Growth Kinetics and Thermal Stress in the Sublimation Growth of Silicon Carbide Ronghui
Ma,†
Hui
Zhang,*,†
Vish
Prasad,§
and Michael
Dudley‡
Department of Mechanical Engineering, Department of Materials Science and Engineering, State University of New York at Stony Brook, Stony Brook, New York 11794-2300, and College of Engineering, Florida International University, Miami, Florida 33174 Received December 11, 2001;
CRYSTAL GROWTH & DESIGN 2002 VOL. 2, NO. 3 213-220
Revised Manuscript Received February 13, 2002
ABSTRACT: The productivity and quality of SiC bulk crystal grown from vapor phase depend strongly on the temperature distribution in a SiC growth chamber. An analytical formulation is proposed to correlate the growth rate with process parameters such as pressure, temperature, and temperature gradient. A growth kinetic model is also developed to predict the growth rate and examine the transport effects on the growth rate and dislocation formation. Simulation and analytical results show that the growth rate increases when the growth temperature increases, argon pressure decreases, and/or the temperature gradient between the source and seed increases. An anisotropic thermoelastic stress model is proposed to study the influence of thermal stress on dislocation density. The method to attach the seed is observed to play an important role in stress distribution in an as-grown silicon carbide ingot. 1. Introduction The growth temperature, temperature distribution, and pressure in a growth chamber can significantly affect the local growth rate along the growth interface. A nonuniform temperature distribution can also induce thermal stresses in the crystals, leading to generation and propagation of dislocations and/or other defects. It is, therefore, important to understand the underlying interrelationship between the SiC crystal growth and thermal conditions. However, this understanding is difficult to achieve solely from the experiments since insitu measurement of temperature and pressure in a growth chamber remains a difficult task. Severe constraints exist on measurement in a SiC system primarily because of high temperature, harsh environment, and opaque graphite wall.1 It is even more challenging to determine only experimentally the right combination of the process conditions to grow a high quality singly crystal. As a result, numerical modeling has been widely used in recent years to simulate the SiC growth process.2 Various models with different levels of complexity have been developed to calculate the induction heating power, heat and mass transfer, and growth kinetics in a SiC physical vapor transport (PVT) growth system.3-6 Ra¨back et al. (1999) proposed an analytical model to estimate the magnitude of the growth rate.7 Thermal stress distributions in an as-grown crystal were predicted by Ha et al. (2000),8 Karpov et al. (1999),9 and Mu¨ller et al. (2000),10 and the dislocation density has been related to thermal stress distribution in SiC ingots by Karpov et al. (1999),9 Mu¨ller et al. (2000),10 and Hoffmann et al. (1999).11 In this paper, a one-dimensional growth model that incorporates thermodynamic analyses, Stefan flow or * To whom correspondence should be addressed: Tel: (631) 6328492. Fax: (631) 632-8544. E-mail:
[email protected]. † Department of Mechanical Engineering, State University of New York at Stony Brook. ‡ Department of Materials Science and Engineering, State University of New York at Stony Brook. § Florida International University.
diffusion transport, and growth kinetics is developed to predict the growth rate in the (0001) direction. On the basis of this model, an analytical formulation is developed to relate the growth rate with the pressure, temperature, and temperature gradient in the growth chamber. The variation of growth rate during the growth run is studied using the growth model. Furthermore, an anisotropic thermal elastic stress model is developed to study the stress distribution in the SiC ingot for different growth stages. 2. SiC PVT Growth System and Growth Kinetics During the SiC growth process, the SiC charge decomposes into several vapor species, e.g., SiC, SiC2, Si2C, and Si.12 When these vapor species flow upward to the top of the crucible, a supersaturated condition is formed in the vicinity of the ingot surface, driving the vapor species to deposit on the seed. To describe the growth process properly, a physical growth model is developed including the chemical reaction, mass transport of vapor species, and kinetics of dissociation and deposition. Because of the very slow growth of the SiC crystal (less than a few millimeters per hour), a local thermodynamic equilibrium state can be easily assumed while calculating the vapor concentration in the growth chamber. The equilibrium vapor pressures of the major vapor species are obtained from Lilov (1993). According to his data, the partial pressures of SiC2 and Si are much higher than those of other species, such as Si2C and SiC, at the commonly used growth temperatures, ranging from 2500 and 3100 K. Therefore, the reaction
SiC2 + Si S 2SiC
(1)
is considered the rate-controlling reaction in this temperature range. SiC2 is taken as the rate-determining species at temperatures below 2900 K and Si as the controlling species at temperatures above 2900 K. The
10.1021/cg015572p CCC: $22.00 © 2002 American Chemical Society Published on Web 03/19/2002
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chemical reaction between the vapor species and the graphite wall is neglected here. The supersaturation of the vapor species is the driving force for deposition of the vapor species at the vapor/ seed surface, as is the supercooling for the decomposition of the SiC charge at the charge/vapor interface. Assuming the mole fluxes of SiC2/Si at the gas/seed surface (Js) and at the gas/charge surface (Jc) proportional to the supersaturation/supercooling of SiC2/Si,13 the fluxes Js and Jc can be described as:
Js )
/ (s)] R[PSiC2/Si (s) - PSiC 2/Si
x2πMSiC /SiRTs
and
2
Jc )
/ (c) - PSiC2/Si (c)] [PSiC 2/Si
x2πMSiC2/SiRTc
(2)
respectively. Here PSiC2/Si (s) and PSiC2/Si (c) are the surface pressures of SiC2/Si at the growth interface and / (s) and at the SiC charge surface, respectively, PSiC 2/Si / PSiC2/Si (c) are the equilibrium pressures at the growth interface and at the charge surface, respectively, R is the sticking coefficient, M is the molar mass, and Ts and Tc are the temperatures at the seed and charge surfaces, respectively. The mass transport rate of SiC2 or Si by Stefan flow can be obtained using a one-dimensional mass transfer equation,13
JSiC2/Si ) (PD/RTavgL)ln{(P - 2PSiC2/Si (s))/ (P - 2PSiC2/Si (c))}/2 (3) where P is the total vapor pressure, Tavg ) (Tc + Ts)/2 is the average temperature of the charge and seed, L is the distance between the source material and seed, and D is the diffusion coefficient, given by Kaldis and Piechotka (1994).
D ) 0.05 × 10-4
( )( ) Tavg 273
1.8
105 P
(4)
If the argon pressure is much higher than that of other vapor species in the chamber, Pargon . PSi and PSiC2, the mass transport will be diffusion dominated and Stefan flow can be neglected. The mass transport rate can then be obtained using the binary diffusion equation13
JSiC2/Si )
PSiC2/Si (c) - PSiC2/Si (s) LRTavg/D
(5)
Thus far, we have three equations and five unknown variables PSiC2/Si (c), PSiC2/Si (s), JSiC2/Si, Js, and Jc. In a closed SiC growth chamber, the mole flux of vapor species through sublimation (Jc), mass transport (JSiC2/Si), and deposition (Js) should be identical, and hence the number of unknown variables can be reduced to three. The two kinetics equations for the dissociation and deposition and one mass transport equation can thus be solved for PSiC2/Si (c), PSiC2/Si (s), and JSiC2/Si. If the mass transport is dominated by Stefan flow, eqs 2 and 3 need to be solved iteratively. If diffusion prevails in mass transport, a simple expression can be obtained from eqs 2 and 5 to predict the mole flux JSiC2/Si
JSiC2/Si ) / / (c) - PSiC (s) PSiC 2/Si 2/Si
x2πRMSiC2/SiTs/R + LRTavg/D + x2πRMSiC2/SiTc
(6)
The growth rate G can then be obtained as
G)
2MSiC J FSiC SiC
(7)
A coefficient 2 in eq 7 is introduced because one SiC2 molecular produces two SiC molecules. The above growth model can account for the major physical mechanisms involved in the SiC growth and can predict the growth rate as a function of the process parameters such as temperature distribution, argon pressure, and growth chamber geometry (the distance between the charge material and seed, L, and the chamber diameter d). However, the interrelationship among these parameters is not straightforward. An analytical model is therefore developed here to correlate the growth rate with the major growth parameters explicitly. If the vapor transport from the source to the seed is controlled by diffusion, we can derive an analytical expression from eq 6. The summation of x2πRMSiC2/SiTc and x2πRMSiC2/SiTs in eq 6 represents the kinetic resistance at the charge and seed surfaces, denoted by Rk, and LRT/D represents the resistance of mass diffusion, denoted by Rd. Since the temperature difference between the charge surface and the seed surface is usually less than 100 K, Rk can be approximated by 2x2πRMSiC2/SiTavg without sacrificing accuracy. An order-of-magnitude analysis can then be easily carried out to examine the significance of growth kinetics and mass transport in the temperature range of 2000 to 3000 K and pressure from 1 Pa to 100 Torr. Listed in Table 1 are the values of Rk and Rd, which show that the kinetic resistance is much smaller than the diffusion resistance for the argon pressure of 100 Pa. Only for very low argon pressures (around 1 Pa) the magnitude of Rk becomes of the same order of that of Rd (not shown here). It is therefore evident that the kinetic resistance at the source or seed plays a minor role at medium and high pressures, and it is reasonable to neglect Rk in eq 6 for the pressure range considered here. The sticking coefficient R is taken as unity following many researchers.7 However, at the vacuum growth condition, R may be much smaller than unity, and the effect of surface kinetics may be as important as that of the transport process. Neglecting the surface kinetics may then lead to an incorrect conclusion regarding to the growth rate. The equilibrium vapor pressures of SiC2 and Si as provided by Lilov (1993) can be approximated by the following formula
PSiC2/Si ) exp(a - b/T)
(8)
where a ) 30.77 and b ) 6.71 × 104 are obtained using a numerical regression. Substituting eqs 4, 6, and 8 into eq 7, and using Taylor series to simplify the resulting formulation, the growth rate can be written as:
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axial components of the unit vector normal to the surface s, respectively. Since SiC crystal is a thermoelastic anisotropic body, the stress-strain relation is given by the following formulation
Table 1. Values of Rk and Rd at 100 Pa ave temp (K)
Rk
Rd
2000 2200 2400 2600 2800 3000
141.56 148.47 155.07 161.41 167.50 173.38
23058.96 21366.11 19929.42 18693.26 17617.21 16671.19
G)
c exp(a - b/T)(∆T/L) T1.2Pargon
(9)
where constant c ) 4.22 × 10-6. This formulation clearly illustrates that the controlling parameters are T, P, and ∆T/L, and the growth rate is an ascending function of temperature gradient and temperature and descending function of argon pressure. Prediction of the growth rate requires accurate representation of the temperature distribution in a growth chamber. The thermal field in the growth chamber is calculated using a two-dimensional model that consists of induction heating, combined heat transfer, sublimation of source material, vapor transport, and crystal deposition in a specific geometry of the growth chamber. The details of this mathematical model and numerical scheme can be found in Chen et al. (2001) and Ma et al. (2000). 3. Thermal Stress in As-Grown Crystal Thermal stress in an as-grown crystal is introduced by the temperature gradient present in the ingot. Although a large axial temperature gradient in the growth chamber favors a high growth rate, higher thermal stress in the ingot can induce more dislocations. The temperature gradient must, therefore, be controlled carefully to obtain a high quality crystal at a reasonable growth rate. In addition, the process to attach the seed to the crucible lid may have a significant influence on thermal stress distribution. A thermoelastic stress analysis can be performed in the axisymmetric crystal using a displacement-based model. Since the gravity is negligible compared with the thermal stress, it is not considered in the equilibrium equations. An axisymmetric thermoelastic body satisfies the following equilibrium equations:15
∂σrz σφφ 1 ∂ (rσrr) + )0 r ∂r ∂z r
(10)
1 ∂ ∂ (rσ ) + σzz ) 0 r ∂r rz ∂z
(11)
where σrr is normal stress in the radial direction, σzz is normal stress in the axial direction, σφφ is normal stress in the azimuthal direction, and σrz is shear stress. Integrating the above equations in a control volume, we can obtain
∫s (σrrnr + σrznz)ds - ∫v
σφφ dv ) 0 r
∫s (σrznr + σzznz)ds ) 0
(12) (13)
where s and v are the surface and volume of the control volume, respectively, and nr and nz are the radial and
()(
σrr c11 c12 σφφ σzz ) c13 σrz 0
c12 c22 c23 0
c13 c23 c33 0
0 0 0 c44
)(
rr - β(T - Tref) φφ - β(T - Tref) zz - β(T - Tref) rz
)
(14)
where i,j is the strain rate, β is the thermal expansion coefficient, and ci,j is the elastic coefficient of SiC single crystal. The strains are calculated as
zz )
∂u ∂v v ∂v ∂u , rr ) , φφ ) , rz ) + (15) ∂z ∂r r ∂z ∂r
where u and v are the displacement in the axial and radial directions, respectively. The final equations are obtained by substituting eqs 14 and 15 into eqs 12 and 13, and the two equations are solved iteratively for u and v. In the present study, the shape of the growth surface is simplified as a flat surface and the ingot geometry as a cylinder. The growing surface is a free moving boundary. The sidewall of the ingot is assumed to be able to move freely by neglecting interaction between the growing crystal and wall of the crucible. In reality, polycrystalline SiC ring exists between the single crystal and the crucible wall. Since polycrystalline SiC is a relatively soft material in the porous form, the assumption of free moving sidewall may not introduce significant errors. Furthermore, an axisymmetric boundary condition is applied to the axial center. The boundary conditions for the central line, bottom growth interface, and side boundaries can be described as:
v ) 0,
∂u ) 0 at r ) 0 ∂r
(16)
σrr ) 0, σrz ) 0 at r ) R
(17)
σzz ) 0, σrz ) 0 at z ) 0
(18)
The boundary condition at the top bonding side of the ingot depends strongly on how a seed is attached to the crucible lid, e.g., mechanical mounting or rigid glue mounting. Here the variation in the seed attachment is considered through three kinds of boundary conditions: (i) free to move in both radial and axial directions, (ii) fixed in the axial direction and free to move in the radial direction, and (iii) totally rigid, although the true situation may lie somewhere between the free moving case and the totally rigid case. One major challenge in performing thermoelastic stress calculation is the unavailability of the properties of SiC single crystal at high temperatures; both the elastic constants and thermal expansion coefficient are strong functions of temperature. Here we use the elastic constants of 6H SiC at room temperature given by Kamitani et al. (1997)16 and thermal expansion coefficient from Li and Bradt (1986).17 The finite volume method (FVM) is employed here to calculate the thermoelastic stress in the as-grown crystal due to its simplicity and effectiveness as well as
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Figure 1. Schematic of a SiC PVT growth system used in simulation.
the convenience to be incorporated into the finite volume solver6,14 for temperature distribution. The second-order central difference scheme is employed to descretize the integral eqs 12 and 13 and an orthogonal mesh of 80 × 40 is used in the simulation. The strong implicit solver (SIP) method is used to solve the linear matrix. 4. Results and Discussion The growth rate, interface shape, and dislocation density vary significantly during the growth process due to the increasing size of the ingot. For example, with the increase of the ingot length, the geometry of the growth cell, e.g., the distance between the growth interface and the charge surface, is varied, resulting in the change of radiative heat transfer, which, in turn, affects the temperature distribution in the growth cell. The powder material is also reported to undergo appreciable changes such as incongruent graphitization and recrystallization.18 To develop a better understanding of the interaction between the thermal field and the crystal growth, the temperature distribution, growth rate, and stress distribution are calculated at several ingot lengths using the integrated, quasi-steady model as described above. The geometry of the charge material is considered unchanged and the detailed decomposition of the charge material is not considered here because of the limited information available on the decomposition process. The growth pressure is assumed to be constant during the crystal growth. Figure 1 shows the schematic of a SiC growth chamber used in this study. The original gap between the charge material and growth interface is taken as 3 cm. The input power is considered as 4988 W and the constant growth pressure as 40 Torr. Figure 2 presents the thermal fields in the crucible for ingot lengths of 0.25, 0.5, 0.75, and 1.0 cm, and Figure 3a shows the variation of ingot/gas interface temperature vs ingot length. While no significant change in the overall thermal profiles has been observed, the growth interface temperature is illustrated to increase substantially as the ingot grows into a larger size, especially at the initial growth stage. In addition to the interface temperature, the variation of maximum growth rate and temperature gradient between the charge surface and the growth interface are presented in Figure 3b. Both the temperature gradient and growth rate decrease at the initial stage of crystal growth, and begin to increase when the ingot length exceeds 0.75 cm. Since the growth temperature remains almost constant during the growth process, this figure clearly shows the strong dependence of growth
Figure 2. Thermal distribution for crystal growth: (a) ingot length of 0.25 cm; (b) ingot length of 0.50 cm; (c) ingot length of 0.75 cm; and (d) ingot length of 1.0 cm.
rate on the temperature gradient. It also demonstrates that increasing the crystal length can possibly enlarge the temperature gradient between the growth interface and the charge, which increases the growth rate at the late stage of the growth. The temperature and growth rate at the growth interface are presented in Figure 4a,b. Since the decomposition of the source material is not considered here, the nonuniformity of the growth rate is caused to a large part by the radial temperature gradient in the ingot, which partially comes from the heat loss from the upper window at the top of the crucible. Thus, the diameter and shape of the upper window must be designed carefully to obtain a flat interface shape of the as-grown crystal. The dominant parameters such as the seed temperature, argon pressure, and temperature gradient between the charge and seed surface are further studied using the following growth models: a one-dimensional growth model considering Stefan flow (denoted as Stefan flow model), growth model only accounting for diffusion transport (denoted as diffusion model), and an analytical formulation. Figure 5 presents the growth rate vs the seed temperature for temperature gradient of 12 K/cm and a pressure of 40 Torr obtained using different models. It shows that increasing the seed temperature will increase the growth rate. The effects of growth temperature on the growth rate can be summarized as follows: first, high-temperature facilitates the chemical reactions in the SiC charge, induces high concentration of vapor species, and results in a large mass flux from the source to the seed. Second, since the diffusion coefficient is an
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Crystal Growth & Design, Vol. 2, No. 3, 2002 217
Figure 3. (a) Variation of maximum temperature at the crystal/vapor interface; (b) variation of maximum growth rate and temperature gradient from charge surface to ingot surface with the crystal length for argon pressure of 40 Torr.
Figure 4. (a) Variation of the temperature and (b) growth rate along the radial direction with different crystal length for argon pressure of 40 Torr.
Figure 5. The growth rates for argon pressure of 40 Torr, powder to seed distance of 2.5 cm, and temperature difference form the powder to seed surface of 30 K.
Figure 6. The growth rate vs temperature gradient for argon pressure of 40 Torr, powder to seed distance of 2.5 cm, and powder surface temperature of 2600 K.
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Figure 7. The growth rate vs pressure for temperature gradient of 12 K/cm, powder to seed distance of 2.5 cm, powder surface temperature of 2800 K.
ascending function of temperature, a high temperature in the chamber can accelerate the mass transport of vapor species. This is important for SiC growth at the medium or high pressure where the growth rate is dominated by mass transport. Finally, the kinetic resistance of sublimation and deposition has a T-0.5 relationship with temperature, and therefore a high temperature will reduce the kinetic resistance. In addition to growth temperature, the temperature gradient from the source material to seed is also a major factor to the growth rate; the growth rate increases almost linearly with the temperature gradient (Figure 6).
Ma et al.
Argon pressure is another important factor because it can affect the mass transport mechanism. Figure 7 shows the dependence of the growth rate on pressure for a seed temperature of 2800 K and temperature gradient of 12 K/cm. We will focus on the results calculated by the Stefan flow model first, and the results calculated from other models will be discussed later. It can be seen that the growth rate remains nearly constant at the low pressure, while it is inversely proportional to the pressure at medium or high pressures. This trend agrees well with the experimental observation. Analytically, at medium or high argon pressures, the mass flux transported by mass diffusion or Stefan flow is low, and hence the mass transport becomes the controlling mechanism for the crystal growth. The 1/p relationship agrees well with the molecular diffusion formulation. When the argon pressure is lower than that of the vapor species, mass transport is dominated by convection and/or molecular transport, leading to large mass flux, and the surface kinetics becomes the controlling parameter. In this case, the growth rate is mainly related to the growth temperature. Solutions from Stefan flow, diffusion, and analytical models are also compared. The analytical model agrees well with the diffusion model, illustrating that the simplification we made is reasonable. However, the diffusion model only agrees well with the Stefan model for the growth condition at low temperature (5000 Pa), and small temperature gradient (
