b A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation. Abstract. The effect of radiative heat transfer ...
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C R Y S T A L G R O W T H
ELSEVIER
Journal of Crystal Growth 146 (1995) 209-213
Development of advanced mathematical models for numerical calculations of radiative heat transfer in metalorganic chemical vapour deposition reactors L. Kadinski a,,, Yu.N. Makarov a, M. Sch~ifer a, M.G. Vasil'ev b, V.S. Yuferev b a Lehrstuhl fiir Str6mungsmechanik, University of Erlangen - Niirnberg, Cauerstrasse 4, D-91058 Erlangen, Germany b A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation
Abstract
The effect of radiative heat transfer in a horizontal chemical vapour deposition (CVD) reactor on the upper wall temperature is studied in detail. A three-band model for the quartz absorption coefficient is introduced and the wall emittance, reflectance, and transmittance are calculated for the cases of specular and diffuse walls, and also for walls covered by a film. Numerical simulation of the heat transfer in the horizontal reactor has shown that the upper wall temperature varies about up to 40-70 K depending on the type of the wall and the emissivity of the susceptor.
I. Introduction
Accurate calculation of the t e m p e r a t u r e field in the reactor wall is extremely important for prediction of heterogeneous decomposition of the species and formation of wall deposits during the metalorganic chemical vapour deposition ( M O C V D ) process, because rates of the decomposition and saturated vapour pressure of the species depend exponentially on the wall temperature. Therefore, even such quite small temperature variations as 30-50 K can affect significantly the wall deposition, and consequently highly accurate heat transfer calculations are required for an optimization of the process [1]. In all numerical studies of radiation processes in M O C V D
* Corresponding author.
reactors published so far rather simple models were used [2-5]. First of all, only diffusively reflective boundary surfaces were considered, although quartz wails are characterized, as a rule, by a specular reflection of radiation and the emittance of specular and diffusive walls can differ very essentially. To account for the assumption of diffuse reflection the effect of extensive multiple reflections of the radiation fluxes inside a reactor chamber was drawn in Ref. [4]. The formation of the deposits on the reactor walls is another factor which could cause the diffuse character of reflection. However, in this case we have to take into account a new effect, the change of radiative properties of the reactor walls due to the deposition of a film on their surfaces. To our knowledge, this effect has also not been considered so far. Another specific feature of the radiative transfer in M O C V D reactors is related to the
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L. Kadinski et al. /Journal of Crystal Growth 146 (1995) 209-213
well-known fact that quartz has an optical window in the wavelength region 0.25-4.0 /xm. Therefore, the grey approximation which was used in Refs. [2,5] turns out to be quite coarse. In Ref. [4] a more accurate two-band model, was formulated. Nevertheless, the analysis of the wavelength dependence of the quartz absorption coefficient allows us to conclude that this model remains too coarse. Finally, it is usually assumed that the susceptor emissivity is equal to the graphite emissivity [3,4]. In fact this assumption is incorrect because during the growth process semiconductor substrates are placed on the top of the susceptor and usually a polycrystalline layer is deposited, which can significantly diminish the susceptor emissivity compared with graphite. The aspects mentioned above mean that rather severe assumptions were used so far to calculate the temperature distribution in the reactor, and the development of an advanced model is required for more adequate simulation of the heat transfer in CVD reactors. In the present paper a three-band model of the absorption coefficient is introduced. Using this model the values of the emittance and reflectance for a 2 mm wall are obtained in each wavelength band in dependence on the wall type and the presence of a film on the wall surface. The mathematical model of radiative-convective heat transfer in a horizontal reactor is described, and results of numerical simulations are given in comparison with results of other studies.
2. Three-band model of quartz absorption coefficient The quartz absorption coefficient strongly depends on the wavelength and temperature. Moreover, this dependence essentially varies for the different types of quartz prepared by different methods. A study of the quartz absorption coefficient dependence on the wavelength [6-9] shows that three spectral bands can be distinguished: (1) A < 3.5 ~m, where the quartz wall of the reactor is almost transparent; (2) 3.5 < A < 4.5 ~m, where the absorption coefficient starts to increase rapidly and the opti-
cal thickness of a quartz wall remains of order of unity; (3) A > 4.5 /~m, where the quartz wall can be considered to be opaque. The Planck average absorption coefficients have been calculated in all bands. It is necessary to emphasize that although the absorption coefficient in band 1 is small the absorption in this band can be essential, because the fraction of the susceptor radiation can reach 44%. It is also interesting to note that if we take the two-band model and carried out the same averaging of the absorption coefficient then we would obtain in band 1 extremely high values of the absorption coefficient, i.e. 0.7-0.9 cm-1.
3. Emittance, reflectance and transmittance of a quartz wall Expressions for the emittance E, reflectance R and transmittance Tr are well-known in the case of specular walls and they can easily be obtained in the case of diffuse reflection or when the inner side of the wall is covered by a film [10]. Calculation of E shows that the emittance of a diffusely transparent wall is essentially higher than the emittance of a specular wall. This can easily be seen for small optical thicknesses where the result can be confirmed analytically: for a specular wall E = 2(n 2 - nnX/~--S- 1 )~" and for a diffuse wall E = 2nZz, where n is the refractive index, ~- = a h is the wall optical thickness with wall thickness h and absorption coefficient a. If one of the wall surfaces is covered by a film, different situations can occur. If an As-film is deposited, the wall becomes opaque with a high emission coefficient. If we are faced to a GaAs or some another semiconductor film deposition, the effect is more complicated and the following specific features can be noted in the case of a diffusely reflective film and a specular reflective quartz wall. First of all, there are two different values of the wall emittance: E~ for radiation directed into the reactor volume and E 2 for radiation directed outside to the ambient, and, as GaAs is characterized by a considerable reflectiv-
L. Kadinski et aL /Journal of Crystal Growth 146 (1995) 209-213
ity, E~ < E 2. Consequently, the radiative exchange in bands 2 and 3 between susceptor and wall will be diminished, which, in turn, can lead to a decrease of the wall temperature. In contrast, the wall emittance in band 1 will increase due to the additional absorption and irradiation in the film. Correspondingly, radiative heat transfer in this band will rise, which can result in an increase of the wall temperature. So, if the semiconductor film is deposited on the wall, one has two effects acting in opposite direction. Which of them will dominate depends on the film optical thickness zf. One can expect that the first one will prevail until the film optical thickness remains extremely small (less than 0.01-0.03). Such small values of the optical thickness are explained by the fact that in the calculations the film surface is assumed to be diffusive. In the case of a specular film, the effect of the heating due to the film deposition will be smaller, and the marginal value of the optical thickness will be larger. To calculate the emission and reflection coefficients for the system wall/film we suppose that the GaAs absorption coefficient is independent of the wavelength. The values of E and R for a 2 mm quartz wall are given in Table 1. One can see that the different models of radiation processes in the reactor walls lead to essentially different values of the emittance and the reflectance in bands 1 and 2.
4. Numerical results
The objective of the present study is to develop a mathematical model which gives accurate prediction of the radiative heat transfer in MOCVD reactors. The mathematical model de-
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quartz upper wall 25 cm Inflow
I 2 cm 9 cm
Outflow 10 cm graphite susceptor
6 cm
Fig. 1. Schematic of a horizontal reactor.
scribed above, which has been introduced into a numerical procedure presented in Ref. [11] is applied to a typical horizontal CVD reactor (see Fig. 1) under the following conditions: the inflow temperature is 300 K, the susceptor temperature is 1000 K, the operating pressure is 0.1 atm, and the inflow velocity is 1 m/s. Numerical simulations are performed to demonstrate the effect of the upper wall temperature by using the different models described above. The values of the emittance and the reflectance have been taken as in Table 1. It is necessary to emphasize that for the calculation of radiative fluxes at the reactor walls the assumption of diffuse reflection is used in all cases including the specular walls. But in latter case the emission and reflection coefficients are calculated by using the assumptions of specularly reflective walls. Fig. 2 shows the temperature distribution along the upper wall for the case of the specular and diffusive walls calculated by using three-band model described above and two-band model given in Ref. [4]. One can see that in accordance with the considerations discussed above the diffuse upper wall is heated stronger, and the two-band model results in a wall temperature which is observably lower, since the absorption in the first band is not taken into account. The effect of the film deposition is demonstrated in Fig. 3. The
Table 1 Radiation properties for the 2 mm quartz wall Band
1 2 3
Specular (pure quartz)
Difussive (pure quartz)
Wall + thin film zf = 0
E
R
E
R
El
E2
R1
R2
E1
E2
R1
R2
0.05 0.65 0.914
0.14 0.095 0.086
0.14 0.83 0.914
0.35 0.095 0.086
0.11 0.43 0.518
0.16 0.78 0.914
0.58 0.50 0.482
0.53 0.15 0.086
0.53 0.62 0.639
0.60 0.88 0.914
0.37 0.37 0.361
0.30 0.11 0.086
Wall + thick film rf = 0.1
L. Kadinski et aL /Journal of Crystal Growth 146 (1995) 209-213
212
1
n,,
1
2
650
650
c~
600
[- 600
550
550 Ed [""1 500
500,-.3 .1 .< ~= 450-
"1
~= 450
400
~x3 400
350
350
13.,
i .05
.10 .15 DISTANCE (m)
.05
.20
Fig. 2. U p p e r wall temperature distribution along the reactor for the three cases: (1) three-band model - specular upper wall; (2) three-band model - diffusive upper wall; (3) two-band model [4] - diffusive upper wall.
deposition of GaAs has been modeled in the following way. It is assumed that the part of the upper wall directly above the susceptor is covered by the film, while other parts still remain clean from the deposits. In the case of an As-film deposition, it is assumed that the upper wall is
750 r~ 700 E-
650 600 550
"1
~
500 450
~
400
~
350,I,
.05
I,Iii
,I
,It
i
.10 .15 DISTANCE (m)
.10
.15
.20
DISTANCE (m)
i
i
i
i
t
i
.20
Fig. 3. U p p e r wall temperature distribution along the reactor: (1) pure quartz without deposited G a A s film; (2) quartz with deposited sufficiently thick film - zf = 0.1; (3) quartz with deposited thin film - rf = 0; (4) quartz with deposited As-film.
Fig. 4. Upper wall temperature distribution along the reactor: (1) pure graphite susceptor; (2) susceptor covered by a GaAs layer.
fully covered by the film. One can see that a sufficiently thick film (optical thickness 0.1) increases the wall temperature, and a thin film (optical thickness equal zero) cools the wall. This means, that the maximum upper wall temperature depends non-monotonically on the film thickness. The As-film is opaque with a high emission coefficient (in our calculations the value E = 0.9 is used) and, therefore, the deposition by the As-film leads to a significantly increased temperature. For this particular case the maximum temperature difference, compared to a pure quartz wall, is approximately 70 K. The nonsmooth temperature dependence for the case of a very thin film is caused by the stepwise change of the radiative properties of the upper wall. The influence of the susceptor emissivity is shown in Fig. 4. As one would expect, the decrease of the susceptor emissivity leads to an essential decreasing of the upper wall temperature. The maximum temperature difference is approximately 40 K. The same trend is observed when the upper wall is covered by a polycrystalline film. All the described effects have been demonstrated for the case of a uniform specified susceptor temperature. It would also be possible to consider more realistic boundary conditions like RF or quartz lamp susceptor heating for the heat
L. Kadinski et al./Journal of Crystal Growth 146 (1995) 209-213
transfer. In these cases the effects are becoming even more pronounced. Moreover, if deposition on the upper wall occurs, the susceptor temperature can change remarkably.
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such undesirable phenomena as parasitic deposition of GaAs and As films.
Acknowledgments 5. Conclusions In general, the following conclusions can be drawn from the results presented in this study: (1) The wall emittance depends significantly on the type of the wall, that, in turn, influences the upper wall temperature. (2) An interesting effect can be observed in the case of a GaAs-film deposition on the reactor wall. This effect has a dual character. On the one hand, the deposited film increases the reflection of a radiation flux coming from the susceptor, and, consequently, tends to decrease the upper wall temperature. On the other hand, the film assists in the absorption of radiation at the wavelength interval corresponding to the quartz transparence window and, therefore, tends to increase the wall heating. The first process dominates at the initial stage of the film deposition until the film remains very thin and absorption in band 1 is small. (3) The presence of GaAs substrates and polycrystalline layers on a graphite susceptor results in a decrease of the susceptor emissivity and, therefore, also in a observable decrease of the upper wall temperature. (4) The differences of the wall temperatures calculated by using the different radiation models can be quite significant (about 40-70 K). This has to be taken into account during simulation and optimization of MOCVD reactors, i.e. to prevent
This study has been carried out within a joint Russian-German project "Development of mathematical models and numerical methods for simulation of growth of A3B5 semiconductor heterostructures" financed by the Volkswagen-Stiftung, Germany. This support is thankfully acknowledged.
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