At low a-, when Ii>> 2A1 and coth(//2A,) (19) transforms into a-

0 0 0

Ai(2CDeIA• 1 V=a-— I—~+—coth---~--~, h A ~A, 2 2A,j and a-

(101) ADPface

~D

©

(20)

VjAju2CeDI~Aj2)’

which gives1~. the proportionality between forAt a-
v

and a<11

I~<<2A 1 coth(l,/2A,)

2A1/11

2(A1/h)p,,

=

Fig. 13. Structure of (101) ADP face showing different “jump” length in direction perpendicular to steps in vicinal sectors. The rectangular holds the building unit of the crystal.

and a-

h

A

A~1

~ flC~D ~

-

A +

f2CeD~”

(21)

and this is exactly the experimentally observed linearity of the a-/v versus p plot for a-> aThe binding condition at a- a- * (a- * is one and the same on the three sectors, which is another confirmation of the validity of the chosen model) (a-/v)11~ (a-/v)~ gives ~ +(h/A,), where A, is the Ghez catch area [19] =

=

A.

=

A~[(1 — Oe)/(1

=

+ a-Oe)] 1/2

For our a- < 0.074, the term in the square brackets is very close to unity, Supposing h 10.66 x 10~ cm (a so far shaky supposition), we are able to determine the A, in the three sectors (see table 2). =

Table 2 Values of A,

1’

x iO~

A, X i04

Sector

p,

(cm)

I II III

2.40 1.70 0.90

0.22 0.32

0.60

According to refs. [1,3], A~ a exp[(~ — =

UD)/2RT]. Since UD ~ ~ and the differences in UD in different directions on the surface should be still smaller, the jumps of the adsorbed partides along the surface will be almost equally probable in all directions. the not equilibrium adsorption sites on theHowever, surface are equidistantly situated, which leads to unequal jumps in different directions. If we now look at the structure of the (101) ADP face (fig. 13), we may see that in the direction perpendicular to the steps in different sectors, the parameters are: a1 5.33 X 10-8 cm, a2 6.53_08 cm and a3 13.06 X 10-8 cm, and A1 : A2 : A3 a1 : a2 : a3. This consideration, if true, connects the anisotropy of the growth hillock to the structure of the crystal. Comparing the theoretical formula (21) with the experimental data (17), we see that the coefficient c accounts for adsorption on the terraces between steps and that is why a- * is one and the same on all three sectors. Knowing c 2.2 x i03 =

=

=

=

s/cm, A

=

2.6 x i03 cm and the kinetic coeffi-

PG. Vekilou et al.

/ Interstep interaction in solution growth;

2.1 x iO~ cient of adsorption f3~d D/A cm/s. The coefficient A~is connected to the surface jump distance [17] and hence should have the same anisotropy as A. Its values are A(S~> 2.2 X i0~ cm, A~2> 3.1 x iO~ cm and A~3> 6.5 x i0~ cm. =

=

—

653

the mean free path of the adsorbed particles on the surface A~> 0.22 x i0” cm, A<~2) 0.32 x iO~ cm, A~S3> 0.760>< ~ cm; =

=

=

(101) ADP face

=

=

=

—

the resistance for incorporation into the step from the surface A~> 2.2 X 10~cm, A~3> 3.1 X iO~ cm, =

5. Conclusions

=

A~3~ 6.5 X i0~ cm. =

Our main results are as follows. A formula that describes the dependence of the slope of an anisotropic growth hillock on a complex dislocation source on the supersaturation is derived for the cases when the step velocity is dependent on (surface diffusion mechanism) and when the step velocity is independent of (direct incorporation mechanism) the step height. The hillock slope proves to be insensitive to the step propagation mechanism. A method based on the experimental curves of the growth hillock slope versus supersaturation for dislocation sources of different strength is proposed. It enables us to determine the free surface energy of the step riser on (101) ADP face and the total Burgers vector of the dislocation sources of the studied crystals. The values obtained are: a 29 erg/cm2, and the Burgers vectors are 5.33 A on a simple dislocation source and 10.66 A and 15.99 A on the double and triple dislocation sources respectively, Experimentally measured hillock slope versus supersaturation dependencies fall into separate groups of coinciding curves according to the dislocation source strength in the whole supersaturation range. This “quantization” means that, in

The anisotropy of the surface features on the (101) ADP face is assumed to be related to the crystal structure through the anisotropic mean free path due to different “jump” lengths in different directions.

Acknowledgements Gratitude for helpful discussions is due to Professor P. Bennema from the Catholic University of Nijmegen and to Mr. I.V. Alexeev.

Appendix A. Supersaturation

=

ADP, dislocation sources of equal strength have one and the same structure. The step propagation on the (101) ADP face is consistent with the surface diffusion mechanism. The following parameters were determined: — the kinetic coefficient of adsorption on the surface Pad

=

2.1 x iO~cm/s;

In experiments where supersaturation was achieved by cooling the solution, it was calculated as o’= [C(Te) — C(T)]/C(T), (A.1) where C(T) [31] is the concentration of the saturated solution at temperature T in mol ADP/ 1000 g H 20 (T is the running temperature and Te the saturation temperature of the solution). We chose molarities since it is not possible to calculate the molarities of the supersaturated solutions, their densities being unknown. In one of the experiments at constant temperature, the supersaturation was changed by adding water to an overcooled solution. Knowing the saturation temperatures before and after the experiment, we get from the solubility curve [31,32]

PG. Vekilol et a!.

654

/ Interstep interaction

the concentration C(T~)and C(T2). For the dissolved substance we have

in solution growth; (101) ADPface

Appendix C. List of notations a

m

=

=

C(T~)V11/1000 C(T2)(V() =

~v[(CT1/CT2)

—

iJ.

+

zlV)/1000, (A.2)

x 1000— C(T2) V{) +

b1

netic coefficient of steps Component of Burgers vector of dislocation perpendicular to studied face Coefficient before argument in linear

(A.3)

For the supersaturation it is now obvious (from eqs. (A.1), (A.2) and (A.3)) that

a-

a b

Free term in experimental linear dependencies Parameters for “jumps” along surface Coefficient of proportionality between step velocity and supersaturation, ki-

c

v

C

=

C(T2)

Ce

where J/ is the sum quantity of water added for each measurement.

Appendix B. Numerical coefficient for the triangular spiral

D D~ h i or

hillock

J or (J) Index to number dislocations and steps J5

In refs. [36—381,a numerical coefficient w1 was introduced that accounts for the delay in the spiral rotation due to elastic stress around the dislocation and the step curvature. Its value depends only on the geometry and is independent of the parameters of the crystal and the medium. For a circular step: =

wr~/V,~. = 0.33,

(B.1)

where w is the frequency of rotation of the spiral that is determined by the parameters of the systern, but does not depend on geometry. We may rewrite ~ =

2~-

(B.2)

—w(j’,

2rrr~

where v/27rr~ is the frequency with which hillock steps reach a fixed point on the face in the case of no delay. For a triangular step: 4, (B.3) =

2~r— 3l~

3.2V’ir~

k

Surface flux in step Bulk flux in step Number of newly formed steps Distance between steps in step train

L~ m

Linear dimensions of dislocation source Quantity of dissolved substance in solution

m

Size of total Burgers vectors of dislocation source in ~(101) units Number of dislocations in growth source Equilibrium concentration of particles per surface unit area Actual concentration of particles per surface unit area

n ne n5 p q R T T Te U

Uniting eqs. (B.1), (B.2) and (B.3), we get 4 0.547. For a square step w~ 0.42 and for a hexagonal step w1 0.36.

U0

=

=

=

(i’i

dependencies Actual concentration of solution Equilibrium concentration of solution Bulk diffusion coefficient Surface diffusion coefficient Step height Index to number sectors of growth

t’

V

Slope of growth hillock Number of elementary steps in really existing step moving along surface Normal growth rate Period for step to make full circle around dislocation source Actual temperature of solution Saturation temperature of solution Flow rate of solution Energy barrier for “jumps” along surface Tangential velocity of steps Volume of solvent in solution

PG. Vekilou eta!.

a

/3

/ Interstep

interaction in solution growth; (101) ADPface

Free surface energy of step riser Kinetic coefficient for adsorption

a

Surface kinetic coefficient Bulk kinetic coefficient Thickness of concentration layer above crystal Activation energy for desorption from surface Ghez catchrnent length

6

A, A5

Mean free path on surface Resistance for adsorption on surface Resistance to enter step from surface Supersaturation Frequency of rotation of growth spiral Numerical coefficient, dimensionless frequency of growth spiral rotation Volume that one effective ADP partidc occupies in crystal

A A a-

w

(2

655

[14] P. Bennema, [15] P. Bennema, [161P. Bennema,

Phys. Status Solidi 17 (1966) 563. J. Crystal Growth 1 (1967) 278. J. Crystal Growth 1 (1967) 287. [17] G.H. Gilmer, R. Ghez and N. Cabrera, J. Crystal Growth 8 (1971) 79.

[181R.

Ghez and G.H. Gilmer, J. Crystal Growth 21(1974)

93.

[19]R. Crystal Growth 22 [20] M. Ghez, RubboJ.and A. Aquilano, J. (1974) Crystal333. Growth 94 (1989) 619. [211 P. Bennema, J. Boom, C. van Leeuwen and G.H. Gilmer, Kristall Tech. 8 (1984) 659.

[22] Yu.G. Kuznetsov, A.A. Chernov, PG. Vekilov and IL. Smol’skii, Soviet Phys.-Cryst. 32 (1987) 584. Crystal Growth 102 (1990) 706. [24] A.A. Chernov, L.N. Rashkovich, IL. Smol’skii, Yu.G. Kuznetsov, A.A. Mkrtchan and AT. Malkin, in: Rost Kristallov (Crystal Growth), Vol. 15, Ed. El. Givargizov (Nauka, Moscow, 1986) p. 43. [25] A.A. Chernov and L.N. Rashkovich, Kristallografiya 32 (1987) 737. [23] PG. Vekilov, Yu.G. Kuznetsov and A.A. Chernov, J.

[26] E. Budevski, G. Staikov and V. Bostanov, J. Crystal

Growth 29 (1975) 316.

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[27] L. Tenzer, B.C. Frazer and R. Pepinski, Structure Rept. 22 (1958) 406. [28] M. Aquilo and C.G. Woensdregt, J. Crystal Growth 69 (1984) 527. [29] A.A. Chernov and Al. Malkin, J. Crystal Growth 92 (1988) 432. [30] Al. Malkin, A.A. Chernov and IV. Alexeev, J. Crystal Growth 97 (1989) 765. [31] J.W. Mullin and A. Amatavivadhana, J. Appl Chem. 17

489.

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[33]

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[38]

J.W.

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