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Hydrodynamic description of epitaxial film growth in a horizontal reactor
,. . . . . . . .
ELSEVIER
CRYSTAL GROW T H
Journal of Crystal Growth 170 (1997) 61-65
Hydrodynamic description of epitaxial film growth in a horizontal reactor Yoshichika Mizuno *, Shin-ichiro Uekusa, Hideyuki Okabe School of Science and Technology, Meiji Unit,ersiO', 1-1-1 Higashimita, Tama-ku, Kawasaki 214, Japan
Abstract
Growth rate uniformity of epitaxial films is an important prerequisite for fabricating semiconductor devices. The correlation between the film thickness distributions and the vorticity distributions in the flow field is investigated in detail. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity around the substrates and the vorticity gradient on the substrates are calculated from the velocity distribution rather than by solving the species equations. The substrate temperature is 1000 K, the total pressures are 2 and 20 Ton" and the Reynolds numbers are 1, 10 and 100. The thicknesses of the films deposited by low-pressure chemical vapor deposition are measured. As a result, it can be shown that the deposited film thickness distributions agree with the vorticity gradient distributions. This suggests that vorticity has an important influence on film deposition.
1. Introduction
Growth rate uniformity of epitaxial films is an important prerequisite for fabricating semiconductor devices. These devices, such as lasers and light emitting diodes, are most often grown by vapor phase epitaxy (VPE). Metalorganic vapor phase epitaxy (MOVPE) of I I I - V compound semiconductors is an efficient method for crystal growth and the formation of steep heterojunctions. MOVPE is usually conducted under atmospheric pressure. Recently low-pressure MOVPE is often taken for growth uniformity [1-3] and crystal growth over large substrate area. Vapor phase growth is traditionally divided into two major categories of physical vapor deposition
Co~esponding author. Fax: + 81 449347909.
(PVD) and chemical vapor deposition (CVD). The growth of compound semiconductors by MOVPE is a special form of CVD. The term CVD will be used in its broadest sense to characterize vapor phase crystal growth processes. CVD is based on the transport of gaseous reagents which undergo a series of gas phase and surface reactions leading to the incorporation of film constituents and the formation of volatile byproducts [4]. The transport of gaseous reagents to the film surface leading to the film growth is governed by complex fluid flows and heat transfer. The fundamental equations for film growth in a flow field are those for conservation of mass, momentum, energy and species. Many models [1,5] have been proposed in order to formulate these equations. The buoyancy effect, recirculation or thermal diffusion has been considered in these models. A number of species equations for gaseous reagents have been solved. Considerable effort is necessary to
0022-0248/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved Pll S0022-0248(96)00566-0
62
K Mizuno et al. / Journal o[' Ct3,stal Growth 170 (I 997) 61-65
solve the species equations, determine the initial and boundary conditions and obtain the physical properties of the gaseous reagents using these models. Some authors [6-8] have tried to simplify the complex numerical calculations by introducing a stagnant or boundary layer or by assuming isothermal flow. However, these simpler models are not sufficient to express the physical and chemical phenomena in the flow field. In this paper, a simpler model is proposed. The gradient of vorticity is introduced as a new physical quantity that can be used to study the influence of the flow field on the growth uniformity. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity distributions in the flow field and the vorticity gradient distributions on the substrate are calculated from the velocity distributions. The calculated vorticity gradient distributions are compared with those obtained experimentally for films deposited on Si substrates by the CVD method.
2. Numerical calculations and experimental procedure The complex flow field around the substrate is calculated [9,10]. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity ( ( = au/ay-ou/ax) is introduced as a simple model. Here, x is the distance from the inlet in the flow direction and y is the distance from the substrate in the direction perpendicular to the flow. u and e are the velocities in the x and y directions, respectively. The dimensionless vorticity equation derived from the momentum equations is analogous to the species equation when neglecting the thermal diffusion terms. The vorticity distributions in the flow field and the vorticity gradient distributions on the substrate are calculated from the velocity distributions. The equation used to calculate the vorticity gradient is analogous to that derived from the species equations used to calculate the growth rate. The flow is assumed to be Newtonian, two-dimensional, steady, incompressible and laminar. Under these assumptions, the conservation equations are solved using a t h r e e - d i m e n s i o n a l analytic system (Fujitsu/c~-flow). The forward finite difference
method is used to calculate the velocity, temperature and pressure distributions of the flow past the thin substrates. Substrates with lengths of 0.2 or 0.4 m are aligned horizontally on the centerline of a channel 1 m long and 0.2 m high; the thickness of the substrates is 0.002 m; the lattice number is 71 X 39 and the way of division in a lattice is via equal space in the flow direction and in a geometric ratio in the direction perpendicular to the flow; the truncated error is 10 6; the CPU time is about 100 rain when the physical properties are constant and 200 rain when they are temperature dependent. The boundary conditions are u = u = 0 at the wall and on the substrates; the velocity distribution is uniform or parabolic at the inlet and du/dx = 0 at the outlet; the temperature is 900 K at the wall and 1000 K on the substrate; the temperature distribution is uniform at the inlet and adiabatic at the outlet; viscosity, heat capacity, thermal conductivity and density are 7/= 1.05 X 10 -5 X ( T / 3 0 0 ) °% (Pa s), C r = 2.10 X 103 X ( T / 3 0 0 ) °3s (J kg ~ K ~), A = 2 . 5 0 X l0 ~X ( T / 3 0 0 ) ls ( W m i K l), and p = 1.82X 10 s X ( T / 3 0 0 ) i (kg m 3) at 2 Torr and 1.82X 10 2X ( T / 3 0 0 ) (kg m 3) at 20 Torr, respectively. When the physical properties are constant, the temperature T is 900 K in these equations. In the experiments, a modified infrared gold image furnace of the cold-wall type was used for the film deposition [11]. It could be used for a rapid heating and cooling and had a digital programmable temperature controller. Thin SiN films were deposited on Si substrates by the low-pressure CVD method. The substrates (0.02 and 0.04 m long, 0.05 m wide and 0.0005 m thick) were put into a chamber (0.09 m in diameter and 0.21 m long). NH 3 and SizH 6 were used as the reaction gases. The substrates, which were aligned horizontally, were heated to 1000 K using the radiant heat of infrared lamps. The wall temperature was about 900 K. The total pressure was 2 or 20 Torr and the Reynolds number (Re) was about 1 or 40. The film thickness was measured on the centerline of the substrate in the flow direction using an ellipsometer.
3. Results and discussions Fig. 1 shows the calculated vorticity distribution around the substrates with a length of 0.4 m at
63
K Mizuno et al. / Journal of Co,stal Growth 170 (1997) 61-65
Re = l, l0 and 100. The contour lines of vorticity are shown in Fig. l a. The lines corresponding to a vortex appear at the c o m e r of the inlet because of the uniform velocity distribution at the inlet and the zero velocity at the wall. The lines disappear when the velocity distribution at the inlet is parabolical, though the vorticity distribution around the substrate for uniform flow is almost the same as that for parabolic flow. At Re = 1, the lines diverge abruptly in front of the leading edge of the substrate and converge abruptly at the back of the rear edge of the substrate. Therefore, the vorticity in the y direction changes gradually over both sides of the substrate. The vorticity is constant in the flow direction over the central part of the substrate. The lines are bilaterally symmetrical here and elsewhere and become asymmetrical with increasing Re number. They diverge slightly in front of the leading edge of the substrate and continue to diverge gradually in the flow direction. Then, the vorticity in the y direction changes greatly near the leading edge and gradually near the rear edge. The vorticity gradient on the substrate is shown
in Fig. lb. The ordinate represents the vorticity normalized by its minimum value on the substrate. The vorticity gradient increases greatly near both edges of the substrate. At Re = 1, the vorticity gradient is bilaterally symmetrical and is constant over the central part of the substrate. It increases near both edges of the substrate, as indicated by the convergence of the contour lines in those regions. The vorticity gradient near the leading edge is equal to that near the rear edge. The vorticity gradient near the leading edge becomes larger than that near the rear edge with increasing Re number. The vorticity gradient is of the order o f l 0 s 1 at R e = l and 102 s I at R e = 1 0 0 . Fig. 2 shows the vorticity distribution around two substrates aligned horizontally on the centerline of the channel at 2 Torr and Re = 1. The spaces s between the substrates which are 0.2 m long are 0.025, 0.05 and 0.1 m. In Fig. 2a, the contour lines are bilaterally symmetrical. At s = 0.025 and 0.05, the contour lines are asymmetrical for each substrate. Each substrate influences the contour lines for the
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Fig. 1. Vorticity distribution around the substrate with a length of 0.4 m. (a) Contour lines of vorticity, (b) vorticity gradient on the substrate.
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E Mizuno et al./Journal of Co'stal Growth 170 (1997) 61-65
other substrate. On the other hand, the substrates have no influence on each other at s = 0.1 m. The contour lines are symmetrical for each substrate. In Fig. 2b, the vorticity gradient near the rear edge of the first substrate and the leading edge of the second substrate is smaller than that near the other edges at s = 0.025. It increases as the space between the substrates increases. At s = 0.1, the vorticity distribution and vorticity gradient are symmetrical for each substrate and are almost the same as the distribution for one substrate with a length of 0.4 m. The contour lines, the vorticity distributions and the vorticity gradients at Re = 10 and 100 show almost the same tendencies as those at Re = 1. Fig. 3 shows the film thickness distributions obtained experimentally at Re = 1 and 40. The ordinate represents the film thickness normalized by the minimum film thickness deposited on the substrates. At Re = 1, the film thickness increases near both edges and is over the central part of the substrate. The distribution is almost the same as the vorticity gradient distribution shown in Fig. lb. The vorticity gradient shows a m i n i m u m over the central part of the substrate, whereas the film thickness increases over that region. It is considered that recirculation due to the buoyancy effect increases the film thickness. Therefore, the conservation equations were calculated numerically taking into account the gravity terms included in the momentum equation and the temperature dependence of the physical properties. However, the film thickness did not increase. To obtain an accurate value, It is necessary to calculate the three-dimensional flow around a rectangular substrate in a circular tube reactor. At Re = 40, the film thickness has a maximum value near the leading edge of the substrate. It decreases abruptly over the first half and reaches a constant value asymptotically over the second half. It reaches a m i n i m u m value and then increases slightly near the rear edge. The normalized film thickness distribution agrees with
65
the normalized vorticity gradient distribution shown in Fig. lb. The same tendency is observed for two substrates.
4. S u m m a r y To simplify the complex transport phenomena in MOVPE, the vorticity distributions around the substrates and the vorticity gradient distributions on the substrates were calculated because the vorticity equation is analogous to the species equations. It was shown that the vorticity gradient distributions agree with the film thickness distributions at a wall temperature of 900 K and a substrate temperature of 1000 K. A strong correlation between the vorticity distribution and the film thickness distribution was shown.
References [1] R.J. Field, J. Crystal Growth 97 (1989) 739. [2] P.B. Chinoy and S.K. Ghandhi,J. Crystal Growth 108 (1991) 105. [3] K.F. Jensen, D.I. Fotiadis and T.J. Mountziaris, J. Crystal Growth 107 (1991) 1. [4] K.F. Jensen, Handbookof Crystal Growth, Vol. 3b (Elsevier, Amsterdam, 1994) p. 541. [5] P.B. Chinoy, D.A. Kaminski and S.K. Ghandhi, J. Electrochem. Soc. 138 (1991) 1452. [6] C. Van Opdorp and M.R. Leys, J. Crystal Growth 84 (1987) 288. [7] J. Ouazzani,K.C. Chiu and F. Rosenberger,J. Crystal Growth 91 (1988) 497. [8] J. Van De Ven, G.M.J. Rutten, M.J. Raaijmakers and L.J. Giling, J. Crystal Growth 76 (1986) 352. [9] Y. Mizuno and S. Uekusa, in: Proc. 3rd Int. Symp. on Process Physics and Modeling in SemiconductorTechnology, Hawaii, USA, 1993 (ECS, Pennington,1993) p. 369. [10] Y. Mizunoand S. Uekusa, Mater. Sci. Eng. B 35 (1995) 156. [11] S. Uekusa and Y. Mizuno, Jpn. J. Appl. Phys. 33 (1994) 2679.
ELSEVIER
CRYSTAL GROW T H
Journal of Crystal Growth 170 (1997) 61-65
Hydrodynamic description of epitaxial film growth in a horizontal reactor Yoshichika Mizuno *, Shin-ichiro Uekusa, Hideyuki Okabe School of Science and Technology, Meiji Unit,ersiO', 1-1-1 Higashimita, Tama-ku, Kawasaki 214, Japan
Abstract
Growth rate uniformity of epitaxial films is an important prerequisite for fabricating semiconductor devices. The correlation between the film thickness distributions and the vorticity distributions in the flow field is investigated in detail. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity around the substrates and the vorticity gradient on the substrates are calculated from the velocity distribution rather than by solving the species equations. The substrate temperature is 1000 K, the total pressures are 2 and 20 Ton" and the Reynolds numbers are 1, 10 and 100. The thicknesses of the films deposited by low-pressure chemical vapor deposition are measured. As a result, it can be shown that the deposited film thickness distributions agree with the vorticity gradient distributions. This suggests that vorticity has an important influence on film deposition.
1. Introduction
Growth rate uniformity of epitaxial films is an important prerequisite for fabricating semiconductor devices. These devices, such as lasers and light emitting diodes, are most often grown by vapor phase epitaxy (VPE). Metalorganic vapor phase epitaxy (MOVPE) of I I I - V compound semiconductors is an efficient method for crystal growth and the formation of steep heterojunctions. MOVPE is usually conducted under atmospheric pressure. Recently low-pressure MOVPE is often taken for growth uniformity [1-3] and crystal growth over large substrate area. Vapor phase growth is traditionally divided into two major categories of physical vapor deposition
Co~esponding author. Fax: + 81 449347909.
(PVD) and chemical vapor deposition (CVD). The growth of compound semiconductors by MOVPE is a special form of CVD. The term CVD will be used in its broadest sense to characterize vapor phase crystal growth processes. CVD is based on the transport of gaseous reagents which undergo a series of gas phase and surface reactions leading to the incorporation of film constituents and the formation of volatile byproducts [4]. The transport of gaseous reagents to the film surface leading to the film growth is governed by complex fluid flows and heat transfer. The fundamental equations for film growth in a flow field are those for conservation of mass, momentum, energy and species. Many models [1,5] have been proposed in order to formulate these equations. The buoyancy effect, recirculation or thermal diffusion has been considered in these models. A number of species equations for gaseous reagents have been solved. Considerable effort is necessary to
0022-0248/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved Pll S0022-0248(96)00566-0
62
K Mizuno et al. / Journal o[' Ct3,stal Growth 170 (I 997) 61-65
solve the species equations, determine the initial and boundary conditions and obtain the physical properties of the gaseous reagents using these models. Some authors [6-8] have tried to simplify the complex numerical calculations by introducing a stagnant or boundary layer or by assuming isothermal flow. However, these simpler models are not sufficient to express the physical and chemical phenomena in the flow field. In this paper, a simpler model is proposed. The gradient of vorticity is introduced as a new physical quantity that can be used to study the influence of the flow field on the growth uniformity. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity distributions in the flow field and the vorticity gradient distributions on the substrate are calculated from the velocity distributions. The calculated vorticity gradient distributions are compared with those obtained experimentally for films deposited on Si substrates by the CVD method.
2. Numerical calculations and experimental procedure The complex flow field around the substrate is calculated [9,10]. Equations for conservation of mass, momentum and energy are solved numerically. The vorticity ( ( = au/ay-ou/ax) is introduced as a simple model. Here, x is the distance from the inlet in the flow direction and y is the distance from the substrate in the direction perpendicular to the flow. u and e are the velocities in the x and y directions, respectively. The dimensionless vorticity equation derived from the momentum equations is analogous to the species equation when neglecting the thermal diffusion terms. The vorticity distributions in the flow field and the vorticity gradient distributions on the substrate are calculated from the velocity distributions. The equation used to calculate the vorticity gradient is analogous to that derived from the species equations used to calculate the growth rate. The flow is assumed to be Newtonian, two-dimensional, steady, incompressible and laminar. Under these assumptions, the conservation equations are solved using a t h r e e - d i m e n s i o n a l analytic system (Fujitsu/c~-flow). The forward finite difference
method is used to calculate the velocity, temperature and pressure distributions of the flow past the thin substrates. Substrates with lengths of 0.2 or 0.4 m are aligned horizontally on the centerline of a channel 1 m long and 0.2 m high; the thickness of the substrates is 0.002 m; the lattice number is 71 X 39 and the way of division in a lattice is via equal space in the flow direction and in a geometric ratio in the direction perpendicular to the flow; the truncated error is 10 6; the CPU time is about 100 rain when the physical properties are constant and 200 rain when they are temperature dependent. The boundary conditions are u = u = 0 at the wall and on the substrates; the velocity distribution is uniform or parabolic at the inlet and du/dx = 0 at the outlet; the temperature is 900 K at the wall and 1000 K on the substrate; the temperature distribution is uniform at the inlet and adiabatic at the outlet; viscosity, heat capacity, thermal conductivity and density are 7/= 1.05 X 10 -5 X ( T / 3 0 0 ) °% (Pa s), C r = 2.10 X 103 X ( T / 3 0 0 ) °3s (J kg ~ K ~), A = 2 . 5 0 X l0 ~X ( T / 3 0 0 ) ls ( W m i K l), and p = 1.82X 10 s X ( T / 3 0 0 ) i (kg m 3) at 2 Torr and 1.82X 10 2X ( T / 3 0 0 ) (kg m 3) at 20 Torr, respectively. When the physical properties are constant, the temperature T is 900 K in these equations. In the experiments, a modified infrared gold image furnace of the cold-wall type was used for the film deposition [11]. It could be used for a rapid heating and cooling and had a digital programmable temperature controller. Thin SiN films were deposited on Si substrates by the low-pressure CVD method. The substrates (0.02 and 0.04 m long, 0.05 m wide and 0.0005 m thick) were put into a chamber (0.09 m in diameter and 0.21 m long). NH 3 and SizH 6 were used as the reaction gases. The substrates, which were aligned horizontally, were heated to 1000 K using the radiant heat of infrared lamps. The wall temperature was about 900 K. The total pressure was 2 or 20 Torr and the Reynolds number (Re) was about 1 or 40. The film thickness was measured on the centerline of the substrate in the flow direction using an ellipsometer.
3. Results and discussions Fig. 1 shows the calculated vorticity distribution around the substrates with a length of 0.4 m at
63
K Mizuno et al. / Journal of Co,stal Growth 170 (1997) 61-65
Re = l, l0 and 100. The contour lines of vorticity are shown in Fig. l a. The lines corresponding to a vortex appear at the c o m e r of the inlet because of the uniform velocity distribution at the inlet and the zero velocity at the wall. The lines disappear when the velocity distribution at the inlet is parabolical, though the vorticity distribution around the substrate for uniform flow is almost the same as that for parabolic flow. At Re = 1, the lines diverge abruptly in front of the leading edge of the substrate and converge abruptly at the back of the rear edge of the substrate. Therefore, the vorticity in the y direction changes gradually over both sides of the substrate. The vorticity is constant in the flow direction over the central part of the substrate. The lines are bilaterally symmetrical here and elsewhere and become asymmetrical with increasing Re number. They diverge slightly in front of the leading edge of the substrate and continue to diverge gradually in the flow direction. Then, the vorticity in the y direction changes greatly near the leading edge and gradually near the rear edge. The vorticity gradient on the substrate is shown
in Fig. lb. The ordinate represents the vorticity normalized by its minimum value on the substrate. The vorticity gradient increases greatly near both edges of the substrate. At Re = 1, the vorticity gradient is bilaterally symmetrical and is constant over the central part of the substrate. It increases near both edges of the substrate, as indicated by the convergence of the contour lines in those regions. The vorticity gradient near the leading edge is equal to that near the rear edge. The vorticity gradient near the leading edge becomes larger than that near the rear edge with increasing Re number. The vorticity gradient is of the order o f l 0 s 1 at R e = l and 102 s I at R e = 1 0 0 . Fig. 2 shows the vorticity distribution around two substrates aligned horizontally on the centerline of the channel at 2 Torr and Re = 1. The spaces s between the substrates which are 0.2 m long are 0.025, 0.05 and 0.1 m. In Fig. 2a, the contour lines are bilaterally symmetrical. At s = 0.025 and 0.05, the contour lines are asymmetrical for each substrate. Each substrate influences the contour lines for the
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Fig. 1. Vorticity distribution around the substrate with a length of 0.4 m. (a) Contour lines of vorticity, (b) vorticity gradient on the substrate.
"gO'O = ~ pue O# 'I = aN le Salealsqns oql uo suo.tmquls!p ssou~Io!ql tUl!J lelUatU!aadx~ "£ "~!d a l e a a s q n ~ aqa j o a ~ p a ~ u w e a I oR1 m o a j o a u e a ~ G
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E Mizuno et al./Journal of Co'stal Growth 170 (1997) 61-65
other substrate. On the other hand, the substrates have no influence on each other at s = 0.1 m. The contour lines are symmetrical for each substrate. In Fig. 2b, the vorticity gradient near the rear edge of the first substrate and the leading edge of the second substrate is smaller than that near the other edges at s = 0.025. It increases as the space between the substrates increases. At s = 0.1, the vorticity distribution and vorticity gradient are symmetrical for each substrate and are almost the same as the distribution for one substrate with a length of 0.4 m. The contour lines, the vorticity distributions and the vorticity gradients at Re = 10 and 100 show almost the same tendencies as those at Re = 1. Fig. 3 shows the film thickness distributions obtained experimentally at Re = 1 and 40. The ordinate represents the film thickness normalized by the minimum film thickness deposited on the substrates. At Re = 1, the film thickness increases near both edges and is over the central part of the substrate. The distribution is almost the same as the vorticity gradient distribution shown in Fig. lb. The vorticity gradient shows a m i n i m u m over the central part of the substrate, whereas the film thickness increases over that region. It is considered that recirculation due to the buoyancy effect increases the film thickness. Therefore, the conservation equations were calculated numerically taking into account the gravity terms included in the momentum equation and the temperature dependence of the physical properties. However, the film thickness did not increase. To obtain an accurate value, It is necessary to calculate the three-dimensional flow around a rectangular substrate in a circular tube reactor. At Re = 40, the film thickness has a maximum value near the leading edge of the substrate. It decreases abruptly over the first half and reaches a constant value asymptotically over the second half. It reaches a m i n i m u m value and then increases slightly near the rear edge. The normalized film thickness distribution agrees with
65
the normalized vorticity gradient distribution shown in Fig. lb. The same tendency is observed for two substrates.
4. S u m m a r y To simplify the complex transport phenomena in MOVPE, the vorticity distributions around the substrates and the vorticity gradient distributions on the substrates were calculated because the vorticity equation is analogous to the species equations. It was shown that the vorticity gradient distributions agree with the film thickness distributions at a wall temperature of 900 K and a substrate temperature of 1000 K. A strong correlation between the vorticity distribution and the film thickness distribution was shown.
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