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The interfacial and volume transport processes during LPE growth of garnets
Journal of Crystal Growth 58 (1982) 537—544 North-Holland Publishing Company
537
THE INTERFACIAL AND VOLUME TRANSPORT PROCESSES DURING LPE GROWTH OF GARNETS B. VAN DER HOEK RIM Laboratory of Solid State Chemistry. Catholic University, Toernoo,veld, 6525 ED Nijmegen, The Netherlands
and W. VAN ERK
*
Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands Received 4 February 1982; manuscript received in final form 27 April 1982
An analysis on the kinetical processes during the epitaxial growth of garnet films is presented. On the rough (Ill) garnet faces the transport process consists of the diffusion of growth units through the mass transfer boundary layer and the interfacial process; both processes have about the same mass transfer resistance. On the flat (110) faces the model of Gilmer. Ghez and Cabrera (GGC) was applicable. With help of growth rate measurements on garnet films grown on spherically shaped gandolinium garnet substrates, we were able to estimate the parameters which determine the growth process. The values of the transport process parameters of the GGC model were A =80 pm, A =70 nm and A~=225 nm. For low misorientations. misorientation angle ~<0.3°, the steps do not interact and the surface process is rate determining. For increasing misorientations (~>0.6°), the step interaction increases and also the volume diffusion becomes important. The volume—surface incorporation resistance is about two times higher for (110) than for (Ill faces.
1. Introduction For magnetic bubble memory devices one needs a thin film of magnetic garnet on a non-magnetic garnet substrate. This film is grown from a solution which primary consists of lead oxide, while the solutes are the garnet-forming rare-earth and ~on oxides. The circular substrate (often Gd 3Ga5O12 (GGG)) is dipped into the supersaturated solution, and rotated about the vertical axis in a horizontal plane. Growth starts im-
mediately and after a few minutes the film has the desired thickness of I p.m. This film growth process is called Liquid Phase Epitaxy (LPE). Many studies about the mechanisms of the LPE growth of garnets have been published. However,
*
no common opinion exists about the growth mechanism or about even the order of magnitude of the parameters which determine the transport processes of growth units from the bulk to the surface and to the step. In terms of the PBC theory the orientations (110), (121) and (100) are flat faces [1], and spiral growth or nucleation growth is to be expected. Indeed, both spiral hillocks and nuclei have been observed by Tsukamoto [2] on the (1 10) surface of LPE-grown Y3Fe5O12 (YIG) garnets.
Present address: Light Development, EWD 431, Eindhoven,
However, different values of the parameters which determine the various transport processes of growth units from the solution into the step are reported for the (110) faces [3—6]. In refs. [3,4] it is concluded that the first desolvation of the growth unit is the rate-determining process. However, in refs. [5,6] it is argued that the rate-determining process is the surface diffu-
The Netherlands.
sion of growth units to the step.
0022-0248/82/0000—0000/$02.75 © 1982 North-Holland
538
B. ian der lloek, ti
ian Erk
Inieu/acul and colunic transport pr~sesio
The widely applied { 111 } orientation produces garnet faces which are rough on atomic scale [I]. Therefore, all growth units which enter this face are immediately attached and no step growth occurs. Görnert et a!. [71assume that this roughness leads to a negligible resistance for entering the surface, so that the only transport resistance is the diffusion of growth units towards the crystal surface. Others [5,8—10] conclude that the resistance of the volume diffusion and interfacial mass transfer processes are of the same order of magnitude. It is the aim of this paper to discuss the growth mechanisms on the (ill) and (110) faces of LPE-grown garnets, and to determine the niagnitude of the various resistances in the transport (i.e. growth) process. We will use data of growth cxperiments on slightly spherically shaped substrates, together with experimental data reported in the literature, 2. Growth of the (111) garnet faces In this section we will discuss the possible mechanisms of LPE growth of (111 } garnet faces. It was shown by periodic bond chain analysis [1] that the { 111) faces of all garnets are rough on atomic scale, indicating that no nucleation harrier is present. In that case, the mass transfer process can be divided into two parts: diffusion of the solute through the diffusion boundary layer (resistance 6/D) and the actual interface process (probably consisting of desolvation of the growth unit) with a resistance of 1/k1~1.Here D is the diffusion constant, 6 is the thickness of the diffusion boundary layer and k111 is the kinetic coefficient for the surface process. By putting the diffusion and interface processes in series, the following expression for the growth rate ~ of the (111) faces was obtained [101: ~T~II1 fiiiRT.,Tg(Cs/Ci~ 1)
+ D
—~--—
(I)
It111
where 1/ 1 and ~XT are saturation temperature, growth temperature and supercooling, respectively: R is the gas constant: ~ is the enthalpy
of solution: and (~ and C~ are the concentration of solute in the film and in the solution. respectivelv. Both for the FuYhJe 5O1. (EYIG) [8 II] and for the Y~Fe5O1,(YIG) [5] system it was shown that eq. (1) gives a good description of the growth rate at various values of the temperature and supercooling. It was also shown, that the interfacial process was first order with respect to the interface supersaturation [5.8—li]. so that the parameter It is a function of temperature only. The hydrodynamic environment of the interface is known, since for all experiments a circular, flat substrate was used, which was rotated around its perpendicular axis. In this case, the Navier Stokes equations can be solved and the thickness of the diffusion boundary layer 6 is [121: I
--
I
I
ô — 1 .61 v I) w . (_) 2/s) and w the where i’ rotation is the kinematic angular velocityviscosity (rad/s). (m By measuring growth rates as a function of the total resistance sum of the growth process (right-handed side of eq. (1)) can be separated and 6 /D and It 111 can he determined. Van Erk [5] introduced the Nusselt number for the growth process of LPE garnets. being the ratio of the volume and surface resistance: Nu It 6 ~L) (~~) III
/
-
If Nu >~1. the growth rate is diffusion limited: if Nu ~ 1, the surface process is rate determining. At w 100 rpm. the value of Nu for the YIG system increases from 0.5 at 800°C to 1.5 at 1000°C[10]. For the EYIG system somewhat higher values for Nu were found [91. Gornert et al. [7] give a different interpretation of their growth rate measurements on { Ill } Onented faces. They use the same horizontal dipping technique with axial rotation. The substrate is Gd~Ga5O12 (GGG) and the film consists of (YSmy~(FeGa)5O12. They assume that the resistance of the interfacial growth process is negligible, i.e. 1/k~11~6/D or Nu>~>I. In their opinion. the transport of growth units is not only conveyed by forced convection, hut also by thermal convec[ion. They introduce an effective boundary layer (6~) which consists of contributions by forced
B. van der Hoek, W. van Erk
/
Interfacial and volume transport processes
convection (sr) and thermal convection (6~).The former is depending on the rotation velocity (eq. (2)), the latter is assumed to be a constant contribution, independent of ~e. An argument against this assumption is that at higher rotation velocities the solution is better homogenized, undoubtedly leading to a strong reduction of the temperature gradients and thus to a decrease of the thermal convection. isIndeed, magnitude of the[13] natural convection usuallythe very low; Morgan estimates a flow of 0.0003 rn/s for a rather large temperature difference of 8°C over 0.03 m. The flow due to forced convection is much larger: 0.0l—0.03 rn/s for the applied rotation velocities of 30—300 rpm [14,15]. Extra transport can occur due to density gradients close to the wafer. The solvent lead oxide has a much larger specific density than the garnet-forming oxides, so that the solution density close to the crystal—solution interface is reduced [16]. This effect contributes to the —
—
natural convection, but difference if it was should important, significant growth rate be cx-a pected for the grown garnet film at the upper side and the film at the lower side of the horizontally dipped substrates. This is usually not observed, however, Therefore, rather than assuming an extra constant mass transfer contribution due to natural convection, we assume that the kinetic coefficient k111 is not necessarily infinite. A physical effect which may lead to a finite k iii value is the following. It has been shown that the metal ions are surrounded by more 02 ions in the solution than in the solid garnet film [17]. During the incorporation of the growth units in the top layer of the crystal, some activation energy is necessary for the stripping (desolvation) of the 02 molecules, leading to a surface incorporation resistance. Regarding the considerations made above, it is in our opinion not justified to neglect the volume—surface incorporation resistance, while the neglect of the natural convection as done in refs. [5,8—11]is not extravagant, According to eqs. (1) and (2), a straight line is to be expected if we plot l/f11~versus ~—I/2 with intercept proportional to 1/k1 and slope proportional to ~/D. Such a plot is shown in fig. 1 for the growth rate data of Gornert et al. [7]. A straight
~ ~
539 3
f
TO
5pTT(~/cl)X10 IS/N) ~
~
~
2 1
x1~(SKIrnI
~
2 1
2 (sIrad)~
2 ~~jlI 4 6 Fig. 1. Growth rate data of ref. [7] for LPE growth on (Ill) oriented substrates versus rotation velocity, plotted according to eq. (1), using ~HI/R = 12500 K.
line fits well with their experimental data and if the reported external experimental parameters 7~ = 1152 K, Cs/CL 62.7, p 2.9 X 106 m2/s are used, we find from the intercept and slope k 111~~H1/R K for rn/stheand ~HID/R 2/s.0.17 Taking value of ~HI/R 3.75 106 12500KKm[5],we find D 3 10b0 rn2/s and k 1 1.3 X 1O~rn/s. If the rotational velocity w is 100 rpm we find 8/D 1.3 X l0~s/rn or Nu = 1.7 at 879°C,i.e. surface and volume transport resistance are about equal. This Nusselt value is somewhat higher than the values found by Van Erk [5,10] or by Ghez and Giess [9]. 3.1. Growth of the (110) faces of LPE garnet films
As mentioned before, the (110) faces of all garnets are flat faces according to a periodic bond chain analysis [1]. On a perfect oriented face, a BCF-type step growth mechanism is expected; the steps originate from spirals or from two-dimensional nucleation. Both spirals and nuclei have been observed by Tsukamoto [2] on surfaces of LPE-grown garnets. A third step source can be obtained by misorientating the GGG substrate. The ideal picture of a misoriented surface is an infinite row of equidistant, parallel and straight steps. Indeed, a regular sequence of parallel steps with a height of 0.9 nm has been observed on films grown on (110) faces of GGG misoriented in the ~1 10) direction [6]. Measurements of the growth rate of (110) faces as a function of the misorientation, and thus as a function of the step spacing,
540
B. van der Hock, iF van Lrk / Inter/acial and volume transport proce
have been reported by Gornert and coworkers [3.18—20]. Two different kinds of misorientated substrate have been employed. One can use several GGG substrates, each having a constant misorientation over its surface, or use a single substrate, which has been spherically shaped, i.e. the misorientation elapses continuously over the surface. We have performed growth experiments using the latter technique and the results will he reported in section 3,3.
solvation). A the mean diffusion length of a growth unit during its stay on the surface and A~ the characteristic length of the second desolvation, i.e. the jump of a growth unit into the step. The difference in chemical potential of fluid and solid p.~— p.~ is given by ~p.. Eq. (4) holds if the coupling factor h A/A -~ I. In the relation between growth rate and misorientation limiting cases for small and large spacings ~.v can he distinguished: (i) ~X >> 2A, so that coth( ~x/2A) I and the
It has been demonstrated that two-dimensional nucleation hardly occurs at low supercoolings (~T <15°C) during LPE growth [5]. Thus, at low supersaturations growth in dislocation-free misori-
growth rate satisfies: I
~ 1) ~H L
f~
C
5RT~T~
ented GGG substrates is only due to the advance of the misorientation steps. 3.2. Growth rate of face covered with an equidistant train of straight steps
The growth mechanism of a stepped face can he described by two models. The Chernov model [21] assumes a direct incorporation of growth units into a step; the mobility of growth units ofl the surface is zero. The second model is the Gilmer, Ghez and Cabrera (GGC) model [22], which is an extension of the BCF [23] model. In this model the path of growth units is diffusion through the boundary layer, entering the surface (possibly ineluding desolvation), diffusion along the surface towards a step, and finally jumping into the step
6](() +
pendicular growth rate f~ of an equidistantly stepped surface (step distance ~x) follows from the GGC model:
L
-
~ + AA~2~X + A ~x coth 2AD °° CL D DA ~H 1 ~T — RTT — ~7’
2A
Ad ±
.
(5)
RTT
~ ~
+
A
+_7
AAd .
(6)
In these two cases, two straight lines are expected if 1/fill) is plotted versus ~.v/d. It can easily he verified that for the intersection point of these two lines ~x 2A. The ratio of slope (i) and (ii) is: slope(i)/slope(ii)
I
I + A/2A.,.
The Chernov model results in: K, 6/D) + ~x/K-j.
//~,
[(
(7)
(8)
if the unstirred layer 6 is much larger than tile step spacing, a condition which is satisfied in our case. Contrary to the GGC model a single straight line is expected for the plot of i/fl]) versus ~.v. ..
3.3. fttttng
f
AAJ ~
where d is the step height. (ii) ~x<2A, so that coth(~x/2A) 2A:~x. and the growth rate satisfies: I C D ~H
(second desolvation). Contrary to the Chernov model direct incorporation of growth units into
the step from the solution is excluded in the GGC model. In the approximation of putting volume and surface transport processes in series, the per-
(
~
of
experimental data
the previous sectionsrate it of becomes thatFrom by measuring the growth garnet clean films
1
(4)
with A being the characteristic length of the transition of the liquid phase to the surface (first de-
on misoriented substrates values of the transport parameters can be obtained. Growth due to spirals or nucleation can he avoided by using carefully polished GGG substrates and keeping the supercooling below 15°C.
B. von der Hock, W. van Erk
/
541
Jnterfacial and volume transport processes
The growth rate of spherically shaped (110) GGG substrates was measured. The radius of curvature was 0.34 m, which means for a I inch wafer a misorientation angle ~ between 0 and 2°. The curvature of the substrate was measured using
25
a stylus technique. The films were grown from a flux melt consisting of 1651.5 g PbO, 36.76 g B2O1, 11.52 g Y2O3, 1l5.05g Fe203, 4.51 g Ga2O3 and l3.30g La203, with a saturation temperature of 938°C. Films were grown by the standard LPE technique, using an axial rotation at 100 rpm and rotation reversal every five revolutions [5]. The first two films were grown at a supercooling well below the critical
20
/ i
C
~
CSTSTSR
1 bT ~HiD
point is formed due to a motion of misorientation steps with orientation This means that the measured growth is too small. order approximation therate correction term Infora first the thickness is t2(8f/~)2/2r, where t is the time and r the ~
radius of curvature. This gives for the first experiment a correction of 0.06 p.m (15%) at small ~‘, and
a/
/
/
~T=83r
/ / // // / //‘ /
//
~
/
/t~T=7~BK
-
/
/ /
10
/7
value for two-dimensional nucleation. The supercoolings were 8.3 and 7.8°C and the dipping times were 1828 and 928 s, respectively. The third dipping was performed at ~T=26.5°C and lasted 328 s. The thickness of the film increases going from the centre of the wafer towards the periphcry. When using a sodium lamp, a fringe pattern is observed and every fringe corresponds to a height difference of 0.1167 p.m. For a film thickness of 4.5 p.m at ~ 1.7° this corresponds to about 40 data points in the growth rate versus misorientation plot, The thickness at a point on the substrate with misorientation ~ is not a direct measure of the growth rate at this misorientation. The layer at this
/
~ 10 ~m)
15
/
/~
/‘‘
/~ ,/
~
,//,
/~‘
//‘
// -
~x1d 100
200
3à0
~00
500
600
Fig. 2. Growth rate of (110) garnet LPE films in dependence on the surface misorientation. Sample a: ~T=8.3°C. = 100 rpm and dipping time is 1828 s, sample b: ~T7.8°C, i~= 100 rpm and dipping time is 928 s; sample c: ~T=26.5°C, sa “100 rpm and dipping time is 328
2/s for D~Hi/R, while Cs/CL = 48. The plot m shows that for experiments a and b not a single straight line is obtained, but two lines in which the two regimes of the GGC model can be dis-
for the second experiment a correction of 0.02 p.m (9%) at small A facet is formed in the centre of the wafer, so that the smallest misorientation angle for which the growth rate can be determined is
tinguished. This shows that the Chernov model is inadequate and the GGC model with the two regimes (eqs. (5) and (6)) can be applied. The lines of the two regimes intersect at ~x 2A and the ratio of the slopes is 1 + A/2A~(eq. (7)), which
0.08° and 0.04° for experiments I and 2, respectively.
leads to the values of A and A., given in table 1, when is assumed that d 0.9 nm. The right-hand
~.
In fig.2, a plot is shown of I/f 1~0versus ~x/d, measured along one the (110) the steps advance theoffastest. Thedirections, lines a, b where and c in fig. 2 refer to experiments with supercoolings of
side of eq. (6) is equal 61i() to 6110+A at ~x=0at + A + 2AA~/X2 (intercept) andintersection equal to point. From these two 2A, the
8.3, 7.8 and 26.5 K, respectively. We used the
value of 6 can also be determined with eq. (2), using D = 3.8>< l0~° m2/s and m’ 3 X 106 m2/s
previously [5] established value of 4.75
X l0~
K
values we find from fig. 2 A and 6~~ (table 1). The
B. van der Hock. ft’. van ErA
542
lnterfm titi and volume transport p101 e
lable I
Values
of
ihe iransport parameters
Experiment a Fxperinient h Ref. [181 .0
of
the GG( model obtained from experimenial (Ito) gross lb ate
A )nni)
,\ (flu)
.\
67 10 77~ 10
225
70 77 65
60
15
230 25))
+
30 30 40
)pnu)
61111 (tIm)
37 23 35
ii)
I)) 15
-
iS (5
2))
data
6 0
~ in)
43.S 43 3 43.3
Computed (eq. 2)).
[5]. For w — 100 rpm we find 6 -- 43 p.m. Without the application of the above mentioned correction for the curvature of the wafer, the difference between the slopes in fig. 2 would be even Iarger.It can be seen that the value of 61])) derived from fig. 2 tends to be somewhat smaller than the cornputed (eq. (2)) value. However, one should realize that eq. (2) is derived on the assumption of a uniformly absorbing surface, i.e. all lateral gradients in the volume concentration field are neglected. On misoriented (110) faces, the steps act as sinks, so lateral gradients are present. Due to the anisotropic nature of (110) garnet faces, the step incorporation process for steps parallel to one of the <110) directions (the steps
where the diffusion fields of neighbouring steps overlap, thus reducing the surface concentration of growth units. Hence, at small step spacing. the surface supersaturation is decreased helo~ the point where two-dimensional nucleation is significant. The transition to the nucleation regime occurs at ~x’~ 120 nm. The catchment area of a step is defined as the width U’i~ of a strip adjacent to the step ~shere surface density gradients are significant [24]. -I his catchment area is not equal to the interaction distance l~.which is defined as the step spacing below which neighbouring steps interact mutuall\. In our case of small coupling coefficient. ~ is equal to 2A[24,25] and
advance along <001)) is more difficult then for steps parallel to one of the <001) directions (steps advance along <110)). Thus, the value of A, will be higher (25—50%) for steps parallel to (110) and the distinction between regime (i) and (ii) will he smaller, since the difference of the slopes of the straight lines is smaller, see eq. (7). This explains why we were not able to find the two cases (i) and (ii) in the plot of the growth rate versus ~x measured along <001). From table 1 it can be seen that the coupling factor b — A/A is about 0.001, which a posteriori justifies the use of eq. (4). The results from experiment c, shown in fig. 2,
demonstrate the influence of too high supercoolings (~T 26.5 K). From fig. 2 it can he coneluded that at high ~x/d (lower misorientation) the growth rate of experiment c is increased with respect to experiments a and b. As is noted in ref. [5], this is due to the occurrence of two-dimensional nucleation between the steps. This phenomenon does not occur at smaller step spacings.
1+6/A 1,
—
2A( I + 2A, A)
3Onm.
(9)
When the two-dimensional nucleation is suppressed due to a reduction in the surface concentration by surface diffusion overlap we conelude that ~x* = 1 2 I3’~,i.e. from experiment c follows A — 30 60 nm. The same phenomenon was reported earlier [5], but for A a value of 20 nm was predicted. 3.4. Comparison with previous/v reported dati 3.4.1. Data of Tsukamoto and Van der Hock Observation of misoriented (1 lO} YIG films showed that large. molecularly’ flat, half-moon shaped terraces occur on the surface [26.6]. These (110) terraces originate from inactivated growth hills; they interrupt the misorientation step sequence. This phenomenon leads to the formation of a semi-infinite initially equidistant step train. ending at the lower side on the terrace. As can also
B. van der Hock. W. van Erk
/
be deduced from eq. (4) the step which faces this terrace will increase its velocity due to extra impact of growth units from the (110) terrace. Indeed, observations show that the step facing the terrace has an increased distance to its neighbour, while the steps far away from the terrace are equidistantly spaced. It was possible to deduce indirectly from the observation the time span t 0
during which the first step has profited from the
terrace area, by measuring the covered distance of the misorientation steps far away from the terrace. X)). We can also measure the covered distance ~l + x0 of the first step, during which a step far
1.5
1
0
0.5~
x~xI0~rn)
50
away from the terrace has covered a distance x11: ~1
=
f
‘v d t
—
J
v~~5 dt
0
dx,
_fXO_V~J_dx_f 0
~111.
543
Interfacial and volume transport processes
(10)
0
00
~S0
Fig. 3. Extra covered distance of first (0) and second (•) step facing the terrace versus total covered distance x0 (corrected -—30%) from refs. [6,261.Fitting curves for both first and second (— — —) step obtained by integration of step kinetics equation (11), are =6250. drawn, with i.~x=0.5 gm. ~ =38 ~im and 2 ±A/2X AA,/A
where v 1 is the velocity of the first step and vmj~ and x = v~1~tthe (constant) velocity and the
covered distance of a misorientation step, situated far from the terrace [26,6]. Introducing v~as the advance rate of an isolated step, the step velocity ~ can be derived from eq. (4), realizing that v = z~xf110/d and A << ~x (= 0.5 p.m): v
v
2+ A/2A AA~/A ‘~ 6/~x + AAS/A2 + A/2A
3.4.2. Data of GOrnert and Hergt The idea of perfoming growth rate measure-
ments on spherically shaped substrates emanates from Hergt and GOrnert [18]. They grew films of (YSm)3(FeGa)5012 at a temperature of 880°C. ____ ____
(11) 1 0~(m) 1~T~H1D 1 f~LcsTsTgRj~ 1 IC
The velocity of the first step can be approximated by averaging the~ velocities which follow from eq.is (11) with ~x= (i.e. the terrace) andatwith the spacing to the neighbouring step the ~Xx back.
// 7/ /
The integral eq. (10) can be solved numerically. obtaining the values of ~l as a function of x 1~.
The growth temperature of the (110) films was 920°C, the rotation velocity 128 rpm (i.e. 6
=
38
p.m), the supercooling was 5°Cand the misorientation step distance was 0.5 p.m. In fig. 3, we present the measured ~l versus the 30% corrected x0 [6,26] for the first and second step facing the
terrace. The curve is the best fit for the 2 +measuring A/2A = points, yielding a value AA.,/A 6250, while we find fromforour experiments described in section 3.3 a value of 3000—6000.
/
10
5
/ ~
200
300
~x/d 4d0
Fig. 4. on Growth rate data of { 110) garnet films in dependence the surface misorientation from LPE ref. [181,plotted in a similar way as fig. 2. Undercooling 14 K, rotation velocity 100 rpm and deposition time 900 s.
544
B. van der Hock, )V van Erk -
Interfacial and volume I/s/il sport processes
They used a supercooling of 14 K and rotation rates of w = 100 rpm and they used spherically shaped GOG substrates with orientation angles between 0 and 2°. Similarly to fig. 2. fig. 4 shows a plot of 1 /111)) versus ~x/d based on the data of Hergt and Görnert of growth experiments with spherical wafers. We used their reported data of (‘ 5,/Cs 62.7 and 1 877°C’.and the slope in fig. I which yielded L~H1D/R = 3.75 X 10 ~ K fir/s. As iii section 3.3. at lower misorientations the growth rate f1 tends to decrease faster than at high misorientations. A transition from region (i) to region (ii) occurs at ~x 120 nm, and the same analysis as iii section 3.3 is performed on fig. 4. The resulting values of A, A ~, A and 6~~ are listed in table I: a good correspondence with our data is obtained.
4.
Acknowledgements We thank Ing. A.H.M. Raayniakers for technical assistance and Professor P. Bennema and I)r. J.P. van der Eerden for helpful discussions. One of the authors (B. v.d. H.) was supported by the Netherlands Foundation of Chemical Research (SON).
References [11 P. Bennema and E.A. Gieo,, to he published. [21 K. Tsukamoto. to be published. 13] P. GOrnert. CU. dAmbly, R. Ilergt and S. Phys Status Saudi (a) 57 (1980) 163
Borninanu.
[4] P. GOrnert, .J. Crystal Grossih 52 (1981) 8%.
151
W. van Erk, H..J.G.J. van Hock-Martens and (3. Bands, .1
Cr%stal Growth 48(1980)621. [6] B. van den Hoek, J.P. s-an der Eerden and P. l3enncma. .1. Crystal Growth 56 (1081) 1)3%. [7] P. Görnert, S. Bornmann. F. Voigi and M. Wentlt. Phys. Status Solidi (a) 41)1977) 505. [8] E.A. Giess. J.D. Kuptsis and E.A.l). White. .1. (. rvsial
Conclusion
With the GGC model we are able to explain the growth kinetics of the (110) faces of LPE-grown garnet. Because of the small coupling coefficient h A/A we may put the volume and surface process in series. For small misorientations (~.v/d> 200) the growth rate is determined by the resistances of the various surface processes. Of these processes, the
[13] A.E. Morgan. .1. Crysial (iron tli 27 (1074)221. [14] W.G. Cochran, Proc. (ambridge Phil. Soc .31) (1934) 365 [151 E. Sinn, Kristall Tech 14 (1979) 117
attachment to the step is the most difficult since
[16] W. van F.rk and H.K. Kuiken,
A/2A.,< 1. For large misorientations (~x< 100) a higher growth rate is possible and also volume diffusion becomes important; neighbouring steps
interact due to the smaller step spacing. For verylarge misorientations (~x/d< 10). the volume and surface diffusion resistance have about the same order of magnitude: Nu 6/A 0.5 at w 100 rpm. The interaction distance 1. is equal to 30 nm. . while the width W of the strip adjacent to a step
with a high surface concentration gradient
is
equal to 150 nm. Comparing the resistance for entering the top layer of the garnet film, for the (111) (eq. (1)) and
(110) (eq. (6)) faces, we find for (111). 1/k l0~ s/rn, and for (110), A/D = 2 X l0~s/rn. As expected, the transition of a growth unit from the volume to the surface is more difficult for the (flat) than for the (rough) (111) garnet faces.
Growth 16 (l972( 36. [9] R. Ghez and E.A. Giess. Mater. Res. Bull. 8 ((973) 31. [10] W. s-an Erk. J. Crystal Growth 43 (1978) 446. [II]
...
..
hA. (ness. MM. Factor and F.( - Frank, .1. C rvsial (~rowilt 46 (l970( 620. -
12] VU. Levich. Physicochemical Flvdrodvnamlcs ( l’ni’ntice Hall.
962).
j.
(‘rystal Growth 5 1>1081)
397. [17] K. Fischer. D. Ltnzen. F. Sinn. 11. Stieff and 1-. Voigi. Krtstall Tech. 14 (1079) 1521. [18] R. Hergi and P. Görnert. Phvs. Status Solidi (a) 40 (1978) 427
[191 P.
Görnert and R. 1-lergi. Phys. Status Solidi (a) 47 (1078)
99. [20] R. Hergt and P. GOrnert. Phys. Staius Solidi (a) 57(108))) [211 A.,A. C’hernov. J. Crystal Growth 24/25 (1974> II. ,.
.
-
-,
[22] G.H. (,iimer, R. (ihez and IN. C abrera. .1. ( rystal (,rowtli 8(1971)79 [23] W.K. Burton. N. Cabrera and F.C. Frank. Phil. Trans. Roy. Soc. London A243 (1951) 299.
1241
R. Ghez and G.H. Gilmer, J. Crystal Growth 21)1974) 93. [25] J.P. van der Eerden, In: Crystals/Growth. Properties and .
.
-
Applications, Volume: Modern Theory of (.rystai (,ron th, Eds. H.M. Müller-Krumhhaar and A.,A. Chernov(Springer. Heidelberg. 1982). [26] K. Tsukamoto and B. van der Flock. J. Crystal (ironth 56 (1982)131.
537
THE INTERFACIAL AND VOLUME TRANSPORT PROCESSES DURING LPE GROWTH OF GARNETS B. VAN DER HOEK RIM Laboratory of Solid State Chemistry. Catholic University, Toernoo,veld, 6525 ED Nijmegen, The Netherlands
and W. VAN ERK
*
Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands Received 4 February 1982; manuscript received in final form 27 April 1982
An analysis on the kinetical processes during the epitaxial growth of garnet films is presented. On the rough (Ill) garnet faces the transport process consists of the diffusion of growth units through the mass transfer boundary layer and the interfacial process; both processes have about the same mass transfer resistance. On the flat (110) faces the model of Gilmer. Ghez and Cabrera (GGC) was applicable. With help of growth rate measurements on garnet films grown on spherically shaped gandolinium garnet substrates, we were able to estimate the parameters which determine the growth process. The values of the transport process parameters of the GGC model were A =80 pm, A =70 nm and A~=225 nm. For low misorientations. misorientation angle ~<0.3°, the steps do not interact and the surface process is rate determining. For increasing misorientations (~>0.6°), the step interaction increases and also the volume diffusion becomes important. The volume—surface incorporation resistance is about two times higher for (110) than for (Ill faces.
1. Introduction For magnetic bubble memory devices one needs a thin film of magnetic garnet on a non-magnetic garnet substrate. This film is grown from a solution which primary consists of lead oxide, while the solutes are the garnet-forming rare-earth and ~on oxides. The circular substrate (often Gd 3Ga5O12 (GGG)) is dipped into the supersaturated solution, and rotated about the vertical axis in a horizontal plane. Growth starts im-
mediately and after a few minutes the film has the desired thickness of I p.m. This film growth process is called Liquid Phase Epitaxy (LPE). Many studies about the mechanisms of the LPE growth of garnets have been published. However,
*
no common opinion exists about the growth mechanism or about even the order of magnitude of the parameters which determine the transport processes of growth units from the bulk to the surface and to the step. In terms of the PBC theory the orientations (110), (121) and (100) are flat faces [1], and spiral growth or nucleation growth is to be expected. Indeed, both spiral hillocks and nuclei have been observed by Tsukamoto [2] on the (1 10) surface of LPE-grown Y3Fe5O12 (YIG) garnets.
Present address: Light Development, EWD 431, Eindhoven,
However, different values of the parameters which determine the various transport processes of growth units from the solution into the step are reported for the (110) faces [3—6]. In refs. [3,4] it is concluded that the first desolvation of the growth unit is the rate-determining process. However, in refs. [5,6] it is argued that the rate-determining process is the surface diffu-
The Netherlands.
sion of growth units to the step.
0022-0248/82/0000—0000/$02.75 © 1982 North-Holland
538
B. ian der lloek, ti
ian Erk
Inieu/acul and colunic transport pr~sesio
The widely applied { 111 } orientation produces garnet faces which are rough on atomic scale [I]. Therefore, all growth units which enter this face are immediately attached and no step growth occurs. Görnert et a!. [71assume that this roughness leads to a negligible resistance for entering the surface, so that the only transport resistance is the diffusion of growth units towards the crystal surface. Others [5,8—10] conclude that the resistance of the volume diffusion and interfacial mass transfer processes are of the same order of magnitude. It is the aim of this paper to discuss the growth mechanisms on the (ill) and (110) faces of LPE-grown garnets, and to determine the niagnitude of the various resistances in the transport (i.e. growth) process. We will use data of growth cxperiments on slightly spherically shaped substrates, together with experimental data reported in the literature, 2. Growth of the (111) garnet faces In this section we will discuss the possible mechanisms of LPE growth of (111 } garnet faces. It was shown by periodic bond chain analysis [1] that the { 111) faces of all garnets are rough on atomic scale, indicating that no nucleation harrier is present. In that case, the mass transfer process can be divided into two parts: diffusion of the solute through the diffusion boundary layer (resistance 6/D) and the actual interface process (probably consisting of desolvation of the growth unit) with a resistance of 1/k1~1.Here D is the diffusion constant, 6 is the thickness of the diffusion boundary layer and k111 is the kinetic coefficient for the surface process. By putting the diffusion and interface processes in series, the following expression for the growth rate ~ of the (111) faces was obtained [101: ~T~II1 fiiiRT.,Tg(Cs/Ci~ 1)
+ D
—~--—
(I)
It111
where 1/ 1 and ~XT are saturation temperature, growth temperature and supercooling, respectively: R is the gas constant: ~ is the enthalpy
of solution: and (~ and C~ are the concentration of solute in the film and in the solution. respectivelv. Both for the FuYhJe 5O1. (EYIG) [8 II] and for the Y~Fe5O1,(YIG) [5] system it was shown that eq. (1) gives a good description of the growth rate at various values of the temperature and supercooling. It was also shown, that the interfacial process was first order with respect to the interface supersaturation [5.8—li]. so that the parameter It is a function of temperature only. The hydrodynamic environment of the interface is known, since for all experiments a circular, flat substrate was used, which was rotated around its perpendicular axis. In this case, the Navier Stokes equations can be solved and the thickness of the diffusion boundary layer 6 is [121: I
--
I
I
ô — 1 .61 v I) w . (_) 2/s) and w the where i’ rotation is the kinematic angular velocityviscosity (rad/s). (m By measuring growth rates as a function of the total resistance sum of the growth process (right-handed side of eq. (1)) can be separated and 6 /D and It 111 can he determined. Van Erk [5] introduced the Nusselt number for the growth process of LPE garnets. being the ratio of the volume and surface resistance: Nu It 6 ~L) (~~) III
/
-
If Nu >~1. the growth rate is diffusion limited: if Nu ~ 1, the surface process is rate determining. At w 100 rpm. the value of Nu for the YIG system increases from 0.5 at 800°C to 1.5 at 1000°C[10]. For the EYIG system somewhat higher values for Nu were found [91. Gornert et al. [7] give a different interpretation of their growth rate measurements on { Ill } Onented faces. They use the same horizontal dipping technique with axial rotation. The substrate is Gd~Ga5O12 (GGG) and the film consists of (YSmy~(FeGa)5O12. They assume that the resistance of the interfacial growth process is negligible, i.e. 1/k~11~6/D or Nu>~>I. In their opinion. the transport of growth units is not only conveyed by forced convection, hut also by thermal convec[ion. They introduce an effective boundary layer (6~) which consists of contributions by forced
B. van der Hoek, W. van Erk
/
Interfacial and volume transport processes
convection (sr) and thermal convection (6~).The former is depending on the rotation velocity (eq. (2)), the latter is assumed to be a constant contribution, independent of ~e. An argument against this assumption is that at higher rotation velocities the solution is better homogenized, undoubtedly leading to a strong reduction of the temperature gradients and thus to a decrease of the thermal convection. isIndeed, magnitude of the[13] natural convection usuallythe very low; Morgan estimates a flow of 0.0003 rn/s for a rather large temperature difference of 8°C over 0.03 m. The flow due to forced convection is much larger: 0.0l—0.03 rn/s for the applied rotation velocities of 30—300 rpm [14,15]. Extra transport can occur due to density gradients close to the wafer. The solvent lead oxide has a much larger specific density than the garnet-forming oxides, so that the solution density close to the crystal—solution interface is reduced [16]. This effect contributes to the —
—
natural convection, but difference if it was should important, significant growth rate be cx-a pected for the grown garnet film at the upper side and the film at the lower side of the horizontally dipped substrates. This is usually not observed, however, Therefore, rather than assuming an extra constant mass transfer contribution due to natural convection, we assume that the kinetic coefficient k111 is not necessarily infinite. A physical effect which may lead to a finite k iii value is the following. It has been shown that the metal ions are surrounded by more 02 ions in the solution than in the solid garnet film [17]. During the incorporation of the growth units in the top layer of the crystal, some activation energy is necessary for the stripping (desolvation) of the 02 molecules, leading to a surface incorporation resistance. Regarding the considerations made above, it is in our opinion not justified to neglect the volume—surface incorporation resistance, while the neglect of the natural convection as done in refs. [5,8—11]is not extravagant, According to eqs. (1) and (2), a straight line is to be expected if we plot l/f11~versus ~—I/2 with intercept proportional to 1/k1 and slope proportional to ~/D. Such a plot is shown in fig. 1 for the growth rate data of Gornert et al. [7]. A straight
~ ~
539 3
f
TO
5pTT(~/cl)X10 IS/N) ~
~
~
2 1
x1~(SKIrnI
~
2 1
2 (sIrad)~
2 ~~jlI 4 6 Fig. 1. Growth rate data of ref. [7] for LPE growth on (Ill) oriented substrates versus rotation velocity, plotted according to eq. (1), using ~HI/R = 12500 K.
line fits well with their experimental data and if the reported external experimental parameters 7~ = 1152 K, Cs/CL 62.7, p 2.9 X 106 m2/s are used, we find from the intercept and slope k 111~~H1/R K for rn/stheand ~HID/R 2/s.0.17 Taking value of ~HI/R 3.75 106 12500KKm[5],we find D 3 10b0 rn2/s and k 1 1.3 X 1O~rn/s. If the rotational velocity w is 100 rpm we find 8/D 1.3 X l0~s/rn or Nu = 1.7 at 879°C,i.e. surface and volume transport resistance are about equal. This Nusselt value is somewhat higher than the values found by Van Erk [5,10] or by Ghez and Giess [9]. 3.1. Growth of the (110) faces of LPE garnet films
As mentioned before, the (110) faces of all garnets are flat faces according to a periodic bond chain analysis [1]. On a perfect oriented face, a BCF-type step growth mechanism is expected; the steps originate from spirals or from two-dimensional nucleation. Both spirals and nuclei have been observed by Tsukamoto [2] on surfaces of LPE-grown garnets. A third step source can be obtained by misorientating the GGG substrate. The ideal picture of a misoriented surface is an infinite row of equidistant, parallel and straight steps. Indeed, a regular sequence of parallel steps with a height of 0.9 nm has been observed on films grown on (110) faces of GGG misoriented in the ~1 10) direction [6]. Measurements of the growth rate of (110) faces as a function of the misorientation, and thus as a function of the step spacing,
540
B. van der Hock, iF van Lrk / Inter/acial and volume transport proce
have been reported by Gornert and coworkers [3.18—20]. Two different kinds of misorientated substrate have been employed. One can use several GGG substrates, each having a constant misorientation over its surface, or use a single substrate, which has been spherically shaped, i.e. the misorientation elapses continuously over the surface. We have performed growth experiments using the latter technique and the results will he reported in section 3,3.
solvation). A the mean diffusion length of a growth unit during its stay on the surface and A~ the characteristic length of the second desolvation, i.e. the jump of a growth unit into the step. The difference in chemical potential of fluid and solid p.~— p.~ is given by ~p.. Eq. (4) holds if the coupling factor h A/A -~ I. In the relation between growth rate and misorientation limiting cases for small and large spacings ~.v can he distinguished: (i) ~X >> 2A, so that coth( ~x/2A) I and the
It has been demonstrated that two-dimensional nucleation hardly occurs at low supercoolings (~T <15°C) during LPE growth [5]. Thus, at low supersaturations growth in dislocation-free misori-
growth rate satisfies: I
~ 1) ~H L
f~
C
5RT~T~
ented GGG substrates is only due to the advance of the misorientation steps. 3.2. Growth rate of face covered with an equidistant train of straight steps
The growth mechanism of a stepped face can he described by two models. The Chernov model [21] assumes a direct incorporation of growth units into a step; the mobility of growth units ofl the surface is zero. The second model is the Gilmer, Ghez and Cabrera (GGC) model [22], which is an extension of the BCF [23] model. In this model the path of growth units is diffusion through the boundary layer, entering the surface (possibly ineluding desolvation), diffusion along the surface towards a step, and finally jumping into the step
6](() +
pendicular growth rate f~ of an equidistantly stepped surface (step distance ~x) follows from the GGC model:
L
-
~ + AA~2~X + A ~x coth 2AD °° CL D DA ~H 1 ~T — RTT — ~7’
2A
Ad ±
.
(5)
RTT
~ ~
+
A
+_7
AAd .
(6)
In these two cases, two straight lines are expected if 1/fill) is plotted versus ~.v/d. It can easily he verified that for the intersection point of these two lines ~x 2A. The ratio of slope (i) and (ii) is: slope(i)/slope(ii)
I
I + A/2A.,.
The Chernov model results in: K, 6/D) + ~x/K-j.
//~,
[(
(7)
(8)
if the unstirred layer 6 is much larger than tile step spacing, a condition which is satisfied in our case. Contrary to the GGC model a single straight line is expected for the plot of i/fl]) versus ~.v. ..
3.3. fttttng
f
AAJ ~
where d is the step height. (ii) ~x<2A, so that coth(~x/2A) 2A:~x. and the growth rate satisfies: I C D ~H
(second desolvation). Contrary to the Chernov model direct incorporation of growth units into
the step from the solution is excluded in the GGC model. In the approximation of putting volume and surface transport processes in series, the per-
(
~
of
experimental data
the previous sectionsrate it of becomes thatFrom by measuring the growth garnet clean films
1
(4)
with A being the characteristic length of the transition of the liquid phase to the surface (first de-
on misoriented substrates values of the transport parameters can be obtained. Growth due to spirals or nucleation can he avoided by using carefully polished GGG substrates and keeping the supercooling below 15°C.
B. von der Hock, W. van Erk
/
541
Jnterfacial and volume transport processes
The growth rate of spherically shaped (110) GGG substrates was measured. The radius of curvature was 0.34 m, which means for a I inch wafer a misorientation angle ~ between 0 and 2°. The curvature of the substrate was measured using
25
a stylus technique. The films were grown from a flux melt consisting of 1651.5 g PbO, 36.76 g B2O1, 11.52 g Y2O3, 1l5.05g Fe203, 4.51 g Ga2O3 and l3.30g La203, with a saturation temperature of 938°C. Films were grown by the standard LPE technique, using an axial rotation at 100 rpm and rotation reversal every five revolutions [5]. The first two films were grown at a supercooling well below the critical
20
/ i
C
~
CSTSTSR
1 bT ~HiD
point is formed due to a motion of misorientation steps with orientation This means that the measured growth is too small. order approximation therate correction term Infora first the thickness is t2(8f/~)2/2r, where t is the time and r the ~
radius of curvature. This gives for the first experiment a correction of 0.06 p.m (15%) at small ~‘, and
a/
/
/
~T=83r
/ / // // / //‘ /
//
~
/
/t~T=7~BK
-
/
/ /
10
/7
value for two-dimensional nucleation. The supercoolings were 8.3 and 7.8°C and the dipping times were 1828 and 928 s, respectively. The third dipping was performed at ~T=26.5°C and lasted 328 s. The thickness of the film increases going from the centre of the wafer towards the periphcry. When using a sodium lamp, a fringe pattern is observed and every fringe corresponds to a height difference of 0.1167 p.m. For a film thickness of 4.5 p.m at ~ 1.7° this corresponds to about 40 data points in the growth rate versus misorientation plot, The thickness at a point on the substrate with misorientation ~ is not a direct measure of the growth rate at this misorientation. The layer at this
/
~ 10 ~m)
15
/
/~
/‘‘
/~ ,/
~
,//,
/~‘
//‘
// -
~x1d 100
200
3à0
~00
500
600
Fig. 2. Growth rate of (110) garnet LPE films in dependence on the surface misorientation. Sample a: ~T=8.3°C. = 100 rpm and dipping time is 1828 s, sample b: ~T7.8°C, i~= 100 rpm and dipping time is 928 s; sample c: ~T=26.5°C, sa “100 rpm and dipping time is 328
2/s for D~Hi/R, while Cs/CL = 48. The plot m shows that for experiments a and b not a single straight line is obtained, but two lines in which the two regimes of the GGC model can be dis-
for the second experiment a correction of 0.02 p.m (9%) at small A facet is formed in the centre of the wafer, so that the smallest misorientation angle for which the growth rate can be determined is
tinguished. This shows that the Chernov model is inadequate and the GGC model with the two regimes (eqs. (5) and (6)) can be applied. The lines of the two regimes intersect at ~x 2A and the ratio of the slopes is 1 + A/2A~(eq. (7)), which
0.08° and 0.04° for experiments I and 2, respectively.
leads to the values of A and A., given in table 1, when is assumed that d 0.9 nm. The right-hand
~.
In fig.2, a plot is shown of I/f 1~0versus ~x/d, measured along one the (110) the steps advance theoffastest. Thedirections, lines a, b where and c in fig. 2 refer to experiments with supercoolings of
side of eq. (6) is equal 61i() to 6110+A at ~x=0at + A + 2AA~/X2 (intercept) andintersection equal to point. From these two 2A, the
8.3, 7.8 and 26.5 K, respectively. We used the
value of 6 can also be determined with eq. (2), using D = 3.8>< l0~° m2/s and m’ 3 X 106 m2/s
previously [5] established value of 4.75
X l0~
K
values we find from fig. 2 A and 6~~ (table 1). The
B. van der Hock. ft’. van ErA
542
lnterfm titi and volume transport p101 e
lable I
Values
of
ihe iransport parameters
Experiment a Fxperinient h Ref. [181 .0
of
the GG( model obtained from experimenial (Ito) gross lb ate
A )nni)
,\ (flu)
.\
67 10 77~ 10
225
70 77 65
60
15
230 25))
+
30 30 40
)pnu)
61111 (tIm)
37 23 35
ii)
I)) 15
-
iS (5
2))
data
6 0
~ in)
43.S 43 3 43.3
Computed (eq. 2)).
[5]. For w — 100 rpm we find 6 -- 43 p.m. Without the application of the above mentioned correction for the curvature of the wafer, the difference between the slopes in fig. 2 would be even Iarger.It can be seen that the value of 61])) derived from fig. 2 tends to be somewhat smaller than the cornputed (eq. (2)) value. However, one should realize that eq. (2) is derived on the assumption of a uniformly absorbing surface, i.e. all lateral gradients in the volume concentration field are neglected. On misoriented (110) faces, the steps act as sinks, so lateral gradients are present. Due to the anisotropic nature of (110) garnet faces, the step incorporation process for steps parallel to one of the <110) directions (the steps
where the diffusion fields of neighbouring steps overlap, thus reducing the surface concentration of growth units. Hence, at small step spacing. the surface supersaturation is decreased helo~ the point where two-dimensional nucleation is significant. The transition to the nucleation regime occurs at ~x’~ 120 nm. The catchment area of a step is defined as the width U’i~ of a strip adjacent to the step ~shere surface density gradients are significant [24]. -I his catchment area is not equal to the interaction distance l~.which is defined as the step spacing below which neighbouring steps interact mutuall\. In our case of small coupling coefficient. ~ is equal to 2A[24,25] and
advance along <001)) is more difficult then for steps parallel to one of the <001) directions (steps advance along <110)). Thus, the value of A, will be higher (25—50%) for steps parallel to (110) and the distinction between regime (i) and (ii) will he smaller, since the difference of the slopes of the straight lines is smaller, see eq. (7). This explains why we were not able to find the two cases (i) and (ii) in the plot of the growth rate versus ~x measured along <001). From table 1 it can be seen that the coupling factor b — A/A is about 0.001, which a posteriori justifies the use of eq. (4). The results from experiment c, shown in fig. 2,
demonstrate the influence of too high supercoolings (~T 26.5 K). From fig. 2 it can he coneluded that at high ~x/d (lower misorientation) the growth rate of experiment c is increased with respect to experiments a and b. As is noted in ref. [5], this is due to the occurrence of two-dimensional nucleation between the steps. This phenomenon does not occur at smaller step spacings.
1+6/A 1,
—
2A( I + 2A, A)
3Onm.
(9)
When the two-dimensional nucleation is suppressed due to a reduction in the surface concentration by surface diffusion overlap we conelude that ~x* = 1 2 I3’~,i.e. from experiment c follows A — 30 60 nm. The same phenomenon was reported earlier [5], but for A a value of 20 nm was predicted. 3.4. Comparison with previous/v reported dati 3.4.1. Data of Tsukamoto and Van der Hock Observation of misoriented (1 lO} YIG films showed that large. molecularly’ flat, half-moon shaped terraces occur on the surface [26.6]. These (110) terraces originate from inactivated growth hills; they interrupt the misorientation step sequence. This phenomenon leads to the formation of a semi-infinite initially equidistant step train. ending at the lower side on the terrace. As can also
B. van der Hock. W. van Erk
/
be deduced from eq. (4) the step which faces this terrace will increase its velocity due to extra impact of growth units from the (110) terrace. Indeed, observations show that the step facing the terrace has an increased distance to its neighbour, while the steps far away from the terrace are equidistantly spaced. It was possible to deduce indirectly from the observation the time span t 0
during which the first step has profited from the
terrace area, by measuring the covered distance of the misorientation steps far away from the terrace. X)). We can also measure the covered distance ~l + x0 of the first step, during which a step far
1.5
1
0
0.5~
x~xI0~rn)
50
away from the terrace has covered a distance x11: ~1
=
f
‘v d t
—
J
v~~5 dt
0
dx,
_fXO_V~J_dx_f 0
~111.
543
Interfacial and volume transport processes
(10)
0
00
~S0
Fig. 3. Extra covered distance of first (0) and second (•) step facing the terrace versus total covered distance x0 (corrected -—30%) from refs. [6,261.Fitting curves for both first and second (— — —) step obtained by integration of step kinetics equation (11), are =6250. drawn, with i.~x=0.5 gm. ~ =38 ~im and 2 ±A/2X AA,/A
where v 1 is the velocity of the first step and vmj~ and x = v~1~tthe (constant) velocity and the
covered distance of a misorientation step, situated far from the terrace [26,6]. Introducing v~as the advance rate of an isolated step, the step velocity ~ can be derived from eq. (4), realizing that v = z~xf110/d and A << ~x (= 0.5 p.m): v
v
2+ A/2A AA~/A ‘~ 6/~x + AAS/A2 + A/2A
3.4.2. Data of GOrnert and Hergt The idea of perfoming growth rate measure-
ments on spherically shaped substrates emanates from Hergt and GOrnert [18]. They grew films of (YSm)3(FeGa)5012 at a temperature of 880°C. ____ ____
(11) 1 0~(m) 1~T~H1D 1 f~LcsTsTgRj~ 1 IC
The velocity of the first step can be approximated by averaging the~ velocities which follow from eq.is (11) with ~x= (i.e. the terrace) andatwith the spacing to the neighbouring step the ~Xx back.
// 7/ /
The integral eq. (10) can be solved numerically. obtaining the values of ~l as a function of x 1~.
The growth temperature of the (110) films was 920°C, the rotation velocity 128 rpm (i.e. 6
=
38
p.m), the supercooling was 5°Cand the misorientation step distance was 0.5 p.m. In fig. 3, we present the measured ~l versus the 30% corrected x0 [6,26] for the first and second step facing the
terrace. The curve is the best fit for the 2 +measuring A/2A = points, yielding a value AA.,/A 6250, while we find fromforour experiments described in section 3.3 a value of 3000—6000.
/
10
5
/ ~
200
300
~x/d 4d0
Fig. 4. on Growth rate data of { 110) garnet films in dependence the surface misorientation from LPE ref. [181,plotted in a similar way as fig. 2. Undercooling 14 K, rotation velocity 100 rpm and deposition time 900 s.
544
B. van der Hock, )V van Erk -
Interfacial and volume I/s/il sport processes
They used a supercooling of 14 K and rotation rates of w = 100 rpm and they used spherically shaped GOG substrates with orientation angles between 0 and 2°. Similarly to fig. 2. fig. 4 shows a plot of 1 /111)) versus ~x/d based on the data of Hergt and Görnert of growth experiments with spherical wafers. We used their reported data of (‘ 5,/Cs 62.7 and 1 877°C’.and the slope in fig. I which yielded L~H1D/R = 3.75 X 10 ~ K fir/s. As iii section 3.3. at lower misorientations the growth rate f1 tends to decrease faster than at high misorientations. A transition from region (i) to region (ii) occurs at ~x 120 nm, and the same analysis as iii section 3.3 is performed on fig. 4. The resulting values of A, A ~, A and 6~~ are listed in table I: a good correspondence with our data is obtained.
4.
Acknowledgements We thank Ing. A.H.M. Raayniakers for technical assistance and Professor P. Bennema and I)r. J.P. van der Eerden for helpful discussions. One of the authors (B. v.d. H.) was supported by the Netherlands Foundation of Chemical Research (SON).
References [11 P. Bennema and E.A. Gieo,, to he published. [21 K. Tsukamoto. to be published. 13] P. GOrnert. CU. dAmbly, R. Ilergt and S. Phys Status Saudi (a) 57 (1980) 163
Borninanu.
[4] P. GOrnert, .J. Crystal Grossih 52 (1981) 8%.
151
W. van Erk, H..J.G.J. van Hock-Martens and (3. Bands, .1
Cr%stal Growth 48(1980)621. [6] B. van den Hoek, J.P. s-an der Eerden and P. l3enncma. .1. Crystal Growth 56 (1081) 1)3%. [7] P. Görnert, S. Bornmann. F. Voigi and M. Wentlt. Phys. Status Solidi (a) 41)1977) 505. [8] E.A. Giess. J.D. Kuptsis and E.A.l). White. .1. (. rvsial
Conclusion
With the GGC model we are able to explain the growth kinetics of the (110) faces of LPE-grown garnet. Because of the small coupling coefficient h A/A we may put the volume and surface process in series. For small misorientations (~.v/d> 200) the growth rate is determined by the resistances of the various surface processes. Of these processes, the
[13] A.E. Morgan. .1. Crysial (iron tli 27 (1074)221. [14] W.G. Cochran, Proc. (ambridge Phil. Soc .31) (1934) 365 [151 E. Sinn, Kristall Tech 14 (1979) 117
attachment to the step is the most difficult since
[16] W. van F.rk and H.K. Kuiken,
A/2A.,< 1. For large misorientations (~x< 100) a higher growth rate is possible and also volume diffusion becomes important; neighbouring steps
interact due to the smaller step spacing. For verylarge misorientations (~x/d< 10). the volume and surface diffusion resistance have about the same order of magnitude: Nu 6/A 0.5 at w 100 rpm. The interaction distance 1. is equal to 30 nm. . while the width W of the strip adjacent to a step
with a high surface concentration gradient
is
equal to 150 nm. Comparing the resistance for entering the top layer of the garnet film, for the (111) (eq. (1)) and
(110) (eq. (6)) faces, we find for (111). 1/k l0~ s/rn, and for (110), A/D = 2 X l0~s/rn. As expected, the transition of a growth unit from the volume to the surface is more difficult for the (flat) than for the (rough) (111) garnet faces.
Growth 16 (l972( 36. [9] R. Ghez and E.A. Giess. Mater. Res. Bull. 8 ((973) 31. [10] W. s-an Erk. J. Crystal Growth 43 (1978) 446. [II]
...
..
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12] VU. Levich. Physicochemical Flvdrodvnamlcs ( l’ni’ntice Hall.
962).
j.
(‘rystal Growth 5 1>1081)
397. [17] K. Fischer. D. Ltnzen. F. Sinn. 11. Stieff and 1-. Voigi. Krtstall Tech. 14 (1079) 1521. [18] R. Hergi and P. Görnert. Phvs. Status Solidi (a) 40 (1978) 427
[191 P.
Görnert and R. 1-lergi. Phys. Status Solidi (a) 47 (1078)
99. [20] R. Hergt and P. GOrnert. Phys. Staius Solidi (a) 57(108))) [211 A.,A. C’hernov. J. Crystal Growth 24/25 (1974> II. ,.
.
-
-,
[22] G.H. (,iimer, R. (ihez and IN. C abrera. .1. ( rystal (,rowtli 8(1971)79 [23] W.K. Burton. N. Cabrera and F.C. Frank. Phil. Trans. Roy. Soc. London A243 (1951) 299.
1241
R. Ghez and G.H. Gilmer, J. Crystal Growth 21)1974) 93. [25] J.P. van der Eerden, In: Crystals/Growth. Properties and .
.
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Applications, Volume: Modern Theory of (.rystai (,ron th, Eds. H.M. Müller-Krumhhaar and A.,A. Chernov(Springer. Heidelberg. 1982). [26] K. Tsukamoto and B. van der Flock. J. Crystal (ironth 56 (1982)131.