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Epitaxy growth kinetics of GaN films
Journal of Crystal Growth 250 (2003) 14–21
Epitaxy growth kinetics of GaN films Bei Wu, Ronghui Ma, Hui Zhang* Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2300, USA
Abstract Group III nitrides, such as GaN, AlN and InGaN, have attracted a lot of attention due to the development of blue– green and ultraviolet light emitting diodes and lasers. A GaN crystal can be grown from the vapor phase by either evaporation of Gallium (Ga) metal or sublimation of GaN powder in ammonia (NH3) atmosphere at a temperaturecontrolled growth furnace. In this paper, an integrated GaN growth model using a sublimation growth model has been developed based on the conservation of momentum, mass, chemical species and energy together with necessary boundary conditions that account for heterogeneous chemical reactions both at the source and seed surfaces. For the growth rate, the effects of the gas-flow rate, source temperature, temperature difference, and the gap width of the growth cell on the growth process have been studied. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.05.Ea; 81.10.Aj; 81.10.Bk Keywords: A1. Computer simulation; A1. Growth models; A2. Growth from vapor; A2. Single crystal growth; A3. Chemical vapor deposition processes; B1. Nitrides
1. Introduction Group III nitrides, such as GaN, AlN and InGaN, have attracted a lot of attention in the past few years due to the development of blue– green light emitting diodes (LEDs) and lasers [1]. Blue–green LEDs can be used in full-color flat display panels with red–blue–green pixels for highdensity optical data storage and high-speed optical communications. GaN, with a direct energy gap of 3.46 eV, is one of the most interesting materials for optoelectronics. GaN thin films have been deposited onto substrates made of sapphire, silicon carbide, LiGaO2 and LiAlO2 [2]. Metalorganic *Corresponding author. Tel.: +1-631-632-8492; fax: +1631-632-8544. E-mail address: [email protected] (H. Zhang).
chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE) techniques are extensively used [3,4]. However, all substrates other than GaN will introduce stresses and dislocations due to lattice mismatch and different thermal expansion coefficients [5] during thin film growth. A dislocation density of 108–1010/cm2 is usually obtained in the GaN thin or thick films. It has been demonstrated that homoepitaxial GaN films reveal better properties in comparison to the heteroepitaxially deposited materials [1]. To obtain such substrates, thick GaN films or substrate crystals of an appropriate size are required. Three techniques are widely used in GaN bulk or thick film growth—high-pressure crystal growth [5], hydride vapor phase epitaxy [6], and vapor growth technique [7,8]. The latter one has been initially developed for growth of bulk or thick film silicon
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-0248(02)02208-X
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
carbide crystals and it has been applied to groupIII nitrides. Recently, the sublimation vapor growth technique also succeeded in growth of large size aluminum nitride (AlN) crystals. Growth of GaN single crystals is possible by reacting gallium vapors with ammonia. The sublimation sandwich technique (SST) is one of the techniques which can be used to grow a thick film. In SST, the source of Ga vapor and the substrate are separated by a clearance of several millimeters forming a growth cell. This growth cell is placed into a temperature-gradient zone to initiate material transport from the source to the substrate. To provide a supply of reactive nitrogen into the growth cell, the cell is purged by ammonia flow. A mixture of liquid gallium and GaN powder is usually used as the source of gallium. Experiments showed that use of pure Ga as the source material provided a maximum growth rate but resulted in poor temporal and spatial stability. In turn, use of pure GaN powder led to partial decomposition of the source material into liquid Ga and N2 vapor immediately after the beginning of the growth process. Some success has been made in control of the transport rate of Ga using metallic Ga diluted with metallic Bi (bismuth) [9]. Both sapphire and SiC were widely used as substrate materials. One of the advantages of this technique is the ability of controlling vapor transport in the growth cell by changing the distance between the source and the substrate and their temperature. Mass transfer from the source material to the substrate can be enhanced and a high growth rate (up to 1 mm/h) is achievable [10]. However, further advancements require a better understanding of the process. Since direct monitoring/in situ observation of the growth process is not feasible, physics-based theoretical modeling and simulation are used to understand the process and identify the relationship between the growth rate and transport kinetics. Few publications are available in modeling of thin and thick film growth of GaN in the open literature. Baranov et al. [11] developed the first model for GaN SST growth. Flow pattern in the furnace and growth rate were predicted. Numerical results were compared with the experimental data. Karpov et al. [12] developed a thermodynamic model for vapor GaN growth. In
15
their model, the sticking/evaporation coefficients of N2 and NH3 extracted from experiments were used for the kinetic effects. The growth rate and the solid phase composition were predicted and compared with available experimental data. In this paper, we will develop a mathematical model for SST GaN growth. The primary physical phenomena considered in the model are: (a) heat and mass in the furnace; (b) species transport between the source and the crystal; and (c) sticking kinetics, reaction and crystallization onto the substrate. The governing equations of Navier– Stokes equations, energy equation and concentration equation are solved. The kinetic parameters of GaN growth rate are obtained from fitting the experimental data in the open literature. To simplify the problem, a one-step reaction model is used here. A growth rate is proposed and compared with the growth rate of SiC.
2. Physical model of GaN growth using SST To analyze in detail the gas-flow dynamics and concentration distribution of the gaseous species in the furnace, a mathematical model is developed considering the conservation of momentum, energy, and chemical species together with necessary boundary conditions that account for heterogeneous chemical reactions. In the sublimation process, Ga metal sublimes at a high temperature, and the resulting gases travel to the cooler seed crystal where chemical reaction and crystallization take place. In an isothermal gas, the mechanisms of mass transport are only by Fickian diffusion and advection (Stefan flow), but if the temperature gradient exists, both thermal and solutal convection will be important. In addition, under conditions with temperature gradient, the thermal diffusion may also be important. A GaN growth model is developed considering gallium vapor and ammonia reaction. The growth cell consists of a pure Ga source and an SiC substrate, both forming a ‘‘sandwich’’ where vapor transport of the main species (Ga and NH3) occurs. The growth cell is placed in a temperature-gradient zone, so that a certain temperature difference DT between the source and the
16
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
substrate is obtained. This temperature difference serves as the driving force for Ga transport inside the growth cell and GaN crystallization on the substrate. Normally, the distance d between the source and the substrate is varied from 2 to 5 mm. Growth of GaN crystals is carried out at atmospheric pressure in the temperature range from 10001C to 13001C. The temperature difference between the substrate and the source is varied from 101C to 1001C [11]. The model based on computational fluid dynamics is capable of simulating multi-component fluid flow, gas/surface reaction, conjugate heat transfer, thermal radiation, and species. Based on the schematic of the growth system (see Fig. 1), we assume that the process is two-dimensional, steady state, and laminar [11,12]. The species transport equation is as following: C r~ u rC ¼ r rD rC þ aT rT ; ð1Þ T where r is the ammonia density, ! u is the velocity vector, C is the ammonia concentration, T is the temperature, and aT is the reactant-carrier gas
Fig. 1. Schematic of an SST growth system for GaN crystal.
(NH3/N2) thermal diffusion factor. Thermal diffusion may be important. Depending on the substrate temperature, deposition on the substrate may be thermodynamics-, diffusion- or kineticscontrolled. The GaN crystal depositing on a substrate is an internal boundary condition in the computational domain. The total net mass flux of species (GaN) normal to the surface of the substrate must be equal to the rate of production of species through chemical reaction: 3 GaðgÞ þ NH3 ðgÞ-GaNðsÞ þ H2 ðgÞ: 2
ð2Þ
The above reaction equation means that the creation of one mole of GaN vapor needs to consume one mole of ammonia. In reality, multiple reaction steps are taken from the reacting gas to final products, and reaction path is important for species and temperature predictions. However, it is not available in the open literature. Since NH3 and Ga vapors will not participate in the chemical reaction and deposit on the substrate completely, sticking coefficients are defined and measured in experiments. The relationship between the production of GaN vapor and the growth rate of the solid crystal can be generally expressed as:At x ¼ xsub ; CNH3 rT ~ n J ¼ D rCNH3 þ aT T rGaN;solid rGaN;solid G ¼ ; ð3Þ asNH3 asGa MGaN rGaN;vapor where J is the molar flux, D is the diffusion coefficient, M is the molecular weight, as is the stick coefficient, with asNH3 ¼ 104 and asGa ¼ 0:03 [11]. Arrhenius formulation, G¼ A expðEa =RTÞC b ; is used as the growth rate of GaN crystals. The coefficient, A; is the Arrhenius pre-exponential factor and Ea is the activation energy. Assuming b ¼ 0, values of A ¼ 6:598 107 mm/h and Ea ¼ 2:205 105 KJ/mol can be obtained from curve fitting of the experimental data of Baranov et al. [11] (see Fig. 2). The source temperature is pre-defined which is determined by input energy from the heater. The substrate temperature is assumed to be constant due to large thermal conductivity and thin substrate thickness, and its value is determined by an energy balance in and out of the control
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
volume surrounding the substrate. It is expected that the substrate temperature will be a strong function of distance between the source and substrate, emissivity of the substrate, and convection heat transfer on the top surface of the substrate. The reaction on the substrate will be treated as a species sink for NH3 vapor. A source term is added into the species transport equation as follows: S¼
JNH3 Ac ; Vc
ð4Þ
where JNH3 is the molar flux of the NH3 species, and Ac and Vc are the surface area and volume of the control volume, respectively.
3. Results and discussion The preliminary simulations are performed using NH3/N2 mixture in a quartz reactor that operates at atmospheric pressure. Nitrogen is the carrier gas. The quartz wall is kept at a temperature of 2001C, and the deposition surface is kept at about 11001C. The Grashof number (Gr ¼ bT gDTL3 =n2 ) in the furnace can be calculated based on bT ¼ 103 =K; g=9.81 m/s2, and DT=900 K. When the diameter of the reactor, L=3.2 cm, is used as the length scale, Gr=9.06 104 is obtained; while when the thickness of the growth cell, d=2–5 mm, is used, Gr=183–346 is obtained. This means that convection heat transfer in the growth cell is weak, and conduction and radiation will be the dominant modes of heat transfer. However, convection will prevent mass transfer of the ammonia vapor transported into the growth cell. The Reynolds number (Re ¼ Uin L=n) will be from 17 to 142 based on U ¼ 3225 cm/s, L ¼ 3:2 cm, and n=56.5 106 m2/s. From dimensional analysis, it is evident that the laminar flow assumption is reasonable. The baseline operating conditions for the calculation are chosen as: Tsub ¼ 10501C;
Tsource ¼ 11001C; 3
Cin ¼ 0:7 mol=cm ; d ¼ 5 mm:
Fig. 2. Growth rate of GaN crystal versus temperature for Tsource ¼ 11001C; DT ¼ 501C and d ¼ 5 mm.
17
TN ¼ Tin ¼ 2001C;
Constant thermophysical properties of materials at 10001C are needed in the simulation except of ammonia (see Table 1) [13]. The model can be easily improved if the parameter variations with
Table 1 Thermophysical properties of the gas components
3
Densityr(kg/m ) Specific heat Cp (J/kg K) Thermal conductivity k (W/m K) Diffusion coefficient Dk (m2/s) Viscosity n (m2/s) Pr a b
Ga
GaNa
NH3b
N2
5910 B372 29–38 NA NA NA
3166 (SiC) 1465 130 NA NA NA
0.3533 2613 63.8 103 6.83 106 in N2 56.5 106 0.817
0.3368 1167 64.7 103 9.21 106 in NH3 118.7 106 0.721
SiC density is used since the density of the thick GaN films is not available in open literatures. Property data for ammonia are at 3001C.
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B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
temperature and pressure are available. A 110 80 mesh with 20 50 grids in the growth cell has been used for all calculations presented in the paper after performing the grid independent study. Figs. 3 and 4 show the stream function, temperature, and concentration distributions in the furnace for incoming velocities of 1 and 10 cm/s, respectively. The other parameters are the same as those in the baseline case. The field variations occur mainly in the vicinity of the growth cell. When the incoming velocity is low, natural convection due to temperature difference is important, and a large cell is formed above the substrate due to the high temperature there. The flow pattern is distorted enough that a cell is also formed at the sidewalls. As the incoming velocity
increases, natural convection becomes less significant. A larger temperature gradient on the top of the substrate is obtained in Fig. 4. The higher incoming velocity has also pushed the temperature contours into the growth cell. The top surface of the source is kept at a constant temperature of 11001C in both calculations. A narrower recirculation is formed inside the gap between two surfaces. The recirculation may prevent the fresh gas from penetrating into the center region. The ammonia concentration is therefore lower at the center. It is observed in experiment that the structural, optical and electrical properties of GaN thin films depend on the NH3 flow rate during growth. The crystalline, luminescent and electrical properties of GaN films can be improved by optimizing the flow rate
Fig. 3. Stream function, temperature, and concentration contours for incoming velocity of 1 cm/s.
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
19
Fig. 4. Stream function, temperature, and concentration contours for incoming velocity of 10 cm/s.
of ammonia [9]. The Stafen flow from the source to the seed has been neglected here since the growth rate is less than 1 mm/h. It, however, will be important if the growth occurs in vacuum or low pressure. The equilibrium constant of reaction (2) can be calculated as PGa PNH3 K¼ : ð5Þ 3=2 PH2 Generally speaking, the reaction is taking place with PNH3 bPGa ; i.e. PTotal EPNH3 : With this condition, diffusion is the limiting step of the reaction. If we neglect the partial pressure of hydrogen, the growth rate can be expressed as
follows: Vg ¼ c
expðA B=TÞ DT ; d P2T T 1:2
ð6Þ
where c ¼ 1:27 106 kg2 K0:2 m4 =s5 ; A ¼ 39:2; and B ¼ 22; 776:7 K are used. The parameters A and B are determined by the reaction free energy directly.For the SiC sublimation growth, Ma et al. [14] proposed a similar formulation of the growth rate as Vg ¼ c
expðA B=TÞ DT PT 1:2 d
with A ¼ 30:77; B ¼ 67; 100K; 4:22 106 kg K0:2 m3 =s3 :
ð7Þ and
c¼
20
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
Since the SiC growth technique is a relatively matured technology, it would be beneficial to compare the growth rates of the two materials and find out the best operating conditions for GaN growth through comparison. The relationship between the growth rate and temperature under different temperature gradients for GaN and SiC growth is shown in Fig. 5. There is a linear relationship between the growth rate and temperature gradient for both GaN and SiC growth. Higher temperature gradient leads to higher growth rate. For GaN growth, the operation pressure is usually from 100 to 760 Torr while the operating temperature is from 10001C to 13001C. For SiC growth, the growth pressure is
usually from 10 to 100 Torr and temperature varies from 1900 to 25001C. From Eq. (6), we can see the role of pressure on the growth rate of GaN becomes smaller when pressure increases. On the other hand, the growth rate is almost a linear function of pressure for SiC growth.
4. Conclusions The GaN thick film growth process using the sublimation sandwich technique is simulated. Governing equations and boundary conditions are proposed and reaction parameters are obtained from experimental data. The stream function, temperature, and concentration fields are presented. When the incoming velocity is lower, natural convection is very important, which will determine the temperature on the top surface of the substrate. As the incoming velocity increases, the cell size above the substrate decreases and temperature gradient of the substrate cooling increases. The ammonia concentration changes mainly occur within the gap, and the concentration variation depends on the incoming velocity. When the total pressure is about the same as the ammonia pressure, diffusion resistance controls the final growth rate. We have also proposed a growth rate formula and compared it with SiC’s growth rate. It is concluded that a higher source temperature, lower pressure and larger temperature gradient will lead to higher growth rate for both GaN and SiC thick film growth.
Acknowledgements
Fig. 5. Relationship between the growth rate and source temperature under different temperature gradients (a) GaN growth rate and (b) SiC growth rate.
We would like to acknowledge the sponsorship of this research by NSF Career award (CTS9876198) and the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Office of Naval Research under Grant N00014-01-1-0716; Dr. Colin E. Wood project monitor. Special thanks are due to V. Prasad of Florida International University at miami, FL, M. Dudley of Stony Brook University, R. Schlesser and Z. Sitar of North Carolina State University, and D.F. Bliss and M. Callahan of
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
USAF research laboratory at Hanscom, MA for helpful discussions.
References [1] S. Nakamura, Solid State Commun. 102 (1997) 237. [2] G. Kamler, J. Zachara, S. Pldsiadlo, L. Adamowicz, W. Gebicki, J. Crystal Growth 212 (2000) 39. [3] Y. Gao, D.A. Gulino, R. Higgins, MRS Internet J. Nitride Semicond. 4S1 (1999) G3.53. [4] S.H. Cho, K. Hata, T. Maruyama, K. Akimoto, J. Cryst. Growth 173 (1997) 260. [5] S. Porowski, J. Crystal Growth 166 (1996) 583. [6] A. Usui, H. Sunakawa, A. Sakai, A.A. Yamaguchi, Jpn. J. Appl. Phys. Part 2-Lett. 36 (1997) L899. [7] M. Callahan, M. Harris, M. Suscavage, D. Bliss, J. Bailey, MRS Internet J. Nitride Semicond. 4 (10) (1999) 1.
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[8] C.M. Balkas, Z. Sitar, L. Bergman, I.K. Shmagin, J.F. Muth, R. Kolbas, R.J. Nemanich, R.F. Davis, J. Crystal Growth 208 (2000) 100. [9] Y. Watanabe, M. Sano, J. Crystal Growth 237 (2002) 1042. [10] E.N. Mokhov, Yu.A. Vodakov, Compound Semicond. 155 (1997) 177. [11] P.G. Baranov, E.N. Mokhov, Yu.N. Makarov, A.O. Ostroumov, M.G. Ramm, M.S. Ramm, V.V. Ratnikov, A.D. Roenkov, Yu.A. Vodakov, A.A. Wolfson, G.V. Saparin, S.Yu. Karpov, D.V. Zimina, H. Juergensen, MRS Internet J. Nitride Semicond. 3 (1998) 50. [12] S.Yu. Karpov, Yu.N. Makarov, M.S. Ramm, Phys. Status Solidi B 202 (1997) 202. [13] R. Poerschke, Data in Science and Technology: Semiconductors Group IV Elements and III–V Compounds, Springer, Berlin, 1991, p. 86. [14] R.-H. Ma, H. Zhang, V. Prasad, M. Dudley, Cryst. Growth Des. 2 (2002) 213.
Epitaxy growth kinetics of GaN films Bei Wu, Ronghui Ma, Hui Zhang* Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2300, USA
Abstract Group III nitrides, such as GaN, AlN and InGaN, have attracted a lot of attention due to the development of blue– green and ultraviolet light emitting diodes and lasers. A GaN crystal can be grown from the vapor phase by either evaporation of Gallium (Ga) metal or sublimation of GaN powder in ammonia (NH3) atmosphere at a temperaturecontrolled growth furnace. In this paper, an integrated GaN growth model using a sublimation growth model has been developed based on the conservation of momentum, mass, chemical species and energy together with necessary boundary conditions that account for heterogeneous chemical reactions both at the source and seed surfaces. For the growth rate, the effects of the gas-flow rate, source temperature, temperature difference, and the gap width of the growth cell on the growth process have been studied. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.05.Ea; 81.10.Aj; 81.10.Bk Keywords: A1. Computer simulation; A1. Growth models; A2. Growth from vapor; A2. Single crystal growth; A3. Chemical vapor deposition processes; B1. Nitrides
1. Introduction Group III nitrides, such as GaN, AlN and InGaN, have attracted a lot of attention in the past few years due to the development of blue– green light emitting diodes (LEDs) and lasers [1]. Blue–green LEDs can be used in full-color flat display panels with red–blue–green pixels for highdensity optical data storage and high-speed optical communications. GaN, with a direct energy gap of 3.46 eV, is one of the most interesting materials for optoelectronics. GaN thin films have been deposited onto substrates made of sapphire, silicon carbide, LiGaO2 and LiAlO2 [2]. Metalorganic *Corresponding author. Tel.: +1-631-632-8492; fax: +1631-632-8544. E-mail address: [email protected] (H. Zhang).
chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE) techniques are extensively used [3,4]. However, all substrates other than GaN will introduce stresses and dislocations due to lattice mismatch and different thermal expansion coefficients [5] during thin film growth. A dislocation density of 108–1010/cm2 is usually obtained in the GaN thin or thick films. It has been demonstrated that homoepitaxial GaN films reveal better properties in comparison to the heteroepitaxially deposited materials [1]. To obtain such substrates, thick GaN films or substrate crystals of an appropriate size are required. Three techniques are widely used in GaN bulk or thick film growth—high-pressure crystal growth [5], hydride vapor phase epitaxy [6], and vapor growth technique [7,8]. The latter one has been initially developed for growth of bulk or thick film silicon
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-0248(02)02208-X
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
carbide crystals and it has been applied to groupIII nitrides. Recently, the sublimation vapor growth technique also succeeded in growth of large size aluminum nitride (AlN) crystals. Growth of GaN single crystals is possible by reacting gallium vapors with ammonia. The sublimation sandwich technique (SST) is one of the techniques which can be used to grow a thick film. In SST, the source of Ga vapor and the substrate are separated by a clearance of several millimeters forming a growth cell. This growth cell is placed into a temperature-gradient zone to initiate material transport from the source to the substrate. To provide a supply of reactive nitrogen into the growth cell, the cell is purged by ammonia flow. A mixture of liquid gallium and GaN powder is usually used as the source of gallium. Experiments showed that use of pure Ga as the source material provided a maximum growth rate but resulted in poor temporal and spatial stability. In turn, use of pure GaN powder led to partial decomposition of the source material into liquid Ga and N2 vapor immediately after the beginning of the growth process. Some success has been made in control of the transport rate of Ga using metallic Ga diluted with metallic Bi (bismuth) [9]. Both sapphire and SiC were widely used as substrate materials. One of the advantages of this technique is the ability of controlling vapor transport in the growth cell by changing the distance between the source and the substrate and their temperature. Mass transfer from the source material to the substrate can be enhanced and a high growth rate (up to 1 mm/h) is achievable [10]. However, further advancements require a better understanding of the process. Since direct monitoring/in situ observation of the growth process is not feasible, physics-based theoretical modeling and simulation are used to understand the process and identify the relationship between the growth rate and transport kinetics. Few publications are available in modeling of thin and thick film growth of GaN in the open literature. Baranov et al. [11] developed the first model for GaN SST growth. Flow pattern in the furnace and growth rate were predicted. Numerical results were compared with the experimental data. Karpov et al. [12] developed a thermodynamic model for vapor GaN growth. In
15
their model, the sticking/evaporation coefficients of N2 and NH3 extracted from experiments were used for the kinetic effects. The growth rate and the solid phase composition were predicted and compared with available experimental data. In this paper, we will develop a mathematical model for SST GaN growth. The primary physical phenomena considered in the model are: (a) heat and mass in the furnace; (b) species transport between the source and the crystal; and (c) sticking kinetics, reaction and crystallization onto the substrate. The governing equations of Navier– Stokes equations, energy equation and concentration equation are solved. The kinetic parameters of GaN growth rate are obtained from fitting the experimental data in the open literature. To simplify the problem, a one-step reaction model is used here. A growth rate is proposed and compared with the growth rate of SiC.
2. Physical model of GaN growth using SST To analyze in detail the gas-flow dynamics and concentration distribution of the gaseous species in the furnace, a mathematical model is developed considering the conservation of momentum, energy, and chemical species together with necessary boundary conditions that account for heterogeneous chemical reactions. In the sublimation process, Ga metal sublimes at a high temperature, and the resulting gases travel to the cooler seed crystal where chemical reaction and crystallization take place. In an isothermal gas, the mechanisms of mass transport are only by Fickian diffusion and advection (Stefan flow), but if the temperature gradient exists, both thermal and solutal convection will be important. In addition, under conditions with temperature gradient, the thermal diffusion may also be important. A GaN growth model is developed considering gallium vapor and ammonia reaction. The growth cell consists of a pure Ga source and an SiC substrate, both forming a ‘‘sandwich’’ where vapor transport of the main species (Ga and NH3) occurs. The growth cell is placed in a temperature-gradient zone, so that a certain temperature difference DT between the source and the
16
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
substrate is obtained. This temperature difference serves as the driving force for Ga transport inside the growth cell and GaN crystallization on the substrate. Normally, the distance d between the source and the substrate is varied from 2 to 5 mm. Growth of GaN crystals is carried out at atmospheric pressure in the temperature range from 10001C to 13001C. The temperature difference between the substrate and the source is varied from 101C to 1001C [11]. The model based on computational fluid dynamics is capable of simulating multi-component fluid flow, gas/surface reaction, conjugate heat transfer, thermal radiation, and species. Based on the schematic of the growth system (see Fig. 1), we assume that the process is two-dimensional, steady state, and laminar [11,12]. The species transport equation is as following: C r~ u rC ¼ r rD rC þ aT rT ; ð1Þ T where r is the ammonia density, ! u is the velocity vector, C is the ammonia concentration, T is the temperature, and aT is the reactant-carrier gas
Fig. 1. Schematic of an SST growth system for GaN crystal.
(NH3/N2) thermal diffusion factor. Thermal diffusion may be important. Depending on the substrate temperature, deposition on the substrate may be thermodynamics-, diffusion- or kineticscontrolled. The GaN crystal depositing on a substrate is an internal boundary condition in the computational domain. The total net mass flux of species (GaN) normal to the surface of the substrate must be equal to the rate of production of species through chemical reaction: 3 GaðgÞ þ NH3 ðgÞ-GaNðsÞ þ H2 ðgÞ: 2
ð2Þ
The above reaction equation means that the creation of one mole of GaN vapor needs to consume one mole of ammonia. In reality, multiple reaction steps are taken from the reacting gas to final products, and reaction path is important for species and temperature predictions. However, it is not available in the open literature. Since NH3 and Ga vapors will not participate in the chemical reaction and deposit on the substrate completely, sticking coefficients are defined and measured in experiments. The relationship between the production of GaN vapor and the growth rate of the solid crystal can be generally expressed as:At x ¼ xsub ; CNH3 rT ~ n J ¼ D rCNH3 þ aT T rGaN;solid rGaN;solid G ¼ ; ð3Þ asNH3 asGa MGaN rGaN;vapor where J is the molar flux, D is the diffusion coefficient, M is the molecular weight, as is the stick coefficient, with asNH3 ¼ 104 and asGa ¼ 0:03 [11]. Arrhenius formulation, G¼ A expðEa =RTÞC b ; is used as the growth rate of GaN crystals. The coefficient, A; is the Arrhenius pre-exponential factor and Ea is the activation energy. Assuming b ¼ 0, values of A ¼ 6:598 107 mm/h and Ea ¼ 2:205 105 KJ/mol can be obtained from curve fitting of the experimental data of Baranov et al. [11] (see Fig. 2). The source temperature is pre-defined which is determined by input energy from the heater. The substrate temperature is assumed to be constant due to large thermal conductivity and thin substrate thickness, and its value is determined by an energy balance in and out of the control
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
volume surrounding the substrate. It is expected that the substrate temperature will be a strong function of distance between the source and substrate, emissivity of the substrate, and convection heat transfer on the top surface of the substrate. The reaction on the substrate will be treated as a species sink for NH3 vapor. A source term is added into the species transport equation as follows: S¼
JNH3 Ac ; Vc
ð4Þ
where JNH3 is the molar flux of the NH3 species, and Ac and Vc are the surface area and volume of the control volume, respectively.
3. Results and discussion The preliminary simulations are performed using NH3/N2 mixture in a quartz reactor that operates at atmospheric pressure. Nitrogen is the carrier gas. The quartz wall is kept at a temperature of 2001C, and the deposition surface is kept at about 11001C. The Grashof number (Gr ¼ bT gDTL3 =n2 ) in the furnace can be calculated based on bT ¼ 103 =K; g=9.81 m/s2, and DT=900 K. When the diameter of the reactor, L=3.2 cm, is used as the length scale, Gr=9.06 104 is obtained; while when the thickness of the growth cell, d=2–5 mm, is used, Gr=183–346 is obtained. This means that convection heat transfer in the growth cell is weak, and conduction and radiation will be the dominant modes of heat transfer. However, convection will prevent mass transfer of the ammonia vapor transported into the growth cell. The Reynolds number (Re ¼ Uin L=n) will be from 17 to 142 based on U ¼ 3225 cm/s, L ¼ 3:2 cm, and n=56.5 106 m2/s. From dimensional analysis, it is evident that the laminar flow assumption is reasonable. The baseline operating conditions for the calculation are chosen as: Tsub ¼ 10501C;
Tsource ¼ 11001C; 3
Cin ¼ 0:7 mol=cm ; d ¼ 5 mm:
Fig. 2. Growth rate of GaN crystal versus temperature for Tsource ¼ 11001C; DT ¼ 501C and d ¼ 5 mm.
17
TN ¼ Tin ¼ 2001C;
Constant thermophysical properties of materials at 10001C are needed in the simulation except of ammonia (see Table 1) [13]. The model can be easily improved if the parameter variations with
Table 1 Thermophysical properties of the gas components
3
Densityr(kg/m ) Specific heat Cp (J/kg K) Thermal conductivity k (W/m K) Diffusion coefficient Dk (m2/s) Viscosity n (m2/s) Pr a b
Ga
GaNa
NH3b
N2
5910 B372 29–38 NA NA NA
3166 (SiC) 1465 130 NA NA NA
0.3533 2613 63.8 103 6.83 106 in N2 56.5 106 0.817
0.3368 1167 64.7 103 9.21 106 in NH3 118.7 106 0.721
SiC density is used since the density of the thick GaN films is not available in open literatures. Property data for ammonia are at 3001C.
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B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
temperature and pressure are available. A 110 80 mesh with 20 50 grids in the growth cell has been used for all calculations presented in the paper after performing the grid independent study. Figs. 3 and 4 show the stream function, temperature, and concentration distributions in the furnace for incoming velocities of 1 and 10 cm/s, respectively. The other parameters are the same as those in the baseline case. The field variations occur mainly in the vicinity of the growth cell. When the incoming velocity is low, natural convection due to temperature difference is important, and a large cell is formed above the substrate due to the high temperature there. The flow pattern is distorted enough that a cell is also formed at the sidewalls. As the incoming velocity
increases, natural convection becomes less significant. A larger temperature gradient on the top of the substrate is obtained in Fig. 4. The higher incoming velocity has also pushed the temperature contours into the growth cell. The top surface of the source is kept at a constant temperature of 11001C in both calculations. A narrower recirculation is formed inside the gap between two surfaces. The recirculation may prevent the fresh gas from penetrating into the center region. The ammonia concentration is therefore lower at the center. It is observed in experiment that the structural, optical and electrical properties of GaN thin films depend on the NH3 flow rate during growth. The crystalline, luminescent and electrical properties of GaN films can be improved by optimizing the flow rate
Fig. 3. Stream function, temperature, and concentration contours for incoming velocity of 1 cm/s.
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
19
Fig. 4. Stream function, temperature, and concentration contours for incoming velocity of 10 cm/s.
of ammonia [9]. The Stafen flow from the source to the seed has been neglected here since the growth rate is less than 1 mm/h. It, however, will be important if the growth occurs in vacuum or low pressure. The equilibrium constant of reaction (2) can be calculated as PGa PNH3 K¼ : ð5Þ 3=2 PH2 Generally speaking, the reaction is taking place with PNH3 bPGa ; i.e. PTotal EPNH3 : With this condition, diffusion is the limiting step of the reaction. If we neglect the partial pressure of hydrogen, the growth rate can be expressed as
follows: Vg ¼ c
expðA B=TÞ DT ; d P2T T 1:2
ð6Þ
where c ¼ 1:27 106 kg2 K0:2 m4 =s5 ; A ¼ 39:2; and B ¼ 22; 776:7 K are used. The parameters A and B are determined by the reaction free energy directly.For the SiC sublimation growth, Ma et al. [14] proposed a similar formulation of the growth rate as Vg ¼ c
expðA B=TÞ DT PT 1:2 d
with A ¼ 30:77; B ¼ 67; 100K; 4:22 106 kg K0:2 m3 =s3 :
ð7Þ and
c¼
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Since the SiC growth technique is a relatively matured technology, it would be beneficial to compare the growth rates of the two materials and find out the best operating conditions for GaN growth through comparison. The relationship between the growth rate and temperature under different temperature gradients for GaN and SiC growth is shown in Fig. 5. There is a linear relationship between the growth rate and temperature gradient for both GaN and SiC growth. Higher temperature gradient leads to higher growth rate. For GaN growth, the operation pressure is usually from 100 to 760 Torr while the operating temperature is from 10001C to 13001C. For SiC growth, the growth pressure is
usually from 10 to 100 Torr and temperature varies from 1900 to 25001C. From Eq. (6), we can see the role of pressure on the growth rate of GaN becomes smaller when pressure increases. On the other hand, the growth rate is almost a linear function of pressure for SiC growth.
4. Conclusions The GaN thick film growth process using the sublimation sandwich technique is simulated. Governing equations and boundary conditions are proposed and reaction parameters are obtained from experimental data. The stream function, temperature, and concentration fields are presented. When the incoming velocity is lower, natural convection is very important, which will determine the temperature on the top surface of the substrate. As the incoming velocity increases, the cell size above the substrate decreases and temperature gradient of the substrate cooling increases. The ammonia concentration changes mainly occur within the gap, and the concentration variation depends on the incoming velocity. When the total pressure is about the same as the ammonia pressure, diffusion resistance controls the final growth rate. We have also proposed a growth rate formula and compared it with SiC’s growth rate. It is concluded that a higher source temperature, lower pressure and larger temperature gradient will lead to higher growth rate for both GaN and SiC thick film growth.
Acknowledgements
Fig. 5. Relationship between the growth rate and source temperature under different temperature gradients (a) GaN growth rate and (b) SiC growth rate.
We would like to acknowledge the sponsorship of this research by NSF Career award (CTS9876198) and the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Office of Naval Research under Grant N00014-01-1-0716; Dr. Colin E. Wood project monitor. Special thanks are due to V. Prasad of Florida International University at miami, FL, M. Dudley of Stony Brook University, R. Schlesser and Z. Sitar of North Carolina State University, and D.F. Bliss and M. Callahan of
B. Wu et al. / Journal of Crystal Growth 250 (2003) 14–21
USAF research laboratory at Hanscom, MA for helpful discussions.
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