- Email: [email protected]
Different macroscopic approaches to the modelling of the sublimation growth of SiC single crystals
MATERIALS SCIERCE & ENCINEERIRG
B
Materials Science and Engineering B46 (1997) 30X-312
Different macroscopic approaches to the modelling of the sublimation growth of Sic single crystals M. Pons a, E. Blanquet a,*, J.M. Dedulle b, R. Madar b, C. Bernard a a Laboratoire b Laboratoire
de Thermod}xamique et Ph,wicochimie M~ialhwgiques, I/AIR CNRS/INPG/ UJF 5614ENSEEG, Institut National Polytechnique de Grenoble, BP 75, 38402 Saint Martin D’Hkes, France de MatPrinux et de GEnie Physique, iJA+fR CNRSjINPG 5628-ENSPG, Instirur Natiortal Poiytechnique de Grenoble, 38402 Saint Martin D’Hkcs, France
BP 46,
Abstract Different macroscopicmodelssuch as thermodynamics,heat transfer and masstransport have beenapplied to the simulation of the growth of singleSic crystals preparedaccordingto the so-called‘modified Lely method’. Thermodynamicmodellinghas beenusedto determinethe most important reactive gaseousand solid speciespresentunder equilibriumconditions. Heat transfer modelling (including induction heating, radiation with multireflection, convection and conduction) has been performed to calculatethe actual temperaturesinside the reactor. Different temperaturefields have been obtained dependingon the level of complexity of the thermal modelling.Finally, masstransport modellingprovided the chemicalfieldsof the processand calculated depositionrates which were found to be closeto the experimentalones.It appearsthat the solid Sic surfaceshapeafter growth dependson the temperaturegradient existing along the seed.Q 1997Elsevier ScienceS.A. Kepords:
Silicon carbide; Single crystals; Sublimation growth; Macroscopic models
1. Introduction Due to the starting applications in electronic and optoelectronic devices involving high temperature, in-
tense radiation, high frequency and high power, intensive work has been done to develop the technology to obtain good quality bulk Sic single crystals [l-3]. Growth of single Sic crystals is performed inside a graphic crucible at temperatures higher than 2300 K and pressures lower than 5 x lo3 Pa [3], using the ‘modified Lely method’ [4,5]. It involves sublimation from a hot source consisting of SIC powder, transport through an inert gas and condensation of SIC single crystals on a sink colder than the source. Numerous studies have been conducted for crystal growth by the physical vapour transport (PVT) inside horizontal and vertical enclosures [6- 131. They have revealed the complexity of this process. Particularly, Hofmann et al. [l l] have already proposed some modelling aspects of the Sic growth. The present study is included in a larger modelling work [13] where different routes with differ-
* Corresponding author. 09X-5107/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5107(96)01995-2
ent levels of complexity have been investigated: firstly, in an uncoupled way, the heat transfer by induction heating, the thermodynamic equilibrium and finally the linking of thermochemical databases and equilibrium with heat and mass transport calculations (LTCE/transport concept) 1141. The purpose of the present study is to extend the previous modelling, propose further refinements in heat transfer evaluation and examine the effects of the temperature gradients existing inside the crucible on the growth of bulk single Sic crystals.
2. Experimental
setup
The modelling of the sublimation growth of Sic bulk crystals has been undertaken on the basis of the experimental configuration of the reactor described in Ref. [3]. Growth experiments have been conducted in an RF heated graphite crucible filled with an inert atmosphere of argon with pressure ranging from 131 to 2630 Pa (l-20 Torr) (Fig, 1). The range of parameters was chosen to get a growth rate from 1 to 4 mm h-l which allows the preparation of good quality (estimated from the FWHM values of rocking curves) crystals.
M. Pans et al. /Materials
Scieme
rind Engineering
B46 (1997)
309
308-312
Hole in graphite foam for pyrometer measurement
SK single crystalline
Induction
Water-cooled
seed
heating coil
quartz tubw
Hole in graphite foam for pyrometer measurement
Fig. 1. Scheme of the experimental configuration for the growth of Sic single crystals by physical vapor transport.
3. Heat transfer modelling A complete simulation using the finite element software package Flux-Expert [15] has consisted of two successive and iterative steps: first the distribution of the electromagnetic field has been computed through the full Maxwell equations; then, from the computed distributions of the generated heat and Joule losses, the temperature distribution was determined with heat transfer modelling. The equations and physical data that have been used for the different models can be found in previous papers [12-141. Most of the thermophysical properties have to be assumed, due to the lack of reliable material data, in the considered temperature and pressure ranges. The temperature distribution obtained with a given induced current density of J = 2.4 x 10’ A m - ’ and a frequency of 125 kHz is shown in Fig. 2. The temperatures of the top and bottom of the graphite crucible on the symmetry axis are equal to 2370 and 2470 K, respectively, which are comparable to the two temperatures (around 2390 + 40 and 2500 i 20 K) experimentally measured by pyrometers. In Fig. 3, the experimental and calculated temperatures at the top and the bottom of the crucible for different tested induced current densities (2.7 x IO7 A mV2 (labelled point A), 2.4 x lo7 A m - ’ (point B) and 2.1 x lo7 A -’ (point C)). The temperature of the seed and Ewder surface at x = 0 are equal to 3275 K, 3406 K for the point A, 2920 K, 3020 K for the point B, 2667 K, 2753 K for the point C. The temperature field, inside the crucible, has been more extensively studied for the point B case, which is the most representative of the current experiments. To simulate the whole process, our previous calculations were performed using an apparent thermal argon conductivity which reproduces the fourth power temperature dependence of radiation. We reduced the computational domain by assuming a temperature of
3000 K along the SIC powder from the results of the first approach. Thus, we refined the knowledge of local temperature gradients which are the driven factors of the deposition rate. Despite some numerical uncertainties in the very next of the seed corner, related to the calculations of view factors, it lead to a more precise temperature field shown on Fig. 4: the temperature is constant on the source (boundary conditions) whereas there is a 100 K temperature difference along the seed
379 521:793 697.207 872.621 1048.04 1223.45 1398.86 1574.28 1749.69 1925.1 2100.52 2275.93 2451.35 2626.76 2802.17 2977.59 3153 3328.42 3503.83 3679.24 3854.66
1 = 3i6
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
= = = = = = = = = = = = = = = = = = = =
u
Fig. 2. Temperature distribution inside the growth chamber obtained from the hear transfer model.
310
M. Pons et
al. /Materials
Science ad
ml0
2500
g 2 E 2400 g & 2300 m 5 % e 3B 2200 g t-, 2100
2000 2200
I
1 2300
I 2.500
2400 Temperalure
of the hofkxn
I 2GOO
I 2700
of the crucible
(K)
2800
Fig. 3. Comparison between experimental (pyrometric measurements) and calculated temperatures at the top and the bottom of the crucible.
instead of the value of 10 K for the previous calculations, and the temperature is lower by more than 100 K on the graphite. In this configuration, the temperature difference between source and seed is about 100 K on the symmetry axis, as obtained with the simplified model. Then, the next modelling step, mass transport linked with thermodynamic calculations was performed starting from the temperature fields inside the crucible calculated with different levels of model complexity.
4. Thermodynamic/mass
transport modelling
A classical thermodynamic
analysis has been per-
Engineering
846 (1997)
305-312
formed on the Si-C-Ar system to provide the composition at equilibrium in the gas phase above silicon carbide in the temperature range 2700-3500 K using the thermodynamic data that came from the recent work of P. Rocabois et al. [ 161. The results indicated that in the considered temperature and pressure range, the free sublimation of Sic is not congruent (enrichment of the condensed phase in carbon) and that the gas phase is essentially composed from Si,(g), Sic,(g) and Si,C(g), which has been confirmed by earlier mass spectrometric data [17], According to these results, in the application of the coupled mass transport-LTCE (local thermochemical equilibrium) model which has been described in detail in previous papers [12--141, the six more important gaseous species (Sil, Si,, Si,, Si,C, SiCJ diluted with argon were considered for both homogeneous and heterogeneous (sublimation and deposition surface) equilibrium. The boundary conditions are: at the source, the mass fractions are specified by the thermodynamic heterogeneous equilibrium on Sic powder at the calculated temperatures, on the walls (graphite surface), negligible deposition was assumed owing to the small area of the walls when compared to the total deposition area, on the sink (deposition surface), the mass fluxes are linked with thermodynamic heterogeneous equilibrium. Calculations are performed in stationary regime. Fig. 5 shows the calculated mass fractions along the centerline of the six considered gaseous species inside the crucible at a total pressure of 30 Torr (3990 Pa), for the temperature field shown on Fig. 4. The most important species are found to be SiC2, Si,C and Si,. For the species Sic, and S&C there is a continuous depletion from the source towards the deposition surface, whereas Si, species concentration increases and remains almost constant. On the seed, the deposition rate in Sic calculated from the surface molar fluxes (essentially the surface fluxes of the 3 predominant species, SIC?, S&C and Si,) is about 2.3 mm T ml F 2759.56 1~2761.97 2~2775.83 3=2789.69 4=2.303.55 5~2617.41 6~2831.27 7 =2845.13 0 = 2858.99 9-2872.85 10=2886.71 11 -2900.57 12 -2914.43 13=2928.29 14z2942.15 15~2956.02 16=2969,88 17-2983.74 16=2997X max=3000
SIC powder Fig. 4. Temperature field inside the crucible for the tested conditions of point B (Fig. 3).
M. Pans et al. f Matevials
Science
and Erlgineering
B46 (1997)
308-312
,~s~
311
.
Graphite
“Lb!
~-0j0iW~
~~.~.~.~.~.~.“~.~.~.,+.~....
. E
0.20
I
(IO K gradient
I
on seed)
3v
Thickness
(mm)
Experimental
profile
Fig. 6. Radial variation of the Sic surface shape after growth or Sic thickness (experimental average value for 1 h run time and calculated from the temperature fields obtained with the refined and simplified (italic) heat transfer models). 0
0.002
0.004
0.006
0.00s
0.01
Distance from the source (m)
perature field in the very small (compared to the whole experimental setup) area around the seed on the magnitude and shape of the growth rate.
Fig. 5. Profile of Si,, Si,, Si,, C,, Sic,, S&C mass fractions along the centerline, for a total pressure of around 3990 Pa.
5. Conclusions h- ‘. This value is in good agreement with the experimental value (average value for 1 h run time) which is 1.55 mm h-r in these conditions. In Fig. 6, the radial variation of the calculated (for the two temperatures distributions) and experimental Sic surface after growth together with the calculated temperatures are reported. We can notice that the experimental and calculated features are comparable all along the seed and graphite headland, whereas there is a discrepancy in the neighbourhood of the step. Two effects contribute to the magnitude of the deposition rate: mass transfer and local thermodynamic equilibrium. The first one by itself would lead to an increase of the total flux near the step. The second one by itself is mainly controlled by temperature. As temperature increases, we should expect a decrease in the deposition rate. To separate each contribution, we can compare the theoretical growth features obtained with the two temperature distributions. With the simplified heat transfer model (where the temperature difference along the seed is around 10 K), the transport effect is predominant and we observe an increase of growth rate at the corner, whereas with the refined model (where the temperature difference along the seed is around 100 K), the growth rate decreases and its shape is closer to the experimental one. However, this last simulation fails in the graphite part for the magnitude of the calculated growth rate. This can be explained by the lower values (100 K difference) of the temperature. It appears that the magnitude of the growth rate is controlled by the ‘vertical’ temperature (from source to sink) gradient while its shape is controlled by the temperature gradient along the sink. These preliminary results seem to show the influence of the actual tem-
Three different macroscopic models have been proposed to simulate the growth of silicon carbide single crystals. The thermodynamic approach showed that the most important species involved in the process are Si,, S&C and SiC2. From the heat transfer, it was possible to visualize the temperature field in such a complex process where sensors cannot be used. Finally, the approach linking mass transport modelling and thermodynamic equilibrium has proved its potential in describing the macroscopic growth of Sic when compared to experimental results.
References Ill M.L. Locatelli, S. Gamal, J. P/Qz. III France, PI P.A. Ivanov, V.E. Chelnokov, Semiconduct.
3 (1993) 1101. Sci. Technol., 7
(1992) 863. I. Garcon, A. Rouault, M. Anikin, C. Jaussaud, R. Madar, Mat. sci. Eng., B29 (1995) 90. [41 Y.M. Tairov, V.F. Tsvetkov, J. Qyst. Grontlz, 43 (1978) 209. [51 G. Ziegler, P. Lanig, D. Theis, C. Weyrich, IEEE Trans. Elec. [31
Deu., ED-30
(1983)
PI D.W. Greenwell,
277.
B.L. Markham, F. Rosenberger, J. Cq~st. Growth, 51 (1981) 413. 171 B.L. Markham, D.W. Greenwell, F. Rosenberger, J. Ciyst. Gl.~wth, 51 (1981) 426. 181 W.M.B. Duval, J. CVD, 2 (1994) 188. PI W.M.B. Duval, J. CT/D, 2 (1994) 289. I101 SK. Lilov, I.Y. Yanchev, Mater. Sci. Eng., 821 (1993) 83. t111 D. Hofmann, M. Heinze, A. Winnacker, F. Durst, L. Kadinski, P. Kaufmann, Y. Makarov, M. Schafer, J. Cryst. Growth, 146 (1995) 214. 1121I. Garcon, Contribution a l’etude de la croissance de monocristaux de carbure de silicium par la methode de Lely modifiee, PIID Thesis, Nat. Polytech. Inst. Grenoble, France, October 1995.
312
M. Pans et al. /Marerials
Science
[13] M. Pans, E. Blanquet, J.M. Dedulle, C, Bernard, R. Madar, L Electrochem. Sot., submitted for publication. 1141 M. Pons, C. Bernard, R. Madar, Sur$ Coni. Technol., 61 (1993) 274. 1151 Flux-Expert Sofware Package Guide, DT2I, 38240 Meylan,
and Engineering
B46 (1997)
308-312
France, 1995. [16] P. Rocabois, C. Chatillon, C. Bernard and F. Genet, High Temp. High Pres., (1995) in press. [17] J. Drowart, G. De Maria, M.G. Inghram, L C!ze/?z.Phys., 29 (1958) 1015.
B
Materials Science and Engineering B46 (1997) 30X-312
Different macroscopic approaches to the modelling of the sublimation growth of Sic single crystals M. Pons a, E. Blanquet a,*, J.M. Dedulle b, R. Madar b, C. Bernard a a Laboratoire b Laboratoire
de Thermod}xamique et Ph,wicochimie M~ialhwgiques, I/AIR CNRS/INPG/ UJF 5614ENSEEG, Institut National Polytechnique de Grenoble, BP 75, 38402 Saint Martin D’Hkes, France de MatPrinux et de GEnie Physique, iJA+fR CNRSjINPG 5628-ENSPG, Instirur Natiortal Poiytechnique de Grenoble, 38402 Saint Martin D’Hkcs, France
BP 46,
Abstract Different macroscopicmodelssuch as thermodynamics,heat transfer and masstransport have beenapplied to the simulation of the growth of singleSic crystals preparedaccordingto the so-called‘modified Lely method’. Thermodynamicmodellinghas beenusedto determinethe most important reactive gaseousand solid speciespresentunder equilibriumconditions. Heat transfer modelling (including induction heating, radiation with multireflection, convection and conduction) has been performed to calculatethe actual temperaturesinside the reactor. Different temperaturefields have been obtained dependingon the level of complexity of the thermal modelling.Finally, masstransport modellingprovided the chemicalfieldsof the processand calculated depositionrates which were found to be closeto the experimentalones.It appearsthat the solid Sic surfaceshapeafter growth dependson the temperaturegradient existing along the seed.Q 1997Elsevier ScienceS.A. Kepords:
Silicon carbide; Single crystals; Sublimation growth; Macroscopic models
1. Introduction Due to the starting applications in electronic and optoelectronic devices involving high temperature, in-
tense radiation, high frequency and high power, intensive work has been done to develop the technology to obtain good quality bulk Sic single crystals [l-3]. Growth of single Sic crystals is performed inside a graphic crucible at temperatures higher than 2300 K and pressures lower than 5 x lo3 Pa [3], using the ‘modified Lely method’ [4,5]. It involves sublimation from a hot source consisting of SIC powder, transport through an inert gas and condensation of SIC single crystals on a sink colder than the source. Numerous studies have been conducted for crystal growth by the physical vapour transport (PVT) inside horizontal and vertical enclosures [6- 131. They have revealed the complexity of this process. Particularly, Hofmann et al. [l l] have already proposed some modelling aspects of the Sic growth. The present study is included in a larger modelling work [13] where different routes with differ-
* Corresponding author. 09X-5107/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5107(96)01995-2
ent levels of complexity have been investigated: firstly, in an uncoupled way, the heat transfer by induction heating, the thermodynamic equilibrium and finally the linking of thermochemical databases and equilibrium with heat and mass transport calculations (LTCE/transport concept) 1141. The purpose of the present study is to extend the previous modelling, propose further refinements in heat transfer evaluation and examine the effects of the temperature gradients existing inside the crucible on the growth of bulk single Sic crystals.
2. Experimental
setup
The modelling of the sublimation growth of Sic bulk crystals has been undertaken on the basis of the experimental configuration of the reactor described in Ref. [3]. Growth experiments have been conducted in an RF heated graphite crucible filled with an inert atmosphere of argon with pressure ranging from 131 to 2630 Pa (l-20 Torr) (Fig, 1). The range of parameters was chosen to get a growth rate from 1 to 4 mm h-l which allows the preparation of good quality (estimated from the FWHM values of rocking curves) crystals.
M. Pans et al. /Materials
Scieme
rind Engineering
B46 (1997)
309
308-312
Hole in graphite foam for pyrometer measurement
SK single crystalline
Induction
Water-cooled
seed
heating coil
quartz tubw
Hole in graphite foam for pyrometer measurement
Fig. 1. Scheme of the experimental configuration for the growth of Sic single crystals by physical vapor transport.
3. Heat transfer modelling A complete simulation using the finite element software package Flux-Expert [15] has consisted of two successive and iterative steps: first the distribution of the electromagnetic field has been computed through the full Maxwell equations; then, from the computed distributions of the generated heat and Joule losses, the temperature distribution was determined with heat transfer modelling. The equations and physical data that have been used for the different models can be found in previous papers [12-141. Most of the thermophysical properties have to be assumed, due to the lack of reliable material data, in the considered temperature and pressure ranges. The temperature distribution obtained with a given induced current density of J = 2.4 x 10’ A m - ’ and a frequency of 125 kHz is shown in Fig. 2. The temperatures of the top and bottom of the graphite crucible on the symmetry axis are equal to 2370 and 2470 K, respectively, which are comparable to the two temperatures (around 2390 + 40 and 2500 i 20 K) experimentally measured by pyrometers. In Fig. 3, the experimental and calculated temperatures at the top and the bottom of the crucible for different tested induced current densities (2.7 x IO7 A mV2 (labelled point A), 2.4 x lo7 A m - ’ (point B) and 2.1 x lo7 A -’ (point C)). The temperature of the seed and Ewder surface at x = 0 are equal to 3275 K, 3406 K for the point A, 2920 K, 3020 K for the point B, 2667 K, 2753 K for the point C. The temperature field, inside the crucible, has been more extensively studied for the point B case, which is the most representative of the current experiments. To simulate the whole process, our previous calculations were performed using an apparent thermal argon conductivity which reproduces the fourth power temperature dependence of radiation. We reduced the computational domain by assuming a temperature of
3000 K along the SIC powder from the results of the first approach. Thus, we refined the knowledge of local temperature gradients which are the driven factors of the deposition rate. Despite some numerical uncertainties in the very next of the seed corner, related to the calculations of view factors, it lead to a more precise temperature field shown on Fig. 4: the temperature is constant on the source (boundary conditions) whereas there is a 100 K temperature difference along the seed
379 521:793 697.207 872.621 1048.04 1223.45 1398.86 1574.28 1749.69 1925.1 2100.52 2275.93 2451.35 2626.76 2802.17 2977.59 3153 3328.42 3503.83 3679.24 3854.66
1 = 3i6
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
= = = = = = = = = = = = = = = = = = = =
u
Fig. 2. Temperature distribution inside the growth chamber obtained from the hear transfer model.
310
M. Pons et
al. /Materials
Science ad
ml0
2500
g 2 E 2400 g & 2300 m 5 % e 3B 2200 g t-, 2100
2000 2200
I
1 2300
I 2.500
2400 Temperalure
of the hofkxn
I 2GOO
I 2700
of the crucible
(K)
2800
Fig. 3. Comparison between experimental (pyrometric measurements) and calculated temperatures at the top and the bottom of the crucible.
instead of the value of 10 K for the previous calculations, and the temperature is lower by more than 100 K on the graphite. In this configuration, the temperature difference between source and seed is about 100 K on the symmetry axis, as obtained with the simplified model. Then, the next modelling step, mass transport linked with thermodynamic calculations was performed starting from the temperature fields inside the crucible calculated with different levels of model complexity.
4. Thermodynamic/mass
transport modelling
A classical thermodynamic
analysis has been per-
Engineering
846 (1997)
305-312
formed on the Si-C-Ar system to provide the composition at equilibrium in the gas phase above silicon carbide in the temperature range 2700-3500 K using the thermodynamic data that came from the recent work of P. Rocabois et al. [ 161. The results indicated that in the considered temperature and pressure range, the free sublimation of Sic is not congruent (enrichment of the condensed phase in carbon) and that the gas phase is essentially composed from Si,(g), Sic,(g) and Si,C(g), which has been confirmed by earlier mass spectrometric data [17], According to these results, in the application of the coupled mass transport-LTCE (local thermochemical equilibrium) model which has been described in detail in previous papers [12--141, the six more important gaseous species (Sil, Si,, Si,, Si,C, SiCJ diluted with argon were considered for both homogeneous and heterogeneous (sublimation and deposition surface) equilibrium. The boundary conditions are: at the source, the mass fractions are specified by the thermodynamic heterogeneous equilibrium on Sic powder at the calculated temperatures, on the walls (graphite surface), negligible deposition was assumed owing to the small area of the walls when compared to the total deposition area, on the sink (deposition surface), the mass fluxes are linked with thermodynamic heterogeneous equilibrium. Calculations are performed in stationary regime. Fig. 5 shows the calculated mass fractions along the centerline of the six considered gaseous species inside the crucible at a total pressure of 30 Torr (3990 Pa), for the temperature field shown on Fig. 4. The most important species are found to be SiC2, Si,C and Si,. For the species Sic, and S&C there is a continuous depletion from the source towards the deposition surface, whereas Si, species concentration increases and remains almost constant. On the seed, the deposition rate in Sic calculated from the surface molar fluxes (essentially the surface fluxes of the 3 predominant species, SIC?, S&C and Si,) is about 2.3 mm T ml F 2759.56 1~2761.97 2~2775.83 3=2789.69 4=2.303.55 5~2617.41 6~2831.27 7 =2845.13 0 = 2858.99 9-2872.85 10=2886.71 11 -2900.57 12 -2914.43 13=2928.29 14z2942.15 15~2956.02 16=2969,88 17-2983.74 16=2997X max=3000
SIC powder Fig. 4. Temperature field inside the crucible for the tested conditions of point B (Fig. 3).
M. Pans et al. f Matevials
Science
and Erlgineering
B46 (1997)
308-312
,~s~
311
.
Graphite
“Lb!
~-0j0iW~
~~.~.~.~.~.~.“~.~.~.,+.~....
. E
0.20
I
(IO K gradient
I
on seed)
3v
Thickness
(mm)
Experimental
profile
Fig. 6. Radial variation of the Sic surface shape after growth or Sic thickness (experimental average value for 1 h run time and calculated from the temperature fields obtained with the refined and simplified (italic) heat transfer models). 0
0.002
0.004
0.006
0.00s
0.01
Distance from the source (m)
perature field in the very small (compared to the whole experimental setup) area around the seed on the magnitude and shape of the growth rate.
Fig. 5. Profile of Si,, Si,, Si,, C,, Sic,, S&C mass fractions along the centerline, for a total pressure of around 3990 Pa.
5. Conclusions h- ‘. This value is in good agreement with the experimental value (average value for 1 h run time) which is 1.55 mm h-r in these conditions. In Fig. 6, the radial variation of the calculated (for the two temperatures distributions) and experimental Sic surface after growth together with the calculated temperatures are reported. We can notice that the experimental and calculated features are comparable all along the seed and graphite headland, whereas there is a discrepancy in the neighbourhood of the step. Two effects contribute to the magnitude of the deposition rate: mass transfer and local thermodynamic equilibrium. The first one by itself would lead to an increase of the total flux near the step. The second one by itself is mainly controlled by temperature. As temperature increases, we should expect a decrease in the deposition rate. To separate each contribution, we can compare the theoretical growth features obtained with the two temperature distributions. With the simplified heat transfer model (where the temperature difference along the seed is around 10 K), the transport effect is predominant and we observe an increase of growth rate at the corner, whereas with the refined model (where the temperature difference along the seed is around 100 K), the growth rate decreases and its shape is closer to the experimental one. However, this last simulation fails in the graphite part for the magnitude of the calculated growth rate. This can be explained by the lower values (100 K difference) of the temperature. It appears that the magnitude of the growth rate is controlled by the ‘vertical’ temperature (from source to sink) gradient while its shape is controlled by the temperature gradient along the sink. These preliminary results seem to show the influence of the actual tem-
Three different macroscopic models have been proposed to simulate the growth of silicon carbide single crystals. The thermodynamic approach showed that the most important species involved in the process are Si,, S&C and SiC2. From the heat transfer, it was possible to visualize the temperature field in such a complex process where sensors cannot be used. Finally, the approach linking mass transport modelling and thermodynamic equilibrium has proved its potential in describing the macroscopic growth of Sic when compared to experimental results.
References Ill M.L. Locatelli, S. Gamal, J. P/Qz. III France, PI P.A. Ivanov, V.E. Chelnokov, Semiconduct.
3 (1993) 1101. Sci. Technol., 7
(1992) 863. I. Garcon, A. Rouault, M. Anikin, C. Jaussaud, R. Madar, Mat. sci. Eng., B29 (1995) 90. [41 Y.M. Tairov, V.F. Tsvetkov, J. Qyst. Grontlz, 43 (1978) 209. [51 G. Ziegler, P. Lanig, D. Theis, C. Weyrich, IEEE Trans. Elec. [31
Deu., ED-30
(1983)
PI D.W. Greenwell,
277.
B.L. Markham, F. Rosenberger, J. Cq~st. Growth, 51 (1981) 413. 171 B.L. Markham, D.W. Greenwell, F. Rosenberger, J. Ciyst. Gl.~wth, 51 (1981) 426. 181 W.M.B. Duval, J. CVD, 2 (1994) 188. PI W.M.B. Duval, J. CT/D, 2 (1994) 289. I101 SK. Lilov, I.Y. Yanchev, Mater. Sci. Eng., 821 (1993) 83. t111 D. Hofmann, M. Heinze, A. Winnacker, F. Durst, L. Kadinski, P. Kaufmann, Y. Makarov, M. Schafer, J. Cryst. Growth, 146 (1995) 214. 1121I. Garcon, Contribution a l’etude de la croissance de monocristaux de carbure de silicium par la methode de Lely modifiee, PIID Thesis, Nat. Polytech. Inst. Grenoble, France, October 1995.
312
M. Pans et al. /Marerials
Science
[13] M. Pans, E. Blanquet, J.M. Dedulle, C, Bernard, R. Madar, L Electrochem. Sot., submitted for publication. 1141 M. Pons, C. Bernard, R. Madar, Sur$ Coni. Technol., 61 (1993) 274. 1151 Flux-Expert Sofware Package Guide, DT2I, 38240 Meylan,
and Engineering
B46 (1997)
308-312
France, 1995. [16] P. Rocabois, C. Chatillon, C. Bernard and F. Genet, High Temp. High Pres., (1995) in press. [17] J. Drowart, G. De Maria, M.G. Inghram, L C!ze/?z.Phys., 29 (1958) 1015.