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Response surface modeling of the composition of AlAsySb1−y alloys grown by molecular beam epitaxy
Journal of Crystal Growth 225 (2001) 556–560
Response surface modeling of the composition of AlAsySb1y alloys grown by molecular beam epitaxy P. Gopaladasua, J.L. Cecchia, K.J. Malloyb, R. Kaspic,* a
Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, NM 87131, USA b Center for High Technology Materials, University of New Mexico, Albuquerque, NM 87106, USA c Air Force Research Laboratory, AFRL/DELS, Kirtland AFB, Albuquerque, NM 87117, USA
Abstract We have developed an empirical model of AlAsySb1y composition during molecular beam epitaxy as a function of growth temperature, growth rate, As2-flux, and Sb2-flux, using response surface methodology. In order to facilitate data collection from a large number of experiments, we have applied desorption mass spectroscopy as an in situ measure of alloy composition. A quadratic model is found to fit the data well, identifying trends and several interactions between growth parameters. Published by Elsevier Science B.V. PACS: 81.15.Hi; 68.55.Nq; 82.80.Ms; 07.05.F Keywords: A1. Design of experiments; A1. Desorption mass spectrometry; A3. Molecular beam epitaxy; B1. Antimonides
1. Introduction During molecular beam epitaxy (MBE) growth of mixed group-III alloys, such as AlxGa1xAs, the alloy composition is directly given by the incident group-III flux ratio. A group-III sticking coefficient of unity across the full range of the desirable growth-parameter space allows for such ease of prediction. In contrast, compositional control of mixed group-V alloys, such as AlAsySb1y, is complicated by the fact that an excess of group-V adatoms must compete for the limited number of anion vacancies. This competi*Corresponding author. AFRL/DELS, c/o CHTM 1313 Goddard, SE, Albuquerque, NM 87106, USA. Tel.: +1-505272-7822; fax: +1-505-272-7801. E-mail address: [email protected] (R. Kaspi). 0022-0248/01/$ - see front matter Published by Elsevier Science B.V. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 0 9 5 2 - 6
tion is known to be strongly influenced by several growth parameters such as temperature [1], groupIII fluxes, and group-V fluxes [2]. It is therefore not possible to predict alloy composition without a detailed understanding of the individual and combined effects of the relevant growth parameters on alloy composition. While many of the observed trends in IIIAsySb1y alloy composition such as an increasing y with increasing substrate temperature (Ts) are well recognized in the crystal growth community, the literature provides only limited guidelines. In attempts to identify a trend in alloy composition, often a single control variable consisting of a ratio of two growth parameters is chosen. For example, Zhao et al. [3] and Chang et al. [1] report y in GaAsySb1y alloys as a function of the Sb to Ga flux ratio at specific Ts. This is clearly incomplete
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in view of Yano et al.’s results which relate the GaAsySb1y alloy composition across the entire range of 05y51, to a temperature dependent function of the Sb-to-As arrival ratio [2]. The latter, in turn, is again inadequate given Klem et al.’s work [4] which demonstrates a growth-rate dependency of GaAsySb1y alloy composition at a fixed Ts . Moreover, reporting in terms of a variable that is a ratio of two growth parameters implies uniqueness, and can be misleading if such uniqueness has not yet been confirmed. For example, it is not known whether increasing the Sb-flux and As-flux while keeping the Sb-to-As ratio fixed will cause a change in alloy composition. We believe that developing a more complete picture which depicts the role of all relevant MBE growth parameters on alloy composition can be a modest step towards a better understanding of alloy formation and possibly making a priori predictions in the future. We have performed systematic experiments based on response surface modeling (RSM) to identify trends in AlAsySb1y alloy composition in terms of each growth parameter while also identifying possible interactions between them.
2. Experimental procedure While RSM is a powerful technique used in design optimization, it has only rarely been used in thin film crystal growth [5–7] In general, RSM allows an empirical model to be built from data collected from a minimal set of systematically designed experiments. This model, referred to as the response function, is a polynomial function of the set of input variables that can best predict the response. In this work, we apply the Box–Behnken design of experiments [8] that allows the fitting of a second-degree polynomial function of four MBE growth variables (factors) to the resultant AlAsySb1y alloy composition. The Box–Behnken scheme we have employed is a design based on four factors, which stipulates 25 unique experiments in which each factor may have one of three possible values. We have selected the four relevant factors to be the growth rate (GR) which is
Table 1 Normalized values of the four factors used in RSM Factor
1
0
1
Units
Ts FAs FSb GR
470 0.4 0.8 0.56
510 1.3 1.4 0.72
550 2.2 2.0 0.88
8C 106 Torr (BEP) of As2 106 Torr (BEP) of Sb2 Monolayers/s
determined by the incident Al-flux, incident As2flux (FAs ), incident Sb2-flux (FSb ), and substrate temperature (Ts ). The levels of the four factors are listed in Table 1. For convenience in analysis, each of the three levels is suitably translated and scaled so that all factors have the normalized values of 1, 0, and 1. The values of each factor were selected to give a large variation in alloy composition, and to be within the range of commonly used MBE growth parameters for AlAsySb1y alloy. In our experiments, the alloy composition, y, is the desired response we seek to model. A standard approach to determining the alloy composition is to use X-ray diffraction (XRD) to measure the average lattice constant and hence determine y. However, ex situ XRD measurements require growth of thick layers, and thus significantly increase the duration of each experiment. This in turn leads to errors introduced by the day-to-day variations in the growth parameters. Rather than using standard blocking techniques to measure this in the RSM analysis, we have used desorption mass spectrometry (DMS) as an in situ measurement of alloy composition. The DMS measurements allowed data collection to be completed in one day, reducing errors associated with temperature and/ or flux re-calibration. Desorption mass spectrometry was shown to lend itself well to determining the GaAsySb1y alloy composition by measuring the Sb-desorption signal and quantifying the relative fraction of the Sb that is displaced in the presence of an As2-flux [9]. It has not previously been applied to AlAsySb1y. To demonstrate the validity of DMS measurements, a set of five 2 mm thick AlAsySb1y alloy samples were grown on GaAs
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Table 2 The terms and coefficients of the fitted quadratic equation modeling y in AlAsySb1y as a function of the four factors listed in order of decreasing significance Term
Coefficient
Constant FAs GR FSb (FAs)2 Ts (Ts)2 GR FAs Ts FAs Ts GR T_S NF_Sb GR NF_Sb F_Sb NF_As NF_Al GR NF_Sb NF_Sb
37.69 23.79 6.39 3.18 3.75 1.59 2.07 2.26 2.22 0 0 0 0 0 0
substrates in a separate experiment. Ex situ measurements of alloy composition using XRD analysis of symmetric (0 0 4) and asymmetric (1 1 5) reflectivity spectra were compared to DMS predictions. The XRD data agree with the DMS data to a compositional error of less than 2% for compositions y between 11% and 67%. The AlAsySb1y alloy films used in the RSM study were deposited on a radiatively heated 3-in diameter GaAs (0 0 1) substrate in a V-80 MBE system equipped with a valved As2-cracker, and an unvalved Sb2-cracker cell. Each test layer was grown to a thickness of approximately 500 A˚, well above that necessary for collecting stable DMS data. Growth rate was measured using reflection high-energy electron diffraction (RHEED) oscillations during AlAs deposition. The incident fluxes were measured by a nude ion gauge and reported as a beam equivalent pressure (BEP) Table 2.
3. Results and discussion The RS-1 statistical software package was used to analyze the data. A model to predict y in AlAsySb1y as a function of all of the factors and
their interactions is obtained from the regression analysis [10]. The terms of the model and their coefficients are tabulated in decreasing order of significance in Table 2. Following standard analysis practice, terms, that are measured by the tstatistic [11] to be insignificant with a confidence level of 95%, are assumed to have a coefficient of zero. Note that the tabulated coefficients are valid for terms that are expressed in normalized units (these can easily be translated using the information in Table 1). The coefficient-of-determination or R-squared value for the model is 0.9925. This implies that 99.25% of the total variation in y is accounted for by this model. Further statistical analysis shows that the ‘‘lack of fit’’, describing the error that cannot be accounted for, either by our model, or by the experimental error, is also insignificant. A quadratic fit to data is therefore adequate to make good predictions to AlAsySb1y alloy composition, in the sampled range of MBE growth parameters. While much information is contained within the model, it is impossible to convey it all at once since it would require five dimensions. We choose to use a set of three-dimensional plots in which a constant-y response surface is shown as a function of three of the four factors. In Fig. 1, a response surface defined by y ¼ 0:4 is shown as a function of three factors, FAs , growth rate, and FSb while Ts is held constant at 470oC. In Fig. 2, a response surface defined by y ¼ 0:66 is shown as a function of a different set of three factors, FSb , growth rate, and Ts while FAs is held constant at 2.2 106 Torr BEP. Two 2-D plots that reproduce the back faces of the cube supplement the 3D plot in each figure. Here, contour plots of constant y are plotted in which compositional trends can be identified as a function of two of the variables. All of the main trends and some of the interactions can be identified in these graphical representations of the model. The data indicates that a small variation in one of the factors can generally be compensated by an appropriate combination of the remaining factors to maintain a constant y in AlAsySb1y. We observe the following basic trends: An increase in FAs or a decrease in FSb , alone, will increase y in AlAsySb1y. This indicates that the
P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
559
Fig. 1. Constant composition (y ¼ 0:4) surface as a function of FAs , FSb , GR, at a fixed Ts=4708C. Contour plots of 100y, which depict the back faces of the cube, are shown above.
Fig. 2. Constant composition (y=0.66) surface as a function of FSb , GR, and Ts at a fixed F Sb=2.2 106 Torr BEP. Contour plots of 100y, which depict the back faces of the cube, are shown above.
rate of incorporation of each group-V species will increase if more of that species is present at the surface, as can be expected in a statistical incorporation process. The data also shows that an increase in Ts alone will favor the incorporation of As over Sb. It is not known whether an increase in thermal desorption rates alone will favor an increase in surface As concentration since Sb molecules tend to have a larger sublimation energy than As molecules. However As, which forms a stronger covalent bond with Al than does Sb, is known to efficiently remove Sb from the surface through an exchange process that is enhanced with increasing Ts [12]. This As-stimulated desorption of Sb may be responsible for the observed trend. Finally, decreasing the growth rate alone is observed to increase y. This means that the ability of Sb adatoms to incorporate is enhanced as more group-V incorporation sites are made available. At higher growth rates, adatoms spend
less time at the surface before being incorporated, perhaps reducing the effectiveness of the As-for-Sb exchange process. A number of interactions between factors are also observed, which result in curvature of the response surfaces. For example the model, as shown in Fig. 2, suggests that at Ts ¼ 4708C, y is quite sensitive to FSb , whereas at Ts ¼ 5508C, y is relatively insensitive to FSb . Similarly, y is measured to be more sensitive to FAs at low growth rates than at high growth rates. Through careful analysis of model data, it is possible to select optimal growth conditions where compositional variation due to unexpected drift in one or several factors can be minimized. The scope of this paper is limited to a demonstration of the applicability of RSM to mixed III-AsySb1y epitaxy. Using models developed in this manner, a priori prediction of alloy composition for any MBE reactor may be
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P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
possible when errors in reproducing growth parameters between different reactors can be eliminated.
4. Summary We have used the response surface modeling technique to model AlAsySb1y alloy composition as a function of four relevant MBE growth parameters: growth rate, As2-flux, Sb2-flux, and substrate temperature. Alloy composition was measured by the desorption mass-spectrometry technique, which helped reduce errors and eliminated the need for blocking. A quadratic model is found to fit the data well, and used to clearly identify the main trends and interactions.
Acknowledgements We gratefully acknowledge the support of the Air Force Office of Scientific Research.
References [1] C.-A. Chang, R. Ludeke, L.L. Chang, L. Esaki, Appl. Phys. Lett. 31 (1977) 759. [2] M. Yano, M. Ashida, A. Kawaguchi, Y. Iwai, M. Inoue, J. Vac. Sci. Technol. B (1989) 199. [3] J.H. Zhao, A.Z. Li, J. Jeong, D. Wong, J.C. Lee, M.L. Milliman, T.E. Schlesinger, A.G. Milnes, J. Vac. Sci. Technol B 6 (1987) 627. [4] J. Klem, R. Fisher, T.J. Drummond, H. Morkoc, A.Y. Cho, Electron. Lett. 19 (1983) 455. [5] K.K. Lee, W.A. Doolittle, A.S. Brown, G.S. May, S.R. Rock, J. Vac. Sci. Technol. B 18 (2000) 1448. [6] M.E. Sherwin, G.O. Munns, M.E. Elta, E.G. Woelk, S.B. Crary, F.L. Terry, G.I. Haddad, J. Crystal Growth 111 (1991) 594. [7] R.N. Bicknell-Tassius, K. Lee, A.S. Brown, G. Dagnall, G. May, J. Crystal Growth 175/176 (1997) 1131. [8] G.E.P. Box, D.W. Behnken, Technometrics 2 (1960) 455. [9] R. Kaspi, W.T. Cooley, K.R. Evans, J. Crystal Growth 173 (1997) 5. [10] R.F. Gunst, R.L. Mason, Regression Analysis and its Application, Marcel Dekker, New York, 1980. [11] G.E.P. Box, W.G. Hunter, J.S. Hunter, Statistics for Experimenters, Wiley, New York, 1978. [12] M. Yano, H. Yokose, Y. Iwai, M. Inoue, J. Crystal Growth 111 (1991) 609.
Response surface modeling of the composition of AlAsySb1y alloys grown by molecular beam epitaxy P. Gopaladasua, J.L. Cecchia, K.J. Malloyb, R. Kaspic,* a
Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, NM 87131, USA b Center for High Technology Materials, University of New Mexico, Albuquerque, NM 87106, USA c Air Force Research Laboratory, AFRL/DELS, Kirtland AFB, Albuquerque, NM 87117, USA
Abstract We have developed an empirical model of AlAsySb1y composition during molecular beam epitaxy as a function of growth temperature, growth rate, As2-flux, and Sb2-flux, using response surface methodology. In order to facilitate data collection from a large number of experiments, we have applied desorption mass spectroscopy as an in situ measure of alloy composition. A quadratic model is found to fit the data well, identifying trends and several interactions between growth parameters. Published by Elsevier Science B.V. PACS: 81.15.Hi; 68.55.Nq; 82.80.Ms; 07.05.F Keywords: A1. Design of experiments; A1. Desorption mass spectrometry; A3. Molecular beam epitaxy; B1. Antimonides
1. Introduction During molecular beam epitaxy (MBE) growth of mixed group-III alloys, such as AlxGa1xAs, the alloy composition is directly given by the incident group-III flux ratio. A group-III sticking coefficient of unity across the full range of the desirable growth-parameter space allows for such ease of prediction. In contrast, compositional control of mixed group-V alloys, such as AlAsySb1y, is complicated by the fact that an excess of group-V adatoms must compete for the limited number of anion vacancies. This competi*Corresponding author. AFRL/DELS, c/o CHTM 1313 Goddard, SE, Albuquerque, NM 87106, USA. Tel.: +1-505272-7822; fax: +1-505-272-7801. E-mail address: [email protected] (R. Kaspi). 0022-0248/01/$ - see front matter Published by Elsevier Science B.V. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 0 9 5 2 - 6
tion is known to be strongly influenced by several growth parameters such as temperature [1], groupIII fluxes, and group-V fluxes [2]. It is therefore not possible to predict alloy composition without a detailed understanding of the individual and combined effects of the relevant growth parameters on alloy composition. While many of the observed trends in IIIAsySb1y alloy composition such as an increasing y with increasing substrate temperature (Ts) are well recognized in the crystal growth community, the literature provides only limited guidelines. In attempts to identify a trend in alloy composition, often a single control variable consisting of a ratio of two growth parameters is chosen. For example, Zhao et al. [3] and Chang et al. [1] report y in GaAsySb1y alloys as a function of the Sb to Ga flux ratio at specific Ts. This is clearly incomplete
557
P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
in view of Yano et al.’s results which relate the GaAsySb1y alloy composition across the entire range of 05y51, to a temperature dependent function of the Sb-to-As arrival ratio [2]. The latter, in turn, is again inadequate given Klem et al.’s work [4] which demonstrates a growth-rate dependency of GaAsySb1y alloy composition at a fixed Ts . Moreover, reporting in terms of a variable that is a ratio of two growth parameters implies uniqueness, and can be misleading if such uniqueness has not yet been confirmed. For example, it is not known whether increasing the Sb-flux and As-flux while keeping the Sb-to-As ratio fixed will cause a change in alloy composition. We believe that developing a more complete picture which depicts the role of all relevant MBE growth parameters on alloy composition can be a modest step towards a better understanding of alloy formation and possibly making a priori predictions in the future. We have performed systematic experiments based on response surface modeling (RSM) to identify trends in AlAsySb1y alloy composition in terms of each growth parameter while also identifying possible interactions between them.
2. Experimental procedure While RSM is a powerful technique used in design optimization, it has only rarely been used in thin film crystal growth [5–7] In general, RSM allows an empirical model to be built from data collected from a minimal set of systematically designed experiments. This model, referred to as the response function, is a polynomial function of the set of input variables that can best predict the response. In this work, we apply the Box–Behnken design of experiments [8] that allows the fitting of a second-degree polynomial function of four MBE growth variables (factors) to the resultant AlAsySb1y alloy composition. The Box–Behnken scheme we have employed is a design based on four factors, which stipulates 25 unique experiments in which each factor may have one of three possible values. We have selected the four relevant factors to be the growth rate (GR) which is
Table 1 Normalized values of the four factors used in RSM Factor
1
0
1
Units
Ts FAs FSb GR
470 0.4 0.8 0.56
510 1.3 1.4 0.72
550 2.2 2.0 0.88
8C 106 Torr (BEP) of As2 106 Torr (BEP) of Sb2 Monolayers/s
determined by the incident Al-flux, incident As2flux (FAs ), incident Sb2-flux (FSb ), and substrate temperature (Ts ). The levels of the four factors are listed in Table 1. For convenience in analysis, each of the three levels is suitably translated and scaled so that all factors have the normalized values of 1, 0, and 1. The values of each factor were selected to give a large variation in alloy composition, and to be within the range of commonly used MBE growth parameters for AlAsySb1y alloy. In our experiments, the alloy composition, y, is the desired response we seek to model. A standard approach to determining the alloy composition is to use X-ray diffraction (XRD) to measure the average lattice constant and hence determine y. However, ex situ XRD measurements require growth of thick layers, and thus significantly increase the duration of each experiment. This in turn leads to errors introduced by the day-to-day variations in the growth parameters. Rather than using standard blocking techniques to measure this in the RSM analysis, we have used desorption mass spectrometry (DMS) as an in situ measurement of alloy composition. The DMS measurements allowed data collection to be completed in one day, reducing errors associated with temperature and/ or flux re-calibration. Desorption mass spectrometry was shown to lend itself well to determining the GaAsySb1y alloy composition by measuring the Sb-desorption signal and quantifying the relative fraction of the Sb that is displaced in the presence of an As2-flux [9]. It has not previously been applied to AlAsySb1y. To demonstrate the validity of DMS measurements, a set of five 2 mm thick AlAsySb1y alloy samples were grown on GaAs
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P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
Table 2 The terms and coefficients of the fitted quadratic equation modeling y in AlAsySb1y as a function of the four factors listed in order of decreasing significance Term
Coefficient
Constant FAs GR FSb (FAs)2 Ts (Ts)2 GR FAs Ts FAs Ts GR T_S NF_Sb GR NF_Sb F_Sb NF_As NF_Al GR NF_Sb NF_Sb
37.69 23.79 6.39 3.18 3.75 1.59 2.07 2.26 2.22 0 0 0 0 0 0
substrates in a separate experiment. Ex situ measurements of alloy composition using XRD analysis of symmetric (0 0 4) and asymmetric (1 1 5) reflectivity spectra were compared to DMS predictions. The XRD data agree with the DMS data to a compositional error of less than 2% for compositions y between 11% and 67%. The AlAsySb1y alloy films used in the RSM study were deposited on a radiatively heated 3-in diameter GaAs (0 0 1) substrate in a V-80 MBE system equipped with a valved As2-cracker, and an unvalved Sb2-cracker cell. Each test layer was grown to a thickness of approximately 500 A˚, well above that necessary for collecting stable DMS data. Growth rate was measured using reflection high-energy electron diffraction (RHEED) oscillations during AlAs deposition. The incident fluxes were measured by a nude ion gauge and reported as a beam equivalent pressure (BEP) Table 2.
3. Results and discussion The RS-1 statistical software package was used to analyze the data. A model to predict y in AlAsySb1y as a function of all of the factors and
their interactions is obtained from the regression analysis [10]. The terms of the model and their coefficients are tabulated in decreasing order of significance in Table 2. Following standard analysis practice, terms, that are measured by the tstatistic [11] to be insignificant with a confidence level of 95%, are assumed to have a coefficient of zero. Note that the tabulated coefficients are valid for terms that are expressed in normalized units (these can easily be translated using the information in Table 1). The coefficient-of-determination or R-squared value for the model is 0.9925. This implies that 99.25% of the total variation in y is accounted for by this model. Further statistical analysis shows that the ‘‘lack of fit’’, describing the error that cannot be accounted for, either by our model, or by the experimental error, is also insignificant. A quadratic fit to data is therefore adequate to make good predictions to AlAsySb1y alloy composition, in the sampled range of MBE growth parameters. While much information is contained within the model, it is impossible to convey it all at once since it would require five dimensions. We choose to use a set of three-dimensional plots in which a constant-y response surface is shown as a function of three of the four factors. In Fig. 1, a response surface defined by y ¼ 0:4 is shown as a function of three factors, FAs , growth rate, and FSb while Ts is held constant at 470oC. In Fig. 2, a response surface defined by y ¼ 0:66 is shown as a function of a different set of three factors, FSb , growth rate, and Ts while FAs is held constant at 2.2 106 Torr BEP. Two 2-D plots that reproduce the back faces of the cube supplement the 3D plot in each figure. Here, contour plots of constant y are plotted in which compositional trends can be identified as a function of two of the variables. All of the main trends and some of the interactions can be identified in these graphical representations of the model. The data indicates that a small variation in one of the factors can generally be compensated by an appropriate combination of the remaining factors to maintain a constant y in AlAsySb1y. We observe the following basic trends: An increase in FAs or a decrease in FSb , alone, will increase y in AlAsySb1y. This indicates that the
P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
559
Fig. 1. Constant composition (y ¼ 0:4) surface as a function of FAs , FSb , GR, at a fixed Ts=4708C. Contour plots of 100y, which depict the back faces of the cube, are shown above.
Fig. 2. Constant composition (y=0.66) surface as a function of FSb , GR, and Ts at a fixed F Sb=2.2 106 Torr BEP. Contour plots of 100y, which depict the back faces of the cube, are shown above.
rate of incorporation of each group-V species will increase if more of that species is present at the surface, as can be expected in a statistical incorporation process. The data also shows that an increase in Ts alone will favor the incorporation of As over Sb. It is not known whether an increase in thermal desorption rates alone will favor an increase in surface As concentration since Sb molecules tend to have a larger sublimation energy than As molecules. However As, which forms a stronger covalent bond with Al than does Sb, is known to efficiently remove Sb from the surface through an exchange process that is enhanced with increasing Ts [12]. This As-stimulated desorption of Sb may be responsible for the observed trend. Finally, decreasing the growth rate alone is observed to increase y. This means that the ability of Sb adatoms to incorporate is enhanced as more group-V incorporation sites are made available. At higher growth rates, adatoms spend
less time at the surface before being incorporated, perhaps reducing the effectiveness of the As-for-Sb exchange process. A number of interactions between factors are also observed, which result in curvature of the response surfaces. For example the model, as shown in Fig. 2, suggests that at Ts ¼ 4708C, y is quite sensitive to FSb , whereas at Ts ¼ 5508C, y is relatively insensitive to FSb . Similarly, y is measured to be more sensitive to FAs at low growth rates than at high growth rates. Through careful analysis of model data, it is possible to select optimal growth conditions where compositional variation due to unexpected drift in one or several factors can be minimized. The scope of this paper is limited to a demonstration of the applicability of RSM to mixed III-AsySb1y epitaxy. Using models developed in this manner, a priori prediction of alloy composition for any MBE reactor may be
560
P. Gopaladasu et al. / Journal of Crystal Growth 225 (2001) 556–560
possible when errors in reproducing growth parameters between different reactors can be eliminated.
4. Summary We have used the response surface modeling technique to model AlAsySb1y alloy composition as a function of four relevant MBE growth parameters: growth rate, As2-flux, Sb2-flux, and substrate temperature. Alloy composition was measured by the desorption mass-spectrometry technique, which helped reduce errors and eliminated the need for blocking. A quadratic model is found to fit the data well, and used to clearly identify the main trends and interactions.
Acknowledgements We gratefully acknowledge the support of the Air Force Office of Scientific Research.
References [1] C.-A. Chang, R. Ludeke, L.L. Chang, L. Esaki, Appl. Phys. Lett. 31 (1977) 759. [2] M. Yano, M. Ashida, A. Kawaguchi, Y. Iwai, M. Inoue, J. Vac. Sci. Technol. B (1989) 199. [3] J.H. Zhao, A.Z. Li, J. Jeong, D. Wong, J.C. Lee, M.L. Milliman, T.E. Schlesinger, A.G. Milnes, J. Vac. Sci. Technol B 6 (1987) 627. [4] J. Klem, R. Fisher, T.J. Drummond, H. Morkoc, A.Y. Cho, Electron. Lett. 19 (1983) 455. [5] K.K. Lee, W.A. Doolittle, A.S. Brown, G.S. May, S.R. Rock, J. Vac. Sci. Technol. B 18 (2000) 1448. [6] M.E. Sherwin, G.O. Munns, M.E. Elta, E.G. Woelk, S.B. Crary, F.L. Terry, G.I. Haddad, J. Crystal Growth 111 (1991) 594. [7] R.N. Bicknell-Tassius, K. Lee, A.S. Brown, G. Dagnall, G. May, J. Crystal Growth 175/176 (1997) 1131. [8] G.E.P. Box, D.W. Behnken, Technometrics 2 (1960) 455. [9] R. Kaspi, W.T. Cooley, K.R. Evans, J. Crystal Growth 173 (1997) 5. [10] R.F. Gunst, R.L. Mason, Regression Analysis and its Application, Marcel Dekker, New York, 1980. [11] G.E.P. Box, W.G. Hunter, J.S. Hunter, Statistics for Experimenters, Wiley, New York, 1978. [12] M. Yano, H. Yokose, Y. Iwai, M. Inoue, J. Crystal Growth 111 (1991) 609.