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LPE growth kinetics of CaGe-Substituted EuTm2Fe5O12 garnet films
Journal of Crystal Growth 42(1977) 328—333 © North-Holland Publishing Company
LPE GROWTH KINETICS OF CaGe-SUBSTITUTED EuTm2Fe5O12 GARNET FILMS E.A. GIESS and C.F. GUERCI T.J. Watson Research Center, Yorktown Heights, New ork 10598, USA
The diffusion-reaction growth model of Ghez and Giess for garnet LPE films, grown with unidirectional axial rotationin isothermal melts fluxed with PbO—B203—Fe203, has been extended to include a thermally-activated diffusion term. The fluxed melt had a lower P1,0 : B203 ratio (11: 1) than that generally used (- 16 : 1) for garnet LPE films. Measured growth rates for LPE films, produced2/s) from = 0.06 meltsexp(—1.0 with foureV/k7); differentreactivity, concentrations K (cm/s) of CaGe-substituted = 8000 exp(—l.61 EuTm2Fe5O12 eV/kT); viscosity, garnet, ~‘ (cm2/s) fit the model 0.01;with: and condiffusivity, D ce centration, (cm(g/cm3) = 4220 exp — 1.0678 eV/kT for rotation rates ~ (rpm) = 36 to 196 in the growth temperature range 890 to 965°C.It is shown that garnet melt concentration terms are the most critical ones in determining the growth rate, hence the activation energy (heat of solution) for the equilibrium concentration is reported to five significant figures. Evaluating the D and the K effects by both a numerical (results given above) and an analytic method demonstrates that these parameters are least critical in determining the growth rate in this case.
1. Introduction
2. Experimental procedure
Understanding the growth kinetics of magnetic garnet films is a prerequisite to the characterization
2.1. Film growth and evaluation
of the segregation effects which result in films having different substituent ratios than those of the melts used to grow the films. It also follows that understanding segregation effects is a prerequisite to being able to control film properties for device applications.
The LPE dipping apparatus employed to grow films was that of Ghez and Giess [1], and their procedures were used to evaluate films as well. Isothermal growth [2] temperatures were in the range 890 to 975°C.Substrate disks of (111) Gd 3Ga5O12 were axially rotated while being held with 95Pt-5Au wire fingers in a horizontal plane. Unidirectional rotation rates w were 100 ±1 rpm except for a special series of films grown at 900 and at 955°Cwith w = 36, 64, 100, 144 and 196 rpm. Growth time was 122 ±2 s. The basic garnet molten solution used was based upon the earlier work of others [3,4] and of the authors [5]. The melt composition is shown in table 1. Compared to the work of refs. [3] and [4] this solution, however, has relatively low PbO : B203 and CaO : Ge02 ratios in order to improve melt stability
Ghez and Giess [1] studied the LPE growth kinetics of EuYb2Fe5O12 garnet films grown by isothermal dipping of Gd3Ga5O12 substrate wafers undergoing unidirectional axial rotation. They employed a kinetics model involving the diffusion of growth units across a hydrodynamically stabilized boundary layer followed by an integration reaction at the growth interface. Their model accounted for the temperature dependence of the garnet equilibrium concentration and of the growth surface reaction con-
stant but not for that of the diffusivity D. Here the model is extended to incorporate a temperature dependent D. Growth rate data for EuTm2Fe5O12 garnet LPE films containing CaO—Ge02 substitutions fit this extended model to within the precision of the experimental rate measurements for four different garnet concentrations over a growth temperature range of 890 to 975°C.
against unwanted crystallization.
This molten garnet solution has a lower PbO: B203 ratio (11: 1) than that generally used (~16: 1) for garnet LPE films. The garnet is nominally EuTm2Fe5O12 substituted with CaO and Ge02. At a growth temperature Tg = 9 10°C,and with axial rotation w °lOOrpm,films are produced whichhave a com328
329
E.A. Giess, CF. Guerci/LPE growth kinetics of garnet films Table 1 The garnet molten solution composition Constituent Eu 203 Tm203 CaCO3 Fe203 Ge02
Moles
Grams
0.210 0.420 0.425 10.500 77:00
0.776 1.702 0.447 17.606 180:465
B203
7.000 97.381
5.117 208.118
fourth (most concentrated) garnet melt composition produced films at the same growth rates as the original corresponding melt from which thirty films had bee own 2.2. The model Ghez and Giess [1] analyzed their data for the growth rate,f, with an equation (1) CL CeD (1 + p 6 6K which simplifies to:
1
P-)~
—
f
CL
—
Ce [(1~)1
.
+ K1],
(2)
position of Eu 0 94Tm1~77Ca0~25Pb0•04Ge0~3Fe4.7O12 3. Film composiand an X-ray density p = 6.5microprobe g/cm tions measured by electron [5] have a relative precision at the 95% confidence level ranging from ±2%for major substituents, such as Eu, Tm and Fe, and ±20%for minor substituents, such as Pb. After studying film growth in the first molten solution, three more solutions were investigated in order to measure the influence of garnet concentration upon film shown growthinrate. theprogresgarnet molten solution table In 1, stages, was made sively more concentrated with respect to garnet solute by three successive additions of 0.6 g of ceramic powder having a composition Eu 094Tm176Ca03Ge0~3Fe4~7O12.In preparing the melt and the ceramic with high-purity constituent oxides, the B203 especially must be kept dry. Calcium is added as the carbonate to avoid the instability of CaO toward ambient H20 and CO2. Removal of CO2 from the CaCO3 was achieved by calcining the crystal (solute) constituents in pressed ceramic form at 1300°C prior to mixing in powdered form and then melting them with the flux constituents. At every stage after each successive garnet nutrient addition, films were grown and their thicknesses were measured interferometrically [1]. A duplicate of the least concentrated melt series of films was 3) produced to check frommelt a duplicate melt (CL = 0.155 g/cm the reproducibility. Finally it was demonstrated that the effects of solvent depletion through successive film growths was not influencing the growth rate data. This was accomplished by determining that a fresh mixture of the
where pp = 6.5 6g/cm3, the solid garnet density; CL = the concentration of garnet nutrient in the melt (g/cm3); Ce = C
0 exp(—E~/kT).
(2.1)
the equilibrium concentration of garnet in the melt at temperature T; D = the diffusion coefficient of the growth species; 3v”6w~”2 (2.2) 6 = 6 D” the diffusion boundary layer thickness; v = 0.01 cm2/s, the kinematic viscosity of the melt; w = the substrate rotation rates in rad/s = 0.1047 rpm; — —
K K0 expk—EK kT
2.3
the growth interface reaction constant at temperature T; and K0 = the preexponential reaction constant. In the present work the diffusion coefficient is assumed to have an Arrhenius temperature dependence D D E kT 3 ~ exp(— D/ ) () rather than being assumed to be constant as in ref. [1]. Also, the present work takes account of the thermal expansion of the melt upon heating to the growth temperature Tg. 2.2. Data analyses — —
First, concentration terms were calculated, viz., CL the concentration of garnet in the melt and Ce the equilibrium concentration of garnet as a function of
E.A. Giess, CF. Guerci / LPE growth kinetics of garnet films
330
temperature. Then, D the diffusivity of the growth species and K the surface integration reaction constant were determined by an iterative numerical procedure and also by an exact solution of Brice [6]. The fluxed melt used as the basis for this study is the molten solution whose composition is shown in table 1. It can be represented as 4.3558 g of solute garnet Eu 0 94Tm 76Ca0, 3Ge0 3Fe4 7012 (the average film composition) with the remainder of the 208 g solution being solvent. This amount of garnet is determined by the amount of Eu, the least abundant of the garnet substituents, in this molten solution. This amount of garnet divided by the melt volume defines CL. In order to express the solute concentration in terms of density, the volume of the melt at the growth temperature must be known. The density of frozen melt at 3room was determined by temperature the Archimedes method, to be 8.0 g/cm Assuming 8% volume expansion upon heating to the growth temperature as will be discussed in the section 3 leads to CL = 0.155 g/cm3 for the melt given in table 1. Thus, the room temperature volume of 26.0 cm3 was assumed to expand to 28.1 cm3 at the growth temperature, and the CL values for melts with successive additions of 0.6 g garnet nutrient were also calculated using a 28.1 cm3 volume. The equilibrium concentration of garnet Ce in these melts was calculated from a linear least squares f-data fit for the growth temperatures near TL in each of the four melt series. Thus, CL = Ce when T is the saturation (liquidus) temperature TL where f goes to zero. From eqs. (1) and (2) it is obvious that no growth will occur in a melt where the temperature is such as to make CL = Ce because the supersaturation term (CL Ce) is then zero. —
Next the preexponential C0 and the heat of solution E~were determined by solving simultaneously eq. (2.1) with Ce = CL and T= the saturation temperatures for the four melts respectively. Values of D and K were first calculated (see eqs. (2.3) and (3)) with assumed activation energies of EDO, 0.6, 1.0, or 1.5 eVandEK= 1.61 eV,respectively using an iterative numerical procedure which minimized (t~ fm) where f~was the calculated and fm the measured value of the growth rate for all the films produced in this study. Then values of D and K were calculated by the analytic method of Brice [6] for the 900 and 955°C
films grown with w = 36 to 196 rpm. This method involves plotting w”2f versus f1’~where n is the order of the interface reaction which Ghez and Giess [1] took to be first order. After finding D and K for 900 and 95 5°C,it was possible to express both terms in the Arrhenius form of eqs. (2.3) and (3).
3. Results and discussion Concentration terms represent the driving force for crystal growth while the D and K terms principally relate to the impedance part of the crystal growth process. As noted earlier in ref. [1], the concentration terms are the most critical ones in determining the growth rate. Sinceeqs.theas concentration the growth a difference CL terms Ce, itappear should in follow that the growth process is sensitive to changes in CL and Ce. Furthermore, the heat of solution E~is a very critical factor. The equilibrium concentration of garnet nutrient in these molten solutions is given in the Arrhenius form of eq. (2.1) as —
Ce4220exp(1.0678eV/kT).
(3.1)
This is based upon the four saturation temperature TL values shown in fig. 1 for each of the respective four melts studied. Arbitrarily changing the heat of solution E~from the calculated 1.0678 eV by 1% up to 1.07848 eV has the effect of increasing TL from
__________________________________________ 3O~i~ CL TL’
-~
2.5~
-~~2O -
/
~ ~
1.5
-
0
-
05 0 990
—
.
~:g~°~4
/
-~
/ ,•
/
/
/
/
/ Jt
A
/ /
01977
-
970.2
-
0.2190 983.1
/
-
At
-
t
/ /
970
f I
/
i’~ I
0’
i 950 930 910 GROWTH TEMPERATURE (°C)
I 890
Fig. I. Growth rate as a function of decreasing growth tern. . 3 perature for garnet melts with concentrations CL (g/cm and with liquidus temperatures TL (°C). The solid lines represent a numerical solution for the growth model.
E.A. Giess, C.F. Guerci / LPE growth kinetics of garnet films
940.5 to 952.7°C for the 0.155 g/cm3 melt. This shows how critical E~is in specifying Ce. To produce the same change in TL, assuming that eq. (3.1) remains unchanged, would require an increase in CL of over 10% from 0.155 to 0.1715 g/cm3. That this is a very large effect can be appreciated by looking at the influence of CL upon the growth rate f. Increasing CL= U.155 g/cm3 by only 1% will increase fby 6% for undercooling i~T 20°and by 26% for ~T= 5°C. Concentrations are expressed as densities and, therefore, require a knowledge of melt volumes Vrn at the growth temperature Tg. Here Vm at Tg was assumed to be 1.08 times the Vm which was measured at room temperature. Davies [7] measured a melt density Pm = 7.5 g/cm3 at Tg by the Archimedes method using Pt to displace the melt. At room ternperature, his melt has Pm 8.1, hence the assumed (8.1 +7.5) 1.08 conversion factor for the effect of heating upon Pm’ The growth of 25 films sequentially from the same melt should have depleted the garnet nutrient by a significant amount. However, here films grew at the same rate from a fresh melt with CL = 0.2 19 as they did from the melt (which after adding garnet nutrient in stages had the same CL) used to grow all of the other films in this study. Presumably some compensating factor, such as solvent vaporization [8], acts to mitigate this depletion effect which should otherwise be observable. In the numerical analyses for D and K, the value of EK = 1.61 eV found in ref. [1] was adopted. Then the best fit for growth rate data was obtained when ED = 1, while worse fits resulted when ED = 0, 0.6 or 1.5 eV. The lines drawn in fig. 1 represent this nu merical solution with D 2/s, ED = 1 eV and 0 = 0.06 cm with K 0 = 8000 cm/s, EK = 1.61 eV. In the model of ref. [1], the relative influence of
2.4 E 2.0
-
w 1.8
-
‘~
.6 .4 +
dence of the growth process was assigned to the K term. Now, however, some of the temperature dependence is accounted for by the D term as well as the K term. Examining eqs. (2) and (2.2) reveals that the influence of the temperature dependent D and K terms can be separated by determining the effects of changing the substrate rotation rate w at two different temperatures. The dependence of f upon wi/2 shown in fig. 2 derives from eq. (2.2) which
CALCULATED
—.
+MEASUR~ I
6
I
7
I
9 10 II 12 13 14 ROTATION frp1fl)~’2 Fig. 2. Growth rate dependence upon the square root of substrate rotation rate. The upper curves are for a melt CL = 0.155 g/cm3 at 900°C,and the lower curves are for a melt CL = 0.1977
8
g/cm3 at 955°C.
expresses 6 the diffusion boundary layer thickness as a function of D, a.,, and v. The curvature of the f versus ~i/2 plot arises from the interplay of the D/6 and K “impedance” terms. At higher rotation rates the diffusion boundary layer becomes thinner and the surface reaction becomes relatively more important. Further evidence of the existence of a surface reaction in addition to the diffusion effect is the crystallographic orientation dependence of growth found by Tolksdorf et al. [9]. Taking the average D of the numerical solution for all of the data points shown in fig. 1 leads to D =
-______________________________________ I
30 2.5
~
I
/
-
I—
~ 2.0 ~ .5
-
~
1.0
-
~
0.5
-
~
I
-
&
was Dheld all be of distingnished the temperature depenthe andconstant K termsand could because D
331
0 990
I
I
I
7 ~
/11 / /
970
I
I
7 ~
——
./—DrQO6exp
l/kT
71 ~_o~43x:o6 I 950 930 910 890 GROWTH TEMPERATURE (CC)
-
Fig. 3. Growth ratea as a function of decreasing growth temperature assuming constant D (dashed lines) and a tempera-
ture dependent D (solid lines) for the four garnet melts
shown in fig. 1.
E.A. Giess, C.F. Guerci / LFE growth kinetics of garnetfilms
332
4.3 X 10—6 cm2/s, the value for Tg 940°C.In fig. 3, the numerical solutions for a constant D and a temperature-dependent D are compared. Since D increases with temperature when its proper temperature-dependence is assigned, the temperature-dependent D solution gives higher f values at high Tg values and lower f values at low Tg values compared to the constant D solution in this particular case. However, had D been kept constant, the best fit for the f data in fig. 1 would not be D = 4.3 X 10_6 cm2/s. Instead a higher value of D 6 X 10_6 and a lower K 0= 6300 cm/s would be obtained because ofofthe ment that the temperature dependence therequiref data needs to be satisfied as discussed above. This latter solution infers that the surface reaction is relatively more important than diffusion in controlling growth; however, the solution predicts too low a dependence off upon w112 because the role of diffusion has been underestimated. This point further illustrates the importance in kinetics studies of determining the influenceofw. The analytic method of Brice [6] applied to the measured f data plotted in fig. 2 gives the exact solution (forn = 1, n being the order of surface reaction): D=0.172 exp(—1.III eV/kJ) and K°625 exp— 1.347 eV/k7’. It can be seen in table 2 that these results are actually not very different from those of the numerical method discussed above. Furthermore, these two methods lead to f values within 2.8% of each other over the entire composition and temperature range studied. Table 2 compares the numerical and analytic solution results. The analytic method allows the order of reaction n to be determined, e.g., by analyzing f data (fm) with n at fixed values from 0.2 to 2.0 at increments of 0.1 and by choosing the resulting D and K parameters for the n value which gives the minimum standard deviation, for the growth rate. For 900°C, n ~ .
0.7 and for 955°C,n> 1, however, this parameter is sensitive to data fluctuations. Furthermore, values of n <0.5 (compared to n = 1) give a greater dependence off upon ~,1/2 especially for low values of n and w112, whereas for n> I, there is even less dependence off upon ~1/2 and n has less effect upon f. Therefore, the f data fit for any n > 0.5 is reasonably good and the assumption of n = 1 would accordingly seem appropriate. Davies et al. [10] remarked that an order of magnitude increase in viscosity could be arises offsetfrom by 4 = 1.78 which increasing D a of factor O” and (2.2). The kinematic the D/6 term eqs. 1 (2) viscosity assumed here, v = 0.01 cm2/s, follows the work of Davies et al. [10] but according to Tolksdorf et al. [9] might be nearly an order of magnitude larger. Davies et al. [10] estimated that viscosity has an activation energy E~= 0.5 eV. The ED = 1.11 eV calculated here incorporates E~/4;therefore, ED should be decreased by 0.125 eV making ED = 0.985 eV.
4. Conclusions Growth rate data for LPE garnet films are well represented by a diffusion-reaction model incorporating temperature-dependent-concentration, -diffusion and -reaction terms. The exact solution for diffusion and reaction constants demonstrates that the interface reaction is more thermally activated than the diffusion boundary layer process, but not markedly so. Concentration terms are more critical than the diffusion and reaction terms in defining the growth process for the molten solution system investigated here.
Acknowledgments Table 2
Values of D and K by numerical and analytic solutions, Method
Numerical
D (X 10_6 cm2/s)
K (X i0~ cm/s)
900
900
~
955
?~
955
The authors are grateful for helpful conversations with J.E. Davies, R. Ghez and M.W. Shafer.
References
[11 R. Ghez and E.A. Giess, I. Crystal Growth 27 (1974)
E.A. Giess, C.F. Guerci/ LPE growth kinetics of garnet films [2] H.J. Levinstein, S. Licht, R.W. Landorf and S.L. Blank, Appl. Phys. Letters 19 (1971) 486. [3] S.L. Blank, i.W. Nielsen and W.A. Biolsi, J. Electrochem. Soc. 123 (1976) 856. [4] W.A. Bonner, I.E. Geusic, D.H. Smith, L.G. Van Uitert and G.P. Vella.Coleiro, Mater. Res. Bull. 8 (1973) 1223. [5] E.A. Giess, C.F. Guerci and F. Cardone, Am. Ceram. Soc., submitted.
333
[6] J.C. Brice, J. Crystal Growth 1(1967)161. [7] I.E. Davies, PhD. Thesis, Department of Electrical Engineering, Imperial College, London (1974). [8] E.A. Giess, J.D. Kuptsis and E.A.D. White, J. Crystal Growth 16 (1972) 36. [9] W. Tolksdorf, G. Bartels, G.P. Espinosa, P. Holst, D. Mateika and F. Welz, J. Crystal Growth 17 (1972) 322. [10] I.E. Davies, E.A.D. White and J.D.C. Wood, I. Crystal Growth 27 (1974) 227.
LPE GROWTH KINETICS OF CaGe-SUBSTITUTED EuTm2Fe5O12 GARNET FILMS E.A. GIESS and C.F. GUERCI T.J. Watson Research Center, Yorktown Heights, New ork 10598, USA
The diffusion-reaction growth model of Ghez and Giess for garnet LPE films, grown with unidirectional axial rotationin isothermal melts fluxed with PbO—B203—Fe203, has been extended to include a thermally-activated diffusion term. The fluxed melt had a lower P1,0 : B203 ratio (11: 1) than that generally used (- 16 : 1) for garnet LPE films. Measured growth rates for LPE films, produced2/s) from = 0.06 meltsexp(—1.0 with foureV/k7); differentreactivity, concentrations K (cm/s) of CaGe-substituted = 8000 exp(—l.61 EuTm2Fe5O12 eV/kT); viscosity, garnet, ~‘ (cm2/s) fit the model 0.01;with: and condiffusivity, D ce centration, (cm(g/cm3) = 4220 exp — 1.0678 eV/kT for rotation rates ~ (rpm) = 36 to 196 in the growth temperature range 890 to 965°C.It is shown that garnet melt concentration terms are the most critical ones in determining the growth rate, hence the activation energy (heat of solution) for the equilibrium concentration is reported to five significant figures. Evaluating the D and the K effects by both a numerical (results given above) and an analytic method demonstrates that these parameters are least critical in determining the growth rate in this case.
1. Introduction
2. Experimental procedure
Understanding the growth kinetics of magnetic garnet films is a prerequisite to the characterization
2.1. Film growth and evaluation
of the segregation effects which result in films having different substituent ratios than those of the melts used to grow the films. It also follows that understanding segregation effects is a prerequisite to being able to control film properties for device applications.
The LPE dipping apparatus employed to grow films was that of Ghez and Giess [1], and their procedures were used to evaluate films as well. Isothermal growth [2] temperatures were in the range 890 to 975°C.Substrate disks of (111) Gd 3Ga5O12 were axially rotated while being held with 95Pt-5Au wire fingers in a horizontal plane. Unidirectional rotation rates w were 100 ±1 rpm except for a special series of films grown at 900 and at 955°Cwith w = 36, 64, 100, 144 and 196 rpm. Growth time was 122 ±2 s. The basic garnet molten solution used was based upon the earlier work of others [3,4] and of the authors [5]. The melt composition is shown in table 1. Compared to the work of refs. [3] and [4] this solution, however, has relatively low PbO : B203 and CaO : Ge02 ratios in order to improve melt stability
Ghez and Giess [1] studied the LPE growth kinetics of EuYb2Fe5O12 garnet films grown by isothermal dipping of Gd3Ga5O12 substrate wafers undergoing unidirectional axial rotation. They employed a kinetics model involving the diffusion of growth units across a hydrodynamically stabilized boundary layer followed by an integration reaction at the growth interface. Their model accounted for the temperature dependence of the garnet equilibrium concentration and of the growth surface reaction con-
stant but not for that of the diffusivity D. Here the model is extended to incorporate a temperature dependent D. Growth rate data for EuTm2Fe5O12 garnet LPE films containing CaO—Ge02 substitutions fit this extended model to within the precision of the experimental rate measurements for four different garnet concentrations over a growth temperature range of 890 to 975°C.
against unwanted crystallization.
This molten garnet solution has a lower PbO: B203 ratio (11: 1) than that generally used (~16: 1) for garnet LPE films. The garnet is nominally EuTm2Fe5O12 substituted with CaO and Ge02. At a growth temperature Tg = 9 10°C,and with axial rotation w °lOOrpm,films are produced whichhave a com328
329
E.A. Giess, CF. Guerci/LPE growth kinetics of garnet films Table 1 The garnet molten solution composition Constituent Eu 203 Tm203 CaCO3 Fe203 Ge02
Moles
Grams
0.210 0.420 0.425 10.500 77:00
0.776 1.702 0.447 17.606 180:465
B203
7.000 97.381
5.117 208.118
fourth (most concentrated) garnet melt composition produced films at the same growth rates as the original corresponding melt from which thirty films had bee own 2.2. The model Ghez and Giess [1] analyzed their data for the growth rate,f, with an equation (1) CL CeD (1 + p 6 6K which simplifies to:
1
P-)~
—
f
CL
—
Ce [(1~)1
.
+ K1],
(2)
position of Eu 0 94Tm1~77Ca0~25Pb0•04Ge0~3Fe4.7O12 3. Film composiand an X-ray density p = 6.5microprobe g/cm tions measured by electron [5] have a relative precision at the 95% confidence level ranging from ±2%for major substituents, such as Eu, Tm and Fe, and ±20%for minor substituents, such as Pb. After studying film growth in the first molten solution, three more solutions were investigated in order to measure the influence of garnet concentration upon film shown growthinrate. theprogresgarnet molten solution table In 1, stages, was made sively more concentrated with respect to garnet solute by three successive additions of 0.6 g of ceramic powder having a composition Eu 094Tm176Ca03Ge0~3Fe4~7O12.In preparing the melt and the ceramic with high-purity constituent oxides, the B203 especially must be kept dry. Calcium is added as the carbonate to avoid the instability of CaO toward ambient H20 and CO2. Removal of CO2 from the CaCO3 was achieved by calcining the crystal (solute) constituents in pressed ceramic form at 1300°C prior to mixing in powdered form and then melting them with the flux constituents. At every stage after each successive garnet nutrient addition, films were grown and their thicknesses were measured interferometrically [1]. A duplicate of the least concentrated melt series of films was 3) produced to check frommelt a duplicate melt (CL = 0.155 g/cm the reproducibility. Finally it was demonstrated that the effects of solvent depletion through successive film growths was not influencing the growth rate data. This was accomplished by determining that a fresh mixture of the
where pp = 6.5 6g/cm3, the solid garnet density; CL = the concentration of garnet nutrient in the melt (g/cm3); Ce = C
0 exp(—E~/kT).
(2.1)
the equilibrium concentration of garnet in the melt at temperature T; D = the diffusion coefficient of the growth species; 3v”6w~”2 (2.2) 6 = 6 D” the diffusion boundary layer thickness; v = 0.01 cm2/s, the kinematic viscosity of the melt; w = the substrate rotation rates in rad/s = 0.1047 rpm; — —
K K0 expk—EK kT
2.3
the growth interface reaction constant at temperature T; and K0 = the preexponential reaction constant. In the present work the diffusion coefficient is assumed to have an Arrhenius temperature dependence D D E kT 3 ~ exp(— D/ ) () rather than being assumed to be constant as in ref. [1]. Also, the present work takes account of the thermal expansion of the melt upon heating to the growth temperature Tg. 2.2. Data analyses — —
First, concentration terms were calculated, viz., CL the concentration of garnet in the melt and Ce the equilibrium concentration of garnet as a function of
E.A. Giess, CF. Guerci / LPE growth kinetics of garnet films
330
temperature. Then, D the diffusivity of the growth species and K the surface integration reaction constant were determined by an iterative numerical procedure and also by an exact solution of Brice [6]. The fluxed melt used as the basis for this study is the molten solution whose composition is shown in table 1. It can be represented as 4.3558 g of solute garnet Eu 0 94Tm 76Ca0, 3Ge0 3Fe4 7012 (the average film composition) with the remainder of the 208 g solution being solvent. This amount of garnet is determined by the amount of Eu, the least abundant of the garnet substituents, in this molten solution. This amount of garnet divided by the melt volume defines CL. In order to express the solute concentration in terms of density, the volume of the melt at the growth temperature must be known. The density of frozen melt at 3room was determined by temperature the Archimedes method, to be 8.0 g/cm Assuming 8% volume expansion upon heating to the growth temperature as will be discussed in the section 3 leads to CL = 0.155 g/cm3 for the melt given in table 1. Thus, the room temperature volume of 26.0 cm3 was assumed to expand to 28.1 cm3 at the growth temperature, and the CL values for melts with successive additions of 0.6 g garnet nutrient were also calculated using a 28.1 cm3 volume. The equilibrium concentration of garnet Ce in these melts was calculated from a linear least squares f-data fit for the growth temperatures near TL in each of the four melt series. Thus, CL = Ce when T is the saturation (liquidus) temperature TL where f goes to zero. From eqs. (1) and (2) it is obvious that no growth will occur in a melt where the temperature is such as to make CL = Ce because the supersaturation term (CL Ce) is then zero. —
Next the preexponential C0 and the heat of solution E~were determined by solving simultaneously eq. (2.1) with Ce = CL and T= the saturation temperatures for the four melts respectively. Values of D and K were first calculated (see eqs. (2.3) and (3)) with assumed activation energies of EDO, 0.6, 1.0, or 1.5 eVandEK= 1.61 eV,respectively using an iterative numerical procedure which minimized (t~ fm) where f~was the calculated and fm the measured value of the growth rate for all the films produced in this study. Then values of D and K were calculated by the analytic method of Brice [6] for the 900 and 955°C
films grown with w = 36 to 196 rpm. This method involves plotting w”2f versus f1’~where n is the order of the interface reaction which Ghez and Giess [1] took to be first order. After finding D and K for 900 and 95 5°C,it was possible to express both terms in the Arrhenius form of eqs. (2.3) and (3).
3. Results and discussion Concentration terms represent the driving force for crystal growth while the D and K terms principally relate to the impedance part of the crystal growth process. As noted earlier in ref. [1], the concentration terms are the most critical ones in determining the growth rate. Sinceeqs.theas concentration the growth a difference CL terms Ce, itappear should in follow that the growth process is sensitive to changes in CL and Ce. Furthermore, the heat of solution E~is a very critical factor. The equilibrium concentration of garnet nutrient in these molten solutions is given in the Arrhenius form of eq. (2.1) as —
Ce4220exp(1.0678eV/kT).
(3.1)
This is based upon the four saturation temperature TL values shown in fig. 1 for each of the respective four melts studied. Arbitrarily changing the heat of solution E~from the calculated 1.0678 eV by 1% up to 1.07848 eV has the effect of increasing TL from
__________________________________________ 3O~i~ CL TL’
-~
2.5~
-~~2O -
/
~ ~
1.5
-
0
-
05 0 990
—
.
~:g~°~4
/
-~
/ ,•
/
/
/
/
/ Jt
A
/ /
01977
-
970.2
-
0.2190 983.1
/
-
At
-
t
/ /
970
f I
/
i’~ I
0’
i 950 930 910 GROWTH TEMPERATURE (°C)
I 890
Fig. I. Growth rate as a function of decreasing growth tern. . 3 perature for garnet melts with concentrations CL (g/cm and with liquidus temperatures TL (°C). The solid lines represent a numerical solution for the growth model.
E.A. Giess, C.F. Guerci / LPE growth kinetics of garnet films
940.5 to 952.7°C for the 0.155 g/cm3 melt. This shows how critical E~is in specifying Ce. To produce the same change in TL, assuming that eq. (3.1) remains unchanged, would require an increase in CL of over 10% from 0.155 to 0.1715 g/cm3. That this is a very large effect can be appreciated by looking at the influence of CL upon the growth rate f. Increasing CL= U.155 g/cm3 by only 1% will increase fby 6% for undercooling i~T 20°and by 26% for ~T= 5°C. Concentrations are expressed as densities and, therefore, require a knowledge of melt volumes Vrn at the growth temperature Tg. Here Vm at Tg was assumed to be 1.08 times the Vm which was measured at room temperature. Davies [7] measured a melt density Pm = 7.5 g/cm3 at Tg by the Archimedes method using Pt to displace the melt. At room ternperature, his melt has Pm 8.1, hence the assumed (8.1 +7.5) 1.08 conversion factor for the effect of heating upon Pm’ The growth of 25 films sequentially from the same melt should have depleted the garnet nutrient by a significant amount. However, here films grew at the same rate from a fresh melt with CL = 0.2 19 as they did from the melt (which after adding garnet nutrient in stages had the same CL) used to grow all of the other films in this study. Presumably some compensating factor, such as solvent vaporization [8], acts to mitigate this depletion effect which should otherwise be observable. In the numerical analyses for D and K, the value of EK = 1.61 eV found in ref. [1] was adopted. Then the best fit for growth rate data was obtained when ED = 1, while worse fits resulted when ED = 0, 0.6 or 1.5 eV. The lines drawn in fig. 1 represent this nu merical solution with D 2/s, ED = 1 eV and 0 = 0.06 cm with K 0 = 8000 cm/s, EK = 1.61 eV. In the model of ref. [1], the relative influence of
2.4 E 2.0
-
w 1.8
-
‘~
.6 .4 +
dence of the growth process was assigned to the K term. Now, however, some of the temperature dependence is accounted for by the D term as well as the K term. Examining eqs. (2) and (2.2) reveals that the influence of the temperature dependent D and K terms can be separated by determining the effects of changing the substrate rotation rate w at two different temperatures. The dependence of f upon wi/2 shown in fig. 2 derives from eq. (2.2) which
CALCULATED
—.
+MEASUR~ I
6
I
7
I
9 10 II 12 13 14 ROTATION frp1fl)~’2 Fig. 2. Growth rate dependence upon the square root of substrate rotation rate. The upper curves are for a melt CL = 0.155 g/cm3 at 900°C,and the lower curves are for a melt CL = 0.1977
8
g/cm3 at 955°C.
expresses 6 the diffusion boundary layer thickness as a function of D, a.,, and v. The curvature of the f versus ~i/2 plot arises from the interplay of the D/6 and K “impedance” terms. At higher rotation rates the diffusion boundary layer becomes thinner and the surface reaction becomes relatively more important. Further evidence of the existence of a surface reaction in addition to the diffusion effect is the crystallographic orientation dependence of growth found by Tolksdorf et al. [9]. Taking the average D of the numerical solution for all of the data points shown in fig. 1 leads to D =
-______________________________________ I
30 2.5
~
I
/
-
I—
~ 2.0 ~ .5
-
~
1.0
-
~
0.5
-
~
I
-
&
was Dheld all be of distingnished the temperature depenthe andconstant K termsand could because D
331
0 990
I
I
I
7 ~
/11 / /
970
I
I
7 ~
——
./—DrQO6exp
l/kT
71 ~_o~43x:o6 I 950 930 910 890 GROWTH TEMPERATURE (CC)
-
Fig. 3. Growth ratea as a function of decreasing growth temperature assuming constant D (dashed lines) and a tempera-
ture dependent D (solid lines) for the four garnet melts
shown in fig. 1.
E.A. Giess, C.F. Guerci / LFE growth kinetics of garnetfilms
332
4.3 X 10—6 cm2/s, the value for Tg 940°C.In fig. 3, the numerical solutions for a constant D and a temperature-dependent D are compared. Since D increases with temperature when its proper temperature-dependence is assigned, the temperature-dependent D solution gives higher f values at high Tg values and lower f values at low Tg values compared to the constant D solution in this particular case. However, had D been kept constant, the best fit for the f data in fig. 1 would not be D = 4.3 X 10_6 cm2/s. Instead a higher value of D 6 X 10_6 and a lower K 0= 6300 cm/s would be obtained because ofofthe ment that the temperature dependence therequiref data needs to be satisfied as discussed above. This latter solution infers that the surface reaction is relatively more important than diffusion in controlling growth; however, the solution predicts too low a dependence off upon w112 because the role of diffusion has been underestimated. This point further illustrates the importance in kinetics studies of determining the influenceofw. The analytic method of Brice [6] applied to the measured f data plotted in fig. 2 gives the exact solution (forn = 1, n being the order of surface reaction): D=0.172 exp(—1.III eV/kJ) and K°625 exp— 1.347 eV/k7’. It can be seen in table 2 that these results are actually not very different from those of the numerical method discussed above. Furthermore, these two methods lead to f values within 2.8% of each other over the entire composition and temperature range studied. Table 2 compares the numerical and analytic solution results. The analytic method allows the order of reaction n to be determined, e.g., by analyzing f data (fm) with n at fixed values from 0.2 to 2.0 at increments of 0.1 and by choosing the resulting D and K parameters for the n value which gives the minimum standard deviation, for the growth rate. For 900°C, n ~ .
0.7 and for 955°C,n> 1, however, this parameter is sensitive to data fluctuations. Furthermore, values of n <0.5 (compared to n = 1) give a greater dependence off upon ~,1/2 especially for low values of n and w112, whereas for n> I, there is even less dependence off upon ~1/2 and n has less effect upon f. Therefore, the f data fit for any n > 0.5 is reasonably good and the assumption of n = 1 would accordingly seem appropriate. Davies et al. [10] remarked that an order of magnitude increase in viscosity could be arises offsetfrom by 4 = 1.78 which increasing D a of factor O” and (2.2). The kinematic the D/6 term eqs. 1 (2) viscosity assumed here, v = 0.01 cm2/s, follows the work of Davies et al. [10] but according to Tolksdorf et al. [9] might be nearly an order of magnitude larger. Davies et al. [10] estimated that viscosity has an activation energy E~= 0.5 eV. The ED = 1.11 eV calculated here incorporates E~/4;therefore, ED should be decreased by 0.125 eV making ED = 0.985 eV.
4. Conclusions Growth rate data for LPE garnet films are well represented by a diffusion-reaction model incorporating temperature-dependent-concentration, -diffusion and -reaction terms. The exact solution for diffusion and reaction constants demonstrates that the interface reaction is more thermally activated than the diffusion boundary layer process, but not markedly so. Concentration terms are more critical than the diffusion and reaction terms in defining the growth process for the molten solution system investigated here.
Acknowledgments Table 2
Values of D and K by numerical and analytic solutions, Method
Numerical
D (X 10_6 cm2/s)
K (X i0~ cm/s)
900
900
~
955
?~
955
The authors are grateful for helpful conversations with J.E. Davies, R. Ghez and M.W. Shafer.
References
[11 R. Ghez and E.A. Giess, I. Crystal Growth 27 (1974)
E.A. Giess, C.F. Guerci/ LPE growth kinetics of garnet films [2] H.J. Levinstein, S. Licht, R.W. Landorf and S.L. Blank, Appl. Phys. Letters 19 (1971) 486. [3] S.L. Blank, i.W. Nielsen and W.A. Biolsi, J. Electrochem. Soc. 123 (1976) 856. [4] W.A. Bonner, I.E. Geusic, D.H. Smith, L.G. Van Uitert and G.P. Vella.Coleiro, Mater. Res. Bull. 8 (1973) 1223. [5] E.A. Giess, C.F. Guerci and F. Cardone, Am. Ceram. Soc., submitted.
333
[6] J.C. Brice, J. Crystal Growth 1(1967)161. [7] I.E. Davies, PhD. Thesis, Department of Electrical Engineering, Imperial College, London (1974). [8] E.A. Giess, J.D. Kuptsis and E.A.D. White, J. Crystal Growth 16 (1972) 36. [9] W. Tolksdorf, G. Bartels, G.P. Espinosa, P. Holst, D. Mateika and F. Welz, J. Crystal Growth 17 (1972) 322. [10] I.E. Davies, E.A.D. White and J.D.C. Wood, I. Crystal Growth 27 (1974) 227.