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An analysis of GaAs LPE growth methods by a diffusion limited growth model
Journal of Crystal Growth 32 (1976) 95—100 © North-Holland Publishing Company
AN ANALYSIS OF GaAs LPE GROWTH METHODS BY A DIFFUSION LIMITED GROWTH MODEL
Stephen I. LONG “, Joseph M. BALLANTYNE and Lester F. EASTMAN School of Electrical Engineering and Material Science ‘Center, Cornell University, Ithaca, New York 14850, U.S.A. Received 3 March 1975; revised manuscripy received 25 June 1975
Experimental data from three types of liquid phase epitaxial growth systems are compared with a diffusion limited growth model, valid for temperature intervals of 50°Cor less. Observed layer thicknesses are analyzed and are used to calculate growth efficiencies. The influence of initial saturation errors, convection, growth competition, solution volume and cooling rate were inferred from the experimental data.
1. Introduction
2. Diffusion limited growth theory
Liquid phase epitaxy (LPE) has been a useful and widely utilized growth technique for microwave [1, 2] and optoelectronic [3—5]applications. As device requirements become increasingly demanding of epitaxial technology, control over layer thickness, carrier concentration, surface and interface morphology, and uniformity of deposition must all be maintained, Layer thickness control to within 0.1 pm is required by such applications as the double heterostructure laser [3], high efficiency GaAs IMPATT devices [6] and integrated optics [7]. Therefore, growth methods and theoretical models which can control and predict deposition thicknesses to this degree are necessary. LPE growth of thin epitaxial layers is generally accomplished by either step or uniform cooling of saturated solutions. The steady-state (isothermal) method has been effective in growth of thick, uniformly doped layers of GaAs [8], however, convection and sensitivity to temperature gradients [9] make it less satisfactory for carefully controlled thin layer growth. In this paper, experimental data from three types of LPE growth systems will be compared with dif. fusion limited growth models. Using this approach, effects due to convection, substrate position, melt volume and cooling rates will be demonstrated. Experimental data on layer uniformity versus cooling rate will also be described.
In many applications of liquid phase epitaxy, when the driving force for growth is provided by a temperature change, a diffusion limited growth model provides either quantitative agreement with experiment, or at least insight into more complicated transport mechanisms or limitations. In this section, some results derived from diffusion models will be briefly presented. Several theoretical models based on diffusion limited growth have been used to predict concentration profiles, growth rates and layer thicknesses [10—16]. These calculations all assume linear cooling rates and most include assumptions or approximations regarding liquidus curves, initial conditions and solution depth. In the case where the semi-infinite approximation is valid (solute diffusion lengths (Dt)1/2 much less than solution depth W), the thickness of the epitaxial layer is proportional to the growth time to the 3/2 power [14—16].In addition, it has been shown that the mathematical solutions of Minden [12], Rode [14] and Ghez [17] are equivalent to a modified Small and Barnes [11] approximation for small growth times and semi-infinite growth solutions [18]: d = (4~/3C(1 k)m) ~D/ir)1I2t3/2 (1)
*
—
.
0
In this equation, d is the layer thickness, ~ is the cooling rate, t is the growth time, D is the diffusivity of As in Ga, C 0 is the initial solvent concentration, k is the segregation coefficient and m represents the slope of the liquidus curve. This is equivalent to
Present address: Varian Associates, Solid State West, 611 Hansen Way, Palo Alto, CA 94303, U.S.A. 95
96
S.I. Long etaL
/ GaAs LPE growth
(2) d = (4o~/3C5m)(D/ir)1/2t3/2 , where C 5 is the As concentration in the solid 3). (2.21 X 1022 atoms/cm When the solute diffusion length is comparable to the solution dimensions, finite boundary conditions
methods
for Dt/W2 < 1. Thus, a normalized coordinate A~Dt/ V2 can be used for plotting fin the case of unequal well and substrate areas. Growth efficiency is calculated from experimental thickness datad by
f°~ d/d~ax = dC 5A5/E~CV.
must be used. One such approach, which emphasizes the relationship between layer thickness, growth efficiency, and growth temperature, utilizes assumptions of constant diffusivity and liquidus slope in the temperature interval under consideration [16] (valid for z.~T< 50°C).If a saturation wafer or crust is present at the solution boundary opposing the substrate, the effective solution depth is one half of the actual depth since the solute concentration reaches maximum in the center of the solution. For growth 2 1 ,layer thickness as a function of growth time approaches d
2 1) (3) 5mD) (Dt/W in which layer thickness is linearly dependent on t. The growth efficiency or melt fraction, f, can be defined as the proportion of total excess solute iNC, present in the solution which actually deposits on the substrate. If all of this solute were to deposit on the =
(aW3/C
—
Eq. (8) is only approximate, because it does not account for layer thickness variations due to lateral diffusion, however good agreement with experimental data has been obtained using this approach.
3. Experimental 3.1. Growth systems In order to evaluate the effectiveness of diffusion limited growth theory in predicting deposition thickness and growth efficiency, and to evaluate the effect of substrate position and melt geometry on the above, LPE GaAs experimental data from several sources will be analyzed. Three separate relationships between substrate and solution are in common usage: substrate
a.
dm~= W the Z~CAw/CsAs. substrate, grown layer thickness would be
(8)
Solution
(4)
If the substrate areaA 5 is exactly equal to the crosssectional area of the solution in contact with it, A then the ideal growth efficiency d/dmax is equal to [14] 1/2 (5) f’°(4/3W) (Dt/ir) in the semi-infinite solution case (Dt/W2 < 1) for small temperature drops, or f= 1
—
W2/3Dt
Graphi?
b.
_-~~ \\\\~SIider ~Substrate
ubstrote Solution
(6)
in the finite solution case (Dt/W2 > 1) [16]. In order to take into account different solution depths, a normalized coordinate,Dt/W2, can be used in plottingf In addition, in many growth system geometries,A 5 is less thanAw. By substituting VAwW (solution volume) into eq. (4), the predicted growth efficiency for the linear cooling becomes: 2Dt/ V2)1!2 (7) f (4/3\/1T) (A5
c.
,~~Z._Sub$trOteholder Substrate boot
“—Solution Fig. 1. Three types of graphite boats used for liquid phase epitaxy: (a) graphite slider boat (substrate beneath solution); substrate, of solution boat.dipped into melt); (c) horizontal (b) verticaltop dipping (substrate
LI. Long etaL
/ GaAs LFEgrowth
under the melt, substrate horizontal on top of the melt, and vertical dipping of the substrate into a large volume melt. These three types of LPE reactor geometries, shown in fig. 1, will be compared. One of the most effective approaches to growing layers under the melt uses the graphite slider boat [2,3, 14, 15, 19,20] in which the substrate is recessed into a slot in the graphite. The substrate either slides under the solution or the solution slides over the substrate to initiate and terminate growth. The vertical dipping method [20—22]uses a large volume cylindrical boat supported in a vertical furnace. A vertical substrate is dipped edgewise into the melt to initiate growth and is withdrawn when the desired deposition has been obtained. The third technique to be discussed requires a horizontal substrate to be located on the top surface of the solution. An undoped source wafer is generally located on the bottom of the well and is used to saturate the solution. The substrate is held on the bottom surface of a graphite plug and is brought into contact with the solution by sliding the plug over the well. Growth is terminated by tipping the plug 90°, thus removing the substrate from the melt. This method was used for growth of GaAs epitaxial layers for the experimental part of this study. The substrate—source wafer distance was 5 mm, but since the source wafer also acts as a substrate during cooling, an effective solution depth of 2.5 mm was used for the following analysis. 3.2. Thickness data Experimental GaAs LPE thickness data both from the authors and from published sources will be compared with diffusion limited growth models [eqs. (2) and (3)]. Diffusivity of As in Ga was assumed to be 4 X iO~cm2/sec at 800°Cand 1.60 X i05 cm2! sec at 750°C[14]; however, other experiments [181 have indicated the possibility of less temperature dependence, so these numbers should not be considered to be absolutely accurate. The empirical liquidus equation of [13] was used to calculate m (the slope of the liquidus curve), In figs. 2, 4 and 6 theoretically predicted growth rates are plotted in solid lines with cooling rate cs as a parameter. In figs. 3, 5 and 7, theoretical growth efficiency [eqs. (5) and (6)] is plotted in solid lines,
100
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Fig. 2. Epitaxial layer thickness as a function of time for published data using graphite slider boat. Initial growth ternperature was 800°C. Solid lines are predicted thicknesses with cooling rate a (°C,’min) as parameter.
Fig. 2 plots experimental data from refs. [14], [19] and [201 on growth under the melt using graphite slider boats. Initial growth temperatures were close to 800°C. From this plot, it is evident that all data is either close to or below the predicted curve, indicating that convective mixing was not present. Deviations below theory at small times and fast cooling rates (curve 1) probably indicate kinetic layer growth limitations, but could also be due to an initially undersaturated solution or to spontaneous nucleation in the solution and subsequent platelet formation, although the latter would likely lead to greater curvature in the data than is observed. Errors in initial solution saturation could be responsible for deviations above (curve 3) and below (curve 5) theory at small times. At larger growth times, the slopes of curves 2, 3 and 5 drop to the 0.2 to 0.5 power range which is far below the 3/2 power dependence for semi-infinite solutions or linear dependence for finite solution dimensions, revealing significant growth competition due either to homogeneous nucleation in the solution or growth on solution surfaces. This would reduce the observed thickness below that predicted by diffusion. Finally, if~Tismuch greater than 50°C,the simplified model breaks down and one which includes the exponential dependence of liquidus and diffusion coefficient must be used, which predicts a smaller slope for larger times. Fig. 3 is a plot of growth efficiency calculated from the slider boat experimental data. For the slider boat, the well area was assumed equal to the substrate area.
5.1. Long et aL / GaAs LFEgrowth methods
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Fig. 3. Calculated growth efficiency as function of normalized growth time for published data using the graphite slider boat. Initial growth temperature was 800°C. Solid line is theoretical growth efficiency. Units of a are ° C/mm.
Fig. 5. Calculated growth efficiency as function of normalized growth time for published thickness data using the vertical dipping system. Initial growth temperature was 800°C. Solid line is the theoretical growth efficiency. Units of a are ° C/mm.
Excess solute available for deposition was calculated
which allows the rejected gallium from the freezing interface to flow down the substrate surface (being greater density than the As saturated melt). This is verified in fig. 4, since the experimental data points are all well above the diffusion limited predictions (solid lines). In addition, wedge-shaped epi-layers have been reported from growth in the vertical orientation [21], which would also confirm the solvent rejection/convectionhypothesis. Fig. 5 represents growth efficiencies calculated from the experimental data. In this case, since the solution area was considerably larger than the substrate area, the data was plotted as a function ofA~Dt/V2 using
from the liquidus expression in ref. [13]. Although curves 2 and 3 appear above the theoretical curve in
fig. 3, on the basis of fig. 2, initial melt saturation errors or experimental error rather than solutal convection is likely. Fig. 4 represents experimental data from refs. [20] and [21] using a vertical dipping system. The initial growth temperature was approximately 800°C.In 2 of subthis case, aarea. 25 g Solutal solutionconvection was reported with 1 certain cm in this strate is almost configuration due to the large well dimensions (17mm diameter) and the vertical orientation of the substrate 100
100
-
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I
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TIME IMIN I
00
I
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Fig. 4. Epitaxial layer thickness as function of time for published data using the vertical dipping system. Initial growth temperature was 800°C. Solid lines are predicted thicknesses with a (°C/min)as a parameter.
—
I
-
10
TIME (MIN I
100
1000
Fig. 6. Epitaxial layer thickness as a function of time for the horizontal top of solution boat. Initial growth temperature was 750°C. Solid lines are predicted thickness witfl a (°C/min) as parameter.
LI. Long et al.
/ GaAs LPE growth methods
eq. (8). Again, convection is revealed by growth efficiencies much larger than expected from theory. Figs. 6 and 7 present experimental LPE growth obtamed with the substrate on top of the solution. The predicted deposition thickness and growth efficiency (plotted in solid lines) utilize a finite solution model [16] necessary in this case because of the small effective solution depth (0.25 cm). The initial growth temperature was 750°C.A 10 g solution was used in a well of 3.75 cm2 cross-sectional area. Since the well area significantly exceeded the substrate area (1.3 cm2), thicknesses larger than predicted would be expected because of the additional solute supplied by lateral diffusion. This effect, evident in fig. 6, is most pronounced at long growth times and slowest cooling rates which combines a long diffusion length needed to provide the excess solute and a smaller temperature drop necessary to avoid excessive growth on solution surfaces. It is possible that competitive growth or homogeneous nucleation is occurring at the larger cooling intervals because layer thicknesses much smaller than theory would predict were observed. Also, a few data points represent temperature intervals of greater than 50°C,which exceeds the limitations of the approximate model. Growth efficiencies, calculated from the experirnental thickness data in fig. 6, were plotted as a function ofA~Dt/V2using eq. (8). This was necessary to compensate for unequal well and substrate areas. From the above data, it is possible to conclude that 0
-
4. Conclusions Observed layer thicknesses and growth efficiencies from three types of LPE growth systems were compared to predicted theoretical limits for diffusion limited growth. Substrates were located beneath the solution (graphite slider boat), on top of the solution, and dipped verti-
04 03
: I
0_i
convection does not occur in this growth system. This is significant because greatly accelerated growth rates have been observed in steady state LPE GaAs growth with substrates on top of the solution [9]. Apparently, convection in the uniform cooling case does not occur even though concentration gradients exist which are sufficient to induce convection under steady state boundary conditions. In other words, the Rayleigh criterion [9] for stable convection cells in the solution is exceeded by these melts, but such cells apparently do not form in the uniform-cooling case. The dependence of layer thickness uniformity on the rate of cooling was investigated in this growth systern by observing typical epilayer cross-sections by cleave and stain and by surface contour measurement using a “Talysurf” profilometer. Thickness uniformity was better at faster rates of cooling as would be expected from the diffusion model (less influence of lateral diffusion). At 0.14°C/mm cooling rates, vanations as high as 60% of the average thickness were obgerved, while at the 5°C,~min rate, only 10 to 24% variation was found across the wafer. Another possible explanation of improved thickness uniformity at higher cooling rates might be the influence of kinetic limitations on the growth rate. Figs. 2 and 6 both indicate thicknesses well below the diffusion limited prediction for a = 5, even for small temperature intervals. This may be evidence that growth rates are being limited by kinetics rather than diffusion at high cooling rates. However, more data would be necessary to verify this.
75~C
0.1
0.01
99
I
I
I
11111 0~
I
I
I I
A2 Dt/V2
III
I
I
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I
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Fig. 7. Calculated growth efficiency as function of normalized growth time for horizontal top of solution boat. Initial growth temperature was 750°C.Solid line is the theoretical growth efficiency. a in °C/min.
cally into a large volume solution. The influences of initial saturation errors, convection, growth competition, melt volume and cooling rates were identified in the above systems. It was shown that convection definitely does not occur when the substrate is located on top of the solution, unlike observed effects in steady state LPE growth. .
100
SI. Long eta!.
I GaAs LFEgrowth methods
Acknowledgements
[8] S. Christensson, D. Woodard and L.F. Eastman, IEEE Trans. Electron Devices ED-17 (1970) 732.
The authors would like to thank R.H. Chilton and J.V. McNamara of the Rome Air Development Center for their advice and assistance throughout this effort. This work was supported by RADC, Griffiss AFB, N.Y., with supplemental support from the National Science Foundation through the Materials Science Center, Cornell University.
[91 S.!.
Long, J.M. Ballantyne and L.F. Eastman, J. Crystal Growth 26 (1974) 13; S.I. Long, Ph.D. Thesis, Cornell Univ. (1974). S.I. Long, Rome Air Development Center Technical Report RADC-TR-74-148 (1974). [101 W.A. Tiller and C.S. Kang, J. Crystal Growth 2 (1968) 345. [11] M.B. Small and J.F. Barnes, J. Crystal GrowthS (1969)9. [12] H.T. Minden, J. Crystal Growth 6 (1970) 228. [13] I. Crossley and M.B. Small, J. Crystal Growth 11(1971) 157.
References [1] 3. Vilms and J.P. Garrett, Solid State Electron. 15 (1972) [2] F.E. Rosztoczy and 3. Kinoshita, J. Electrochem. Soc. 121 (1974) 439. [3] I. Hayashi, M.B. Panish, P.W. Foy and S. Sum~ki,App!. Phys. Letters 17 (1970) 109. [4] R.A. Logan, H.G. White and W. Weigman, App!. Phys. Letters 13(1968)139. [5] L.W. James, GA. Antypas, R.L. Moon, 3. Edgecumbe and R.L. Bell, AppI. Phys. Letters 22 (1973) 270. [6] R.E. Goldwasser and FE. Rosztoczy, Appi. Phys. Letters 25 (1974) 92. [71 S.E. Miller eta!., Proc. IEEE 61(1973)1703.
[14] D.L. Rode, J. Crystal Growth 20 (1973) 13. [15] R.L. Moon and J. Kinoshita, J. Crystal Growth 21(1974) 149. [16] R.L. Moon, J. Crystal Growth 27 (1974) 62. [17] R. Ghez, I. Crystal Growth 19 (1973) 153. [18] R.L. Moon and S.!. Long, J. Crystal Growth 30 (1975). [19] M. Migataka, A. Doj and M. Miyazaki, in: Proc. 4th Intern. Symp. on GaAs, 1972, p. 249. [20] I. Crossley and M.B. Small, 3. Crystal Growth 19 (1973) 160. [211 I. Crossley and MB. Small, J. Crystal Growth 15 (1972) 275. [22] L.R. Dawson and 3. Whelan, Bull. Am. Phys. Soc. Ser. II, 13 (1968) 3.
AN ANALYSIS OF GaAs LPE GROWTH METHODS BY A DIFFUSION LIMITED GROWTH MODEL
Stephen I. LONG “, Joseph M. BALLANTYNE and Lester F. EASTMAN School of Electrical Engineering and Material Science ‘Center, Cornell University, Ithaca, New York 14850, U.S.A. Received 3 March 1975; revised manuscripy received 25 June 1975
Experimental data from three types of liquid phase epitaxial growth systems are compared with a diffusion limited growth model, valid for temperature intervals of 50°Cor less. Observed layer thicknesses are analyzed and are used to calculate growth efficiencies. The influence of initial saturation errors, convection, growth competition, solution volume and cooling rate were inferred from the experimental data.
1. Introduction
2. Diffusion limited growth theory
Liquid phase epitaxy (LPE) has been a useful and widely utilized growth technique for microwave [1, 2] and optoelectronic [3—5]applications. As device requirements become increasingly demanding of epitaxial technology, control over layer thickness, carrier concentration, surface and interface morphology, and uniformity of deposition must all be maintained, Layer thickness control to within 0.1 pm is required by such applications as the double heterostructure laser [3], high efficiency GaAs IMPATT devices [6] and integrated optics [7]. Therefore, growth methods and theoretical models which can control and predict deposition thicknesses to this degree are necessary. LPE growth of thin epitaxial layers is generally accomplished by either step or uniform cooling of saturated solutions. The steady-state (isothermal) method has been effective in growth of thick, uniformly doped layers of GaAs [8], however, convection and sensitivity to temperature gradients [9] make it less satisfactory for carefully controlled thin layer growth. In this paper, experimental data from three types of LPE growth systems will be compared with dif. fusion limited growth models. Using this approach, effects due to convection, substrate position, melt volume and cooling rates will be demonstrated. Experimental data on layer uniformity versus cooling rate will also be described.
In many applications of liquid phase epitaxy, when the driving force for growth is provided by a temperature change, a diffusion limited growth model provides either quantitative agreement with experiment, or at least insight into more complicated transport mechanisms or limitations. In this section, some results derived from diffusion models will be briefly presented. Several theoretical models based on diffusion limited growth have been used to predict concentration profiles, growth rates and layer thicknesses [10—16]. These calculations all assume linear cooling rates and most include assumptions or approximations regarding liquidus curves, initial conditions and solution depth. In the case where the semi-infinite approximation is valid (solute diffusion lengths (Dt)1/2 much less than solution depth W), the thickness of the epitaxial layer is proportional to the growth time to the 3/2 power [14—16].In addition, it has been shown that the mathematical solutions of Minden [12], Rode [14] and Ghez [17] are equivalent to a modified Small and Barnes [11] approximation for small growth times and semi-infinite growth solutions [18]: d = (4~/3C(1 k)m) ~D/ir)1I2t3/2 (1)
*
—
.
0
In this equation, d is the layer thickness, ~ is the cooling rate, t is the growth time, D is the diffusivity of As in Ga, C 0 is the initial solvent concentration, k is the segregation coefficient and m represents the slope of the liquidus curve. This is equivalent to
Present address: Varian Associates, Solid State West, 611 Hansen Way, Palo Alto, CA 94303, U.S.A. 95
96
S.I. Long etaL
/ GaAs LPE growth
(2) d = (4o~/3C5m)(D/ir)1/2t3/2 , where C 5 is the As concentration in the solid 3). (2.21 X 1022 atoms/cm When the solute diffusion length is comparable to the solution dimensions, finite boundary conditions
methods
for Dt/W2 < 1. Thus, a normalized coordinate A~Dt/ V2 can be used for plotting fin the case of unequal well and substrate areas. Growth efficiency is calculated from experimental thickness datad by
f°~ d/d~ax = dC 5A5/E~CV.
must be used. One such approach, which emphasizes the relationship between layer thickness, growth efficiency, and growth temperature, utilizes assumptions of constant diffusivity and liquidus slope in the temperature interval under consideration [16] (valid for z.~T< 50°C).If a saturation wafer or crust is present at the solution boundary opposing the substrate, the effective solution depth is one half of the actual depth since the solute concentration reaches maximum in the center of the solution. For growth 2
2 1) (3) 5mD) (Dt/W in which layer thickness is linearly dependent on t. The growth efficiency or melt fraction, f, can be defined as the proportion of total excess solute iNC, present in the solution which actually deposits on the substrate. If all of this solute were to deposit on the =
(aW3/C
—
Eq. (8) is only approximate, because it does not account for layer thickness variations due to lateral diffusion, however good agreement with experimental data has been obtained using this approach.
3. Experimental 3.1. Growth systems In order to evaluate the effectiveness of diffusion limited growth theory in predicting deposition thickness and growth efficiency, and to evaluate the effect of substrate position and melt geometry on the above, LPE GaAs experimental data from several sources will be analyzed. Three separate relationships between substrate and solution are in common usage: substrate
a.
dm~= W the Z~CAw/CsAs. substrate, grown layer thickness would be
(8)
Solution
(4)
If the substrate areaA 5 is exactly equal to the crosssectional area of the solution in contact with it, A then the ideal growth efficiency d/dmax is equal to [14] 1/2 (5) f’°(4/3W) (Dt/ir) in the semi-infinite solution case (Dt/W2 < 1) for small temperature drops, or f= 1
—
W2/3Dt
Graphi?
b.
_-~~ \\\\~SIider ~Substrate
ubstrote Solution
(6)
in the finite solution case (Dt/W2 > 1) [16]. In order to take into account different solution depths, a normalized coordinate,Dt/W2, can be used in plottingf In addition, in many growth system geometries,A 5 is less thanAw. By substituting VAwW (solution volume) into eq. (4), the predicted growth efficiency for the linear cooling becomes: 2Dt/ V2)1!2 (7) f (4/3\/1T) (A5
c.
,~~Z._Sub$trOteholder Substrate boot
“—Solution Fig. 1. Three types of graphite boats used for liquid phase epitaxy: (a) graphite slider boat (substrate beneath solution); substrate, of solution boat.dipped into melt); (c) horizontal (b) verticaltop dipping (substrate
LI. Long etaL
/ GaAs LFEgrowth
under the melt, substrate horizontal on top of the melt, and vertical dipping of the substrate into a large volume melt. These three types of LPE reactor geometries, shown in fig. 1, will be compared. One of the most effective approaches to growing layers under the melt uses the graphite slider boat [2,3, 14, 15, 19,20] in which the substrate is recessed into a slot in the graphite. The substrate either slides under the solution or the solution slides over the substrate to initiate and terminate growth. The vertical dipping method [20—22]uses a large volume cylindrical boat supported in a vertical furnace. A vertical substrate is dipped edgewise into the melt to initiate growth and is withdrawn when the desired deposition has been obtained. The third technique to be discussed requires a horizontal substrate to be located on the top surface of the solution. An undoped source wafer is generally located on the bottom of the well and is used to saturate the solution. The substrate is held on the bottom surface of a graphite plug and is brought into contact with the solution by sliding the plug over the well. Growth is terminated by tipping the plug 90°, thus removing the substrate from the melt. This method was used for growth of GaAs epitaxial layers for the experimental part of this study. The substrate—source wafer distance was 5 mm, but since the source wafer also acts as a substrate during cooling, an effective solution depth of 2.5 mm was used for the following analysis. 3.2. Thickness data Experimental GaAs LPE thickness data both from the authors and from published sources will be compared with diffusion limited growth models [eqs. (2) and (3)]. Diffusivity of As in Ga was assumed to be 4 X iO~cm2/sec at 800°Cand 1.60 X i05 cm2! sec at 750°C[14]; however, other experiments [181 have indicated the possibility of less temperature dependence, so these numbers should not be considered to be absolutely accurate. The empirical liquidus equation of [13] was used to calculate m (the slope of the liquidus curve), In figs. 2, 4 and 6 theoretically predicted growth rates are plotted in solid lines with cooling rate cs as a parameter. In figs. 3, 5 and 7, theoretical growth efficiency [eqs. (5) and (6)] is plotted in solid lines,
100
‘
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97
800°C
a~
_.
~,
)775
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-
~
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/
,,•
cr7Jf a•
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I ‘~
re~
3
025
20
5
.0
20 I
TIME (MINI
Fig. 2. Epitaxial layer thickness as a function of time for published data using graphite slider boat. Initial growth ternperature was 800°C. Solid lines are predicted thicknesses with cooling rate a (°C,’min) as parameter.
Fig. 2 plots experimental data from refs. [14], [19] and [201 on growth under the melt using graphite slider boats. Initial growth temperatures were close to 800°C. From this plot, it is evident that all data is either close to or below the predicted curve, indicating that convective mixing was not present. Deviations below theory at small times and fast cooling rates (curve 1) probably indicate kinetic layer growth limitations, but could also be due to an initially undersaturated solution or to spontaneous nucleation in the solution and subsequent platelet formation, although the latter would likely lead to greater curvature in the data than is observed. Errors in initial solution saturation could be responsible for deviations above (curve 3) and below (curve 5) theory at small times. At larger growth times, the slopes of curves 2, 3 and 5 drop to the 0.2 to 0.5 power range which is far below the 3/2 power dependence for semi-infinite solutions or linear dependence for finite solution dimensions, revealing significant growth competition due either to homogeneous nucleation in the solution or growth on solution surfaces. This would reduce the observed thickness below that predicted by diffusion. Finally, if~Tismuch greater than 50°C,the simplified model breaks down and one which includes the exponential dependence of liquidus and diffusion coefficient must be used, which predicts a smaller slope for larger times. Fig. 3 is a plot of growth efficiency calculated from the slider boat experimental data. For the slider boat, the well area was assumed equal to the substrate area.
5.1. Long et aL / GaAs LFEgrowth methods
98 10
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io~
~IIIIIII
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I
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~:
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2
10
10
A5 Dt/V .
Fig. 3. Calculated growth efficiency as function of normalized growth time for published data using the graphite slider boat. Initial growth temperature was 800°C. Solid line is theoretical growth efficiency. Units of a are ° C/mm.
Fig. 5. Calculated growth efficiency as function of normalized growth time for published thickness data using the vertical dipping system. Initial growth temperature was 800°C. Solid line is the theoretical growth efficiency. Units of a are ° C/mm.
Excess solute available for deposition was calculated
which allows the rejected gallium from the freezing interface to flow down the substrate surface (being greater density than the As saturated melt). This is verified in fig. 4, since the experimental data points are all well above the diffusion limited predictions (solid lines). In addition, wedge-shaped epi-layers have been reported from growth in the vertical orientation [21], which would also confirm the solvent rejection/convectionhypothesis. Fig. 5 represents growth efficiencies calculated from the experimental data. In this case, since the solution area was considerably larger than the substrate area, the data was plotted as a function ofA~Dt/V2 using
from the liquidus expression in ref. [13]. Although curves 2 and 3 appear above the theoretical curve in
fig. 3, on the basis of fig. 2, initial melt saturation errors or experimental error rather than solutal convection is likely. Fig. 4 represents experimental data from refs. [20] and [21] using a vertical dipping system. The initial growth temperature was approximately 800°C.In 2 of subthis case, aarea. 25 g Solutal solutionconvection was reported with 1 certain cm in this strate is almost configuration due to the large well dimensions (17mm diameter) and the vertical orientation of the substrate 100
100
-
10
I
I!;~~
0
TIME IMIN I
00
I
000
Fig. 4. Epitaxial layer thickness as function of time for published data using the vertical dipping system. Initial growth temperature was 800°C. Solid lines are predicted thicknesses with a (°C/min)as a parameter.
—
I
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10
TIME (MIN I
100
1000
Fig. 6. Epitaxial layer thickness as a function of time for the horizontal top of solution boat. Initial growth temperature was 750°C. Solid lines are predicted thickness witfl a (°C/min) as parameter.
LI. Long et al.
/ GaAs LPE growth methods
eq. (8). Again, convection is revealed by growth efficiencies much larger than expected from theory. Figs. 6 and 7 present experimental LPE growth obtamed with the substrate on top of the solution. The predicted deposition thickness and growth efficiency (plotted in solid lines) utilize a finite solution model [16] necessary in this case because of the small effective solution depth (0.25 cm). The initial growth temperature was 750°C.A 10 g solution was used in a well of 3.75 cm2 cross-sectional area. Since the well area significantly exceeded the substrate area (1.3 cm2), thicknesses larger than predicted would be expected because of the additional solute supplied by lateral diffusion. This effect, evident in fig. 6, is most pronounced at long growth times and slowest cooling rates which combines a long diffusion length needed to provide the excess solute and a smaller temperature drop necessary to avoid excessive growth on solution surfaces. It is possible that competitive growth or homogeneous nucleation is occurring at the larger cooling intervals because layer thicknesses much smaller than theory would predict were observed. Also, a few data points represent temperature intervals of greater than 50°C,which exceeds the limitations of the approximate model. Growth efficiencies, calculated from the experirnental thickness data in fig. 6, were plotted as a function ofA~Dt/V2using eq. (8). This was necessary to compensate for unequal well and substrate areas. From the above data, it is possible to conclude that 0
-
4. Conclusions Observed layer thicknesses and growth efficiencies from three types of LPE growth systems were compared to predicted theoretical limits for diffusion limited growth. Substrates were located beneath the solution (graphite slider boat), on top of the solution, and dipped verti-
04 03
: I
0_i
convection does not occur in this growth system. This is significant because greatly accelerated growth rates have been observed in steady state LPE GaAs growth with substrates on top of the solution [9]. Apparently, convection in the uniform cooling case does not occur even though concentration gradients exist which are sufficient to induce convection under steady state boundary conditions. In other words, the Rayleigh criterion [9] for stable convection cells in the solution is exceeded by these melts, but such cells apparently do not form in the uniform-cooling case. The dependence of layer thickness uniformity on the rate of cooling was investigated in this growth systern by observing typical epilayer cross-sections by cleave and stain and by surface contour measurement using a “Talysurf” profilometer. Thickness uniformity was better at faster rates of cooling as would be expected from the diffusion model (less influence of lateral diffusion). At 0.14°C/mm cooling rates, vanations as high as 60% of the average thickness were obgerved, while at the 5°C,~min rate, only 10 to 24% variation was found across the wafer. Another possible explanation of improved thickness uniformity at higher cooling rates might be the influence of kinetic limitations on the growth rate. Figs. 2 and 6 both indicate thicknesses well below the diffusion limited prediction for a = 5, even for small temperature intervals. This may be evidence that growth rates are being limited by kinetics rather than diffusion at high cooling rates. However, more data would be necessary to verify this.
75~C
0.1
0.01
99
I
I
I
11111 0~
I
I
I I
A2 Dt/V2
III
I
I
038 I
I
I
11111 0
Fig. 7. Calculated growth efficiency as function of normalized growth time for horizontal top of solution boat. Initial growth temperature was 750°C.Solid line is the theoretical growth efficiency. a in °C/min.
cally into a large volume solution. The influences of initial saturation errors, convection, growth competition, melt volume and cooling rates were identified in the above systems. It was shown that convection definitely does not occur when the substrate is located on top of the solution, unlike observed effects in steady state LPE growth. .
100
SI. Long eta!.
I GaAs LFEgrowth methods
Acknowledgements
[8] S. Christensson, D. Woodard and L.F. Eastman, IEEE Trans. Electron Devices ED-17 (1970) 732.
The authors would like to thank R.H. Chilton and J.V. McNamara of the Rome Air Development Center for their advice and assistance throughout this effort. This work was supported by RADC, Griffiss AFB, N.Y., with supplemental support from the National Science Foundation through the Materials Science Center, Cornell University.
[91 S.!.
Long, J.M. Ballantyne and L.F. Eastman, J. Crystal Growth 26 (1974) 13; S.I. Long, Ph.D. Thesis, Cornell Univ. (1974). S.I. Long, Rome Air Development Center Technical Report RADC-TR-74-148 (1974). [101 W.A. Tiller and C.S. Kang, J. Crystal Growth 2 (1968) 345. [11] M.B. Small and J.F. Barnes, J. Crystal GrowthS (1969)9. [12] H.T. Minden, J. Crystal Growth 6 (1970) 228. [13] I. Crossley and M.B. Small, J. Crystal Growth 11(1971) 157.
References [1] 3. Vilms and J.P. Garrett, Solid State Electron. 15 (1972) [2] F.E. Rosztoczy and 3. Kinoshita, J. Electrochem. Soc. 121 (1974) 439. [3] I. Hayashi, M.B. Panish, P.W. Foy and S. Sum~ki,App!. Phys. Letters 17 (1970) 109. [4] R.A. Logan, H.G. White and W. Weigman, App!. Phys. Letters 13(1968)139. [5] L.W. James, GA. Antypas, R.L. Moon, 3. Edgecumbe and R.L. Bell, AppI. Phys. Letters 22 (1973) 270. [6] R.E. Goldwasser and FE. Rosztoczy, Appi. Phys. Letters 25 (1974) 92. [71 S.E. Miller eta!., Proc. IEEE 61(1973)1703.
[14] D.L. Rode, J. Crystal Growth 20 (1973) 13. [15] R.L. Moon and J. Kinoshita, J. Crystal Growth 21(1974) 149. [16] R.L. Moon, J. Crystal Growth 27 (1974) 62. [17] R. Ghez, I. Crystal Growth 19 (1973) 153. [18] R.L. Moon and S.!. Long, J. Crystal Growth 30 (1975). [19] M. Migataka, A. Doj and M. Miyazaki, in: Proc. 4th Intern. Symp. on GaAs, 1972, p. 249. [20] I. Crossley and M.B. Small, 3. Crystal Growth 19 (1973) 160. [211 I. Crossley and MB. Small, J. Crystal Growth 15 (1972) 275. [22] L.R. Dawson and 3. Whelan, Bull. Am. Phys. Soc. Ser. II, 13 (1968) 3.