Bullock, D.

UARK (University of Arkansas )

Journal of the Arkansas Academy of Science Volume 62 Article 22 2008 Growth Rate Calculations for Epitaxially Grown Thin Films D. Bullock Arkansas Tech University, dbullock@atu.edu Follow this and additional works at: https://scholarworks.uark.edu/jaas  Part of the Semiconductor and Optical Materials Commons Recommended Citation Bullock, D. (2008) "Growth Rate Calculations for Epitaxially Grown Thin Films," Journal of the Arkansas Academy of Science: Vol. 62 , Article 22. Available at: https://scholarworks.uark.edu/jaas/vol62/iss1/22 This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This General Note is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu. Journal of the Arkansas Academy of Science, Vol. 62, 2008142Growth Rate Calculations for Epitaxially Grown Thin FilmsD. BullockDepartment of Physical Science, Arkansas Tech University, Russellville, AR 72801Correspondence: dbullock@atu.eduArtificially fabricated semiconductor structures wereintroduced more than 20 years ago with somespectacular results in both basic physics discoveriesand in commercial applications (Maclean 2001).Quantum well lasers, for example, are now found inevery compact disc (CD) and digital video disc (DVD)player. The quantum well laser is a prime example ofwhat has been termed a first-generation quantumdevice that is a device that reproduces the function ofits respective conventional bulk device but with higherperformance specifications (e.g., more stability over abroader temperature range). Second-generationquantum devices (still in the research stage) are multi-functional; a single structure can accomplish a task thatwould normally require as many as 10 conventionaldevices. A factor of 10 reduction in the number ofcomponents naturally leads to a significant increase inspeed as well as a reduction in power consumption.These multi-functional devices will require theproduction of complex structures with very shortperiod superlattices, non-linear graded compositionprofiles, gross alterations in the lattice constant, etc.The key to accomplishing this goal is to have precise,atomic-scale control of the growth process. Unlike Si-based devices, which are primarily fabricated usingimplantation techniques, the III-V semiconductorsrequire atomic layer-by-layer growth of theheterostructures (Current 1992). The remarkableadvances that have been made over the past decade in avariety of advanced structures involving III-Vcompound semiconductors has come about primarilydue to the advent of, and refinements in, the primaryfabrication technique known as molecular beamepitaxy (MBE).Semiconductor device fabrication via MBE growthoccurs in an ultra-high vacuum (UHV) environment(~10-11 Torr) where a molecular beam of group III andgroup V molecules impinges on a single crystallinesubstrate. In MBE the mean free path for the impingingflux is much greater than the distance from source tosample, consequently the growth kinetics aredetermined by the relative sticking coefficients of thetwo species, by the diffusion rate of the two speciesonce they are in contact with the substrate surface, andby the dissociation rate of the molecules.Although MBE produces the highest qualitysamples, it has one major drawback; it is a veryexpensive technique to implement on a productionscale. A basic production MBE system can costupwards of ten million dollars. In addition to theequipment cost, both the consumables associated withrunning the machine as well as the required manpowercreate large overhead costs for the manufacturer.Consequently, it is of great interest to make the MBEproduction process as efficient as possible. One majorsource of production downtime comes whilecalibrating the most important aspect of the production,namely, the growth rate. It can take several days for amanufacturer to calibrate the growth rate from a sourcematerial in order to get the appropriate atomic ratios.Naturally, a better understanding of the physicsgoverning the growth rate of these materials couldenhance the efficiency of the MBE manufacturingprocess. What is needed is a model that can accuratelypredict the fraction of atoms that evaporate from thesource material and arrive at the sample substrate tocontribute to the growth of the structure.Accurately predicting the growth rate is a difficultproblem to solve because it is related to both static(such as the temperature of the material and theresidual vacuum chamber pressure) and dynamicvariables (such as the source-sample distance and solidangle) (Herman et al. 1989). However, with recentadvances in both dynamic computational techniquesand microcomputers, solution of this problem ispossible. Our approach is to develop a model thatdescribes the growth rate as it relates to the sourcetemperature and geometry (i.e. solid angle and source-sample distance). Ideally, this model will have sub-monolayer accuracy and be computationally efficient.The growth rate model will describe the molecularbeam flux evaporating from the metallic Gallium (Ga)source and impinging on the surface of a GaAssubstrate. This is directly related to Ga’s vaporpressure. A material’s vapor pressure, P, determinesthe number of atoms that evaporate from a sample’ssurface as a function of temperature (Tsao 1993). Thecomplete temperature dependence of vapor pressurerequires a formula with four adjustable constants.Many formulas have been suggested, but the one found142Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22Published by Arkansas Academy of Science, 2008Growth Rate Calculations for Epitaxially Grown Thin FilmsJournal of the Arkansas Academy of Science, Vol. 62, 2008143to be the most accurate by Nesmeyanov (Nesmeyanov1963) isTBCTTADP loglog  . (1)In this equation, T is the temperature (in Kelvin) andA, B, C, and D are fitting parameters.In order to develop an accurate model of the growthrate, it is necessary to determine which terms in thevapor pressure equation are the most dominate. InFigure 1 each individual term is plotted with itsappropriate fitting constants for Ga. The log P is alsoplotted on the graph. From examining Figure 1 we seethat the vapor pressure curve is dominated by the A/Tterm from equation 1. The next dominant term isdetermined to be the B log T term. In order tosimultaneously maximize the accuracy of our modelbut minimized computation time it is necessary to keeponly the two most dominant terms from equation (1).Equation (1) then becomesTBTAP loglog  . (2)From equation (2) the vapor pressure equation is foundto beBTATP 10 . (3)Equation (3) assumes a uniform spherical distributionof evaporating material. This assumption will not applyin MBE since it takes advantage of highly directionalKnudsen effusion cells (or K-cells) (Herman et al.1989). To correct for this a new constant, GR0, isintroduced that scales the modified vapor pressure forthe geometry of the K-cell and the source-sampledistance. Equation (3) becomes the geometry correctedvapor pressure, PGC,BTAGC TGRPGRP 1000 . (4)It is important to note that GR0 will be constant onlyfor a single growth since the source-sample distanceand the solid angle between the source and sample willchange for each successive production run requiring anew value for GR0 to be calculated. This can beaccomplished by measuring the temperature for aselected growth rate and using this information to solvefor GR0. In equation (4) the constant A = Ea/kB whereEa is the activation energy and kB is Boltzmann’sconstant with a value of 8.617 × 10−5 eV/K. Theactivation energy is a parameter that describes theamount of energy required to completely remove anatom from the surface of the bulk material, for Galliummetal Ea = -1.457 eV. The fitting parameter B is relatedto the latent heat of the material and is found to have avalue of -9.4101 (Nesmeyanov 1963). The growth rateequation is then expressed asBT TGRTGR Bak-E10)( 0 . (5)Figure 2 shows the derived growth rate equation hasexcellent agreement with experimentally determinedgrowth rates (LaBella 2000).Figure 1. Graph illustrating the dependence of each term in thevapor pressure equation using the appropriate constants forGallium as a function of absolute temperature. From this thedominating term (A/T) is ascertained.Measured Growth RateGrowth Rate EquationFigure 2. Graph illustrating the GaAs growth rate as a functionof Gallium cell temperature. The line represents the developedformula which is graphed with actual growth rate data.143Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22https://scholarworks.uark.edu/jaas/vol62/iss1/22D. BullockJournal of the Arkansas Academy of Science, Vol. 62, 2008144To further enhance the productivity of MBEsystems it is necessary to provide immediate growthrate feedback to the user. Using equation (5), it waspossible to develop a program to calculate theappropriate K-cell temperature for a desired growthrate. To do this the user inputs values for the currentcell temperature, current growth rate, and the desiredgrowth rate. Using this information the computer firstcalculates GR0. Next, the calculation of the K-celltemperature is performed unfortunately due to thetranscendental nature of equation (5) it is impossible tosolve for the temperature algebraically, instead anumerical technique must be utilized that finds the rootof the equation given the input parameters (Press et al.2002). This root corresponds to the K-cell temperaturethat yields the desired growth rate.In conclusion we have been successful atdeveloping a model that predicts the growth rate as afunction of temperature. This model agrees well withexperimentally measured growth rates. Additionally,we have also developed software that can greatlyreduce the time needed for growth rate calibrations.Acknowledgments: The author would like to expressgratitude to the Arkansas Space Grant Consortium andthe Arkansas Tech University Undergraduate ResearchCouncil for their financial support of this project.Literature CitedLabella VP. 2000. Microscopic View of a Two-Dimensional Lattice-Gas Ising System within theGrand Canonical Ensemble. Physical ReviewLetters 84:4152.Current MI. 1992. Ion implantation for silicondevice manufacturing: A vacuum perspective.Journal of Vacuum Science and Technology A14:1115.Herman M and H Sitter. 1989. MBE, Fundamentalsand Current Status, New York: Springer-Velag.Maclean J. 2001. III-Vs Innovate for Future OpticalCommunications. Compound Semiconductor 7:12.Nesmeyanov AN. 1963. Vapor Pressure of theChemical Elements. New York: AcademicPress.197 p.Press W, S Teukolsky, W Vetterling, and BFlannery. 2002. Numerical Recipes in C++.Cambridge: Cambridge Univ. Press. 351 p.Tsao J. 1993. Materials Fundamentals of MolecularBeam Epitaxy. New York: Academic Press. 332 p144Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22Published by Arkansas Academy of Science, 2008

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Growth Rate Calculations for Epitaxially Grown Thin Films

ScholarWorks@UARK

Journal of the Arkansas Academy of ScienceVolume 62 Article 222008Growth Rate Calculations for Epitaxially GrownThin FilmsD. BullockArkansas Tech University, dbullock@atu.eduFollow this and additional works at: http://scholarworks.uark.edu/jaasPart of the Semiconductor and Optical Materials CommonsThis article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able toread, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking priorpermission from the publisher or the author.This General Note is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the ArkansasAcademy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu.Recommended CitationBullock, D. (2008) "Growth Rate Calculations for Epitaxially Grown Thin Films," Journal of the Arkansas Academy of Science: Vol. 62 ,Article 22.Available at: http://scholarworks.uark.edu/jaas/vol62/iss1/22Journal of the Arkansas Academy of Science, Vol. 62, 2008142Growth Rate Calculations for Epitaxially Grown Thin FilmsD. BullockDepartment of Physical Science, Arkansas Tech University, Russellville, AR 72801Correspondence: dbullock@atu.eduArtificially fabricated semiconductor structures wereintroduced more than 20 years ago with somespectacular results in both basic physics discoveriesand in commercial applications (Maclean 2001).Quantum well lasers, for example, are now found inevery compact disc (CD) and digital video disc (DVD)player. The quantum well laser is a prime example ofwhat has been termed a first-generation quantumdevice that is a device that reproduces the function ofits respective conventional bulk device but with higherperformance specifications (e.g., more stability over abroader temperature range). Second-generationquantum devices (still in the research stage) are multi-functional; a single structure can accomplish a task thatwould normally require as many as 10 conventionaldevices. A factor of 10 reduction in the number ofcomponents naturally leads to a significant increase inspeed as well as a reduction in power consumption.These multi-functional devices will require theproduction of complex structures with very shortperiod superlattices, non-linear graded compositionprofiles, gross alterations in the lattice constant, etc.The key to accomplishing this goal is to have precise,atomic-scale control of the growth process. Unlike Si-based devices, which are primarily fabricated usingimplantation techniques, the III-V semiconductorsrequire atomic layer-by-layer growth of theheterostructures (Current 1992). The remarkableadvances that have been made over the past decade in avariety of advanced structures involving III-Vcompound semiconductors has come about primarilydue to the advent of, and refinements in, the primaryfabrication technique known as molecular beamepitaxy (MBE).Semiconductor device fabrication via MBE growthoccurs in an ultra-high vacuum (UHV) environment(~10-11 Torr) where a molecular beam of group III andgroup V molecules impinges on a single crystallinesubstrate. In MBE the mean free path for the impingingflux is much greater than the distance from source tosample, consequently the growth kinetics aredetermined by the relative sticking coefficients of thetwo species, by the diffusion rate of the two speciesonce they are in contact with the substrate surface, andby the dissociation rate of the molecules.Although MBE produces the highest qualitysamples, it has one major drawback; it is a veryexpensive technique to implement on a productionscale. A basic production MBE system can costupwards of ten million dollars. In addition to theequipment cost, both the consumables associated withrunning the machine as well as the required manpowercreate large overhead costs for the manufacturer.Consequently, it is of great interest to make the MBEproduction process as efficient as possible. One majorsource of production downtime comes whilecalibrating the most important aspect of the production,namely, the growth rate. It can take several days for amanufacturer to calibrate the growth rate from a sourcematerial in order to get the appropriate atomic ratios.Naturally, a better understanding of the physicsgoverning the growth rate of these materials couldenhance the efficiency of the MBE manufacturingprocess. What is needed is a model that can accuratelypredict the fraction of atoms that evaporate from thesource material and arrive at the sample substrate tocontribute to the growth of the structure.Accurately predicting the growth rate is a difficultproblem to solve because it is related to both static(such as the temperature of the material and theresidual vacuum chamber pressure) and dynamicvariables (such as the source-sample distance and solidangle) (Herman et al. 1989). However, with recentadvances in both dynamic computational techniquesand microcomputers, solution of this problem ispossible. Our approach is to develop a model thatdescribes the growth rate as it relates to the sourcetemperature and geometry (i.e. solid angle and source-sample distance). Ideally, this model will have sub-monolayer accuracy and be computationally efficient.The growth rate model will describe the molecularbeam flux evaporating from the metallic Gallium (Ga)source and impinging on the surface of a GaAssubstrate. This is directly related to Ga’s vaporpressure. A material’s vapor pressure, P, determinesthe number of atoms that evaporate from a sample’ssurface as a function of temperature (Tsao 1993). Thecomplete temperature dependence of vapor pressurerequires a formula with four adjustable constants.Many formulas have been suggested, but the one found142Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22Published by Arkansas Academy of Science, 2008Growth Rate Calculations for Epitaxially Grown Thin FilmsJournal of the Arkansas Academy of Science, Vol. 62, 2008143to be the most accurate by Nesmeyanov (Nesmeyanov1963) isTBCTTADP loglog  . (1)In this equation, T is the temperature (in Kelvin) andA, B, C, and D are fitting parameters.In order to develop an accurate model of the growthrate, it is necessary to determine which terms in thevapor pressure equation are the most dominate. InFigure 1 each individual term is plotted with itsappropriate fitting constants for Ga. The log P is alsoplotted on the graph. From examining Figure 1 we seethat the vapor pressure curve is dominated by the A/Tterm from equation 1. The next dominant term isdetermined to be the B log T term. In order tosimultaneously maximize the accuracy of our modelbut minimized computation time it is necessary to keeponly the two most dominant terms from equation (1).Equation (1) then becomesTBTAP loglog  . (2)From equation (2) the vapor pressure equation is foundto beBTATP 10 . (3)Equation (3) assumes a uniform spherical distributionof evaporating material. This assumption will not applyin MBE since it takes advantage of highly directionalKnudsen effusion cells (or K-cells) (Herman et al.1989). To correct for this a new constant, GR0, isintroduced that scales the modified vapor pressure forthe geometry of the K-cell and the source-sampledistance. Equation (3) becomes the geometry correctedvapor pressure, PGC,BTAGC TGRPGRP 1000 . (4)It is important to note that GR0 will be constant onlyfor a single growth since the source-sample distanceand the solid angle between the source and sample willchange for each successive production run requiring anew value for GR0 to be calculated. This can beaccomplished by measuring the temperature for aselected growth rate and using this information to solvefor GR0. In equation (4) the constant A = Ea/kB whereEa is the activation energy and kB is Boltzmann’sconstant with a value of 8.617 × 10−5 eV/K. Theactivation energy is a parameter that describes theamount of energy required to completely remove anatom from the surface of the bulk material, for Galliummetal Ea = -1.457 eV. The fitting parameter B is relatedto the latent heat of the material and is found to have avalue of -9.4101 (Nesmeyanov 1963). The growth rateequation is then expressed asBT TGRTGR Bak-E10)( 0 . (5)Figure 2 shows the derived growth rate equation hasexcellent agreement with experimentally determinedgrowth rates (LaBella 2000).Figure 1. Graph illustrating the dependence of each term in thevapor pressure equation using the appropriate constants forGallium as a function of absolute temperature. From this thedominating term (A/T) is ascertained.Measured Growth RateGrowth Rate EquationFigure 2. Graph illustrating the GaAs growth rate as a functionof Gallium cell temperature. The line represents the developedformula which is graphed with actual growth rate data.143Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22http://scholarworks.uark.edu/jaas/vol62/iss1/22D. BullockJournal of the Arkansas Academy of Science, Vol. 62, 2008144To further enhance the productivity of MBEsystems it is necessary to provide immediate growthrate feedback to the user. Using equation (5), it waspossible to develop a program to calculate theappropriate K-cell temperature for a desired growthrate. To do this the user inputs values for the currentcell temperature, current growth rate, and the desiredgrowth rate. Using this information the computer firstcalculates GR0. Next, the calculation of the K-celltemperature is performed unfortunately due to thetranscendental nature of equation (5) it is impossible tosolve for the temperature algebraically, instead anumerical technique must be utilized that finds the rootof the equation given the input parameters (Press et al.2002). This root corresponds to the K-cell temperaturethat yields the desired growth rate.In conclusion we have been successful atdeveloping a model that predicts the growth rate as afunction of temperature. This model agrees well withexperimentally measured growth rates. Additionally,we have also developed software that can greatlyreduce the time needed for growth rate calibrations.Acknowledgments: The author would like to expressgratitude to the Arkansas Space Grant Consortium andthe Arkansas Tech University Undergraduate ResearchCouncil for their financial support of this project.Literature CitedLabella VP. 2000. Microscopic View of a Two-Dimensional Lattice-Gas Ising System within theGrand Canonical Ensemble. Physical ReviewLetters 84:4152.Current MI. 1992. Ion implantation for silicondevice manufacturing: A vacuum perspective.Journal of Vacuum Science and Technology A14:1115.Herman M and H Sitter. 1989. MBE, Fundamentalsand Current Status, New York: Springer-Velag.Maclean J. 2001. III-Vs Innovate for Future OpticalCommunications. Compound Semiconductor 7:12.Nesmeyanov AN. 1963. Vapor Pressure of theChemical Elements. New York: AcademicPress.197 p.Press W, S Teukolsky, W Vetterling, and BFlannery. 2002. Numerical Recipes in C++.Cambridge: Cambridge Univ. Press. 351 p.Tsao J. 1993. Materials Fundamentals of MolecularBeam Epitaxy. New York: Academic Press. 332 p144Journal of the Arkansas Academy of Science, Vol. 62 [2008], Art. 22Published by Arkansas Academy of Science, 2008

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Growth Rate Calculations for Epitaxially Grown Thin Films

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