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Temperature field simulation and correlation to the structural quality of semi-insulating GaAs crystals grown by the vapour pressure controlled Czochralski method (VCz)
Journal of Crystal Growth 213 (2000) 10}18
Temperature "eld simulation and correlation to the structural quality of semi-insulating GaAs crystals grown by the vapour pressure controlled Czochralski method (VCz) Ch. Frank , K. Jacob , M. Neubert *, P. Rudolph , J. Fainberg, G. MuK ller Institut fu( r Kristallzu( chtung, Max-Born-Str. 2, D-12489 Berlin, Germany Institut fu( r Werkstowwissenschaften WW6, Universita( t Erlangen-Nu( rnberg, Martensstr. 7, D-91058 Erlangen, Germany Received 14 October 1999; accepted 21 January 2000 Communicated by K.W. Benz
Abstract Recent results of global thermal "eld modelling using the software programs CrysVUN# # and STHAMAS are presented for 3- and 4-inch VCz SI GaAs growth assemblies. For the "rst time global simulations including the gas and melt convection in a VCz arrangement are shown. In contrast to conventional LEC the impact of gas convection on the temperature distribution in the growing crystal is not dominant in the VCz case. Contrary to that melt convection cannot be neglected and needs to be controlled by crucible and crystal rotation rates. Simulations were used to optimise the temperature "elds in the VCz arrangements and thus, to reduce the von Mises stress within the growing crystal. Doing this, the EPD could be reduced by approximately one order of magnitude compared to conventional LEC. Minimum EPD's were found to be in the range of some 10 cm\ for 4 crystals. The observed nearly #at (slightly convex) interfaces are in agreement with the simulations. The achieved good radial uniformity of the carbon incorporation and electrical parameters is due to the improved thermal boundary conditions. 2000 Elsevier Science B.V. All rights reserved. PACS: 02.70.Fj; 02.60.Pn; 07.05.Tp; 44.40; 68.45.!v; 81.05.Ea; 81.10.Fq Keywords: GaAs; VCz; Global simulation; Interface; Homogeneity
1. Introduction The demand for semi-insulating (SI) GaAs crystals was rapidly increasing during the last decade because of the strongly growing market of microwave devices. Consequently, the rise of mass production is accompanied by the demand for higher
* Corresponding author. Fax: #49-30-6392-3003. E-mail address: [email protected] (M. Neubert).
productivity, i.e. longer crystals and transition from the well-established 4 production line towards wafer dimensions of 6 (150 mm). Key issues being connected with are the reduction of structural and electrical inhomogeneities in the as-grown crystals. Additionally, many e!orts are aimed to maintain the 4 crystal qualities, i.e. dislocation densities of some 10 cm\ passing to the 6 production line. This is not possible to be achieved by the conventional liquid encapsulation Czochralski (LEC) technique, the major industrial SI GaAs
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 2 0 8 - 6
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
production method with a share of nearly 90%. The way to achieve this goal is to reduce the thermally induced stress in order to minimise or avoid dislocation multiplication within the growing crystal. This again can be achieved by markedly lowering the temperature gradients in axial and radial directions. Additionally, to obtain a high uniformity of structural and, consequently, electrical properties, the temperature "eld in the crystal needs to be homogenized. Among others, the so-called vapour pressure controlled Czochralski (VCz) technique is a powerful method to provide such temperature "elds. The axial temperature gradients are in the range of 15}35 K/cm. General features of VCz can be found in Refs. [1,2]. Typical VCz pullers have a very sophisticated and complex set-up of the inner growth chamber which makes it di$cult to measure temperature distributions inside the high pressure vessels and within the growing crystal. Numerical simulations are therefore of high importance for the design and `tailoringa of such facilities. This work compiles global temperature "eld simulations of VCz arrangements carried out by using the "nite volume codes CrysVUN# # [3] and STHAMAS [4]. The present study is mainly concerned with the correlation between the temperature "eld and structural crystal properties, but also some electrical properties.
2. Experimental procedure The main constructive feature of a VCz arrangement is the presence of a gas-tight inner chamber shielding the growing crystal from the water-cooled walls of the outer high-pressure vessel. This is the precondition to grow GaAs under markedly reduced temperature gradients because of the tendency of arsenic to evaporate from the relatively hot surface of the crystals. A separate, temperature controlled, source of solid arsenic establishes a certain partial pressure within the inner chamber to keep the crystal in equilibrium with its gas phase. The inert gas pressure (Ar or N ) and boric oxide en capsulant are still employed like in the conventional LEC process.
11
Fig. 1. Typical as-grown 4 SI VCz GaAs crystal.
In order to avoid constitutional supercooling due to the low temperature gradients in the melt (2}10 K/cm) depending on the particular growth situation) the pulling rate has to be reduced compared to LEC down to a value of )5 mm/h. Fig. 1 shows a typical 4 VCz crystal grown in [0 0 1] direction applying an arsenic partial pressure of about 0.05 MPa. The mirror-like phenotype of the crystal gives evidence that its surface was in thermodynamic equilibrium with the gas phase throughout the whole run. The diameter variations are caused by the lack of su$ciently high radial temperature gradients and thus, complexity with diameter control. To analyse the dislocation density polished crystal wafers were etched in a KOH melt and the developed etch pits counted automatically. Growth striations were investigated from cuts parallel to the crystal axis using DSL etchant. The content of substitutional carbon [C ] was measured by local
12
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
vibrational mode (LVM) IR-absorption at 80 K. Electrical resistivity and free carrier concentration were obtained from Hall measurements.
3. Simulation of thermal and stress 5elds inside the growing crystal Simulations of the global temperature "eld were carried out with the two software programs CrysVUN# # and STHAMAS for both 3 and 4 VCz growth arrangements installed in commercial LPA Mark 3 and CI 358 pullers, respectively. Both programs are considering axisymmetric geometries (2D). CrysVUN# # is a "nite volume code with unstructured grid currently comprising heat transfer by conduction and radiation and the release of the latent heat. STHAMAS is also a "nite volume code, but it works with structured grids and can additionally simulate the heat and momentum transfer in #uid phases by convection. In both programs the melt temperature at the tri-junction crystal, melt and boric oxide is established by "tting the heating power. The thermal boundary conditions at the outer high-pressure vessel are a constant temperature of 300 K (see also Ref. [5]). The material properties applied in the simulations are shown in the appendix. The calculated temperature "elds within the crystals are used to calculate the thermo-elastic stress distributions in terms of the scalar von Mises stress. It is used to estimate the in#uence of the thermal "eld on the structural quality of the growing crystals. At "rst, calculations were made without considering convective heat transport with both codes (CrysVUN# # and STHAMAS) in order to analyse their comparability concerning the basic model of radiative and conductive transport of heat. As can be seen from Figs. 2a and b the results are quite similar and the calculated heater powers of 11.5 kW (CrysVUN) and 11.4 kW (STHAMAS) agree well. In comparison, the experimental value for the same growth situation was measured to be 12.5 kW. This is a fortunately low deviation of about 9% only. In order to study the impact of gas convection on the temperature distribution the LPA Mark 3 set-up was simulated using the STHAMAS code, includ-
ing turbulent gas convection (k}e model with wall functions; for more details see Ref. [4] and the appendix). This is to our knowledge the "rst numerical study of this kind of a VCz set-up (Fig. 2c). Comparing Figs. 2a or b with c the isotherms, i.e. the temperature "eld in the gas within the VCz chamber is altered considerably, although nearly the same heater power was calculated. Fig. 3 shows a line scan of the temperature distribution along the axis of the apparatus for both cases: with and without gas convection. Surprisingly, apart from little absolute temperature di!erences, nearly identical temperature gradients of about 37 K/cm are calculated within the growing crystal. This is a very important result because in the conventional LEC case calculations with and without gas convection di!er markedly each to the other [4]. This may be understood because the gas-"lled volume surrounding the crystal is much larger in the LEC case than in the VCz case and, as it is well known, the Grashof number (Gr) scales with the 3rd power of the length but only with the 1st power of the temperature di!erences. However, the temperature differences in both cases also di!er signi"cantly. They are in the range of 1200}1300 K under LEC conditions, much higher than those of VCz where the inner chamber exhibits temperature di!erences of about 400}500 K (Fig. 3) and in a "nally optimised system even only 200}300 K (CI 358 puller) [6]. All in all, the impact of heat transport due to gas convection to the temperature "eld inside the growing crystal should be very di!erent for LEC and VCz. The observed small di!erences in the VCz case are the major argument for carrying out the following optimisation of the thermal "eld using CrysVUN# #, i.e. without considering the convective transport of heat by the inert gas. Another question is of which e!ect is the melt convection on the shape of the phase boundary and the resulting temperature "eld inside the crystal. Generally, it was found that the ratio and absolute values of crucible and crystal rotation rates in#uence the interface curvature considerably. This was shown by experiments and by some local FIDAP calculations of another paper of the IKZ considering melt and crystal only [7]. In connection with this, Fig. 4a shows the stream lines and isotherms in the melt of a 4 VCz growth situation in the CI 358
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
13
Fig. 2. Calculated global temperature "elds of a VCz arrangement inside the puller LPA Mark 3 carried out with the codes CrysVUN# # (a) and STHAMAS (b), in both cases without consideration of convective transport and with STHAMAS (c) including gas convection of argon gas at 0.9 MPa. The calculated stream "eld is added on the left side of (c). P indicates the calculated heater powers, the measured power consumption was 12.5 kW. The distances of the isotherms are 50 K in all the three cases.
puller obtained from global simulations by STHAMAS. In the demonstrated case, the rotation rates were chosen close to the results in Ref. [7] with 26 and 21 rpm for crystal and crucible, respectively. This was guided by the intention to nearly suppress the buoyancy-driven convection, i.e. that heat transfer by conduction becomes dominant within the melt. In fact, the isotherm pattern of the CrysVUN# # simulation of the same situation in Fig. 4b shows a quite similar result like in Fig. 4a although convection is not considered in this
model. In other words: for the presented VCz snapshot one can "nd appropriate rotation rates of crystal and crucible where the melt #ow plays only a minor role for the heat transport in the melt. Choosing such a priori conditions, the application of CrysVUN## is justi"ed because the in#uence of melt convection can be nearly neglected. This, in fact, was done in all of the simulations presented in the following. Additionally, the use of CrysVUN# # is advisable because of its better practicability compared to STHAMAS.
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Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
After learning the above facts, the thermal "eld inside the growing crystals was optimised for the relevant VCz set-ups using the CrysVUN# #
Fig. 3. Calculated axial temperature distribution (left) taken from Figs. 2b and c and values of the local temperature gradients. The global temperature "eld is added on the right side to correlate the axial coordinate with the interior of the VCz arrangement. The isotherms are separated by 25 K.
program. The main objectives were (i) reduction of the thermal stress due to the #attening of the interface boundary and (ii) homogenisation of the heat #ow through the crystal by adjusting it to be nearly uniaxial. To achieve this, the design of the thermal insulation of the VCz chamber, the height of boric oxide, the crucible position etc. were varied in wide ranges during the simulation procedure. Finally, results like illustrated in Fig. 5a were obtained. As expected, the nearly uniaxial heat #ow (nearly equidistant and #at isotherms) through the crystal guarantees relatively low values of the von Mises stress as demonstrated in Fig. 5a. The main part of the interface area and the core region of the growing crystal are exposed to very low stress, often
Fig. 5. 4 crystal in "nally optimised thermal conditions: (a) calculated using CrysVUN# #, the "gure shows isotherms, *¹"10 K (left) and lines of equal von Mises stress, separated by 0.25 MPa (right). (b) striation patterns of a real 4 GaAs crystal, grown under those optimised conditions, cut parallel to its growth axis [0 0 1].
Fig. 4. Comparison of a typical 4 growth arrangement calculated with: (a) STHAMAS; crystal and crucible rotation rates are 26 and 21, respectively, the "gure shows convection stream lines (left) and isotherms (right). (b) CrysVUN# #; without convective transport of heat, isotherm are shown. In both (a) and (b) the isotherms are separated by 12 K. The stream lines in left side of (a) are normalised between 0 and 1.
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
15
below the critical resolved shear stress (CRSS [kPa]"22.72 exp(4334/¹ ) [8], i.e. approximately 0.5 MPa near the melting point of GaAs). However, near the crystal edge stress peaks of 1.8 and 3.7 MPa appear. They are mainly caused by the discontinuities of the material properties like conduction and emissivity (solid and liquid GaAs, boric oxide). These e!ects are well known from the conventional LEC process [9] where the absolute stress values are, however, higher by of about one order of magnitude than in VCz crystals.
4. Experimental veri5cation To verify the above results experimentally, the inner VCz constructions were manufactured according to this. As shown in Fig. 5b the real curvature of the interface, obtained by the striation technique, is close to the calculated one in Fig. 5a. Please note that several stages of growth were simulated while only one stage is shown in Fig. 5 exemplarly. Fig. 6 shows two typical radial EPD distributions of 4 VCz crystals (lower curves) pulled in the computer-tailored growth arrangement. For comparison the EPD distribution of a conventional 3 LEC crystal is added. Obviously, the optimisation in the VCz case leads to an EPD reduction of about one order of magnitude. Further, the radial distribution is found to be more U-shaped instead of the well-known W-shape in conventional LEC crystals. The EPDs in Fig. 6 show a good qualitative correlation with the calculated stress "eld of Fig. 5a, i.e. within the main area of the slices the EPD is rather low and homogeneously distributed. The lowest dislocation densities obtained are at about some 10 cm\. In these areas the most dislocations are randomly distributed and only rarely polygonised. Only near the crystal edge the dislocation density increases to maximum values of about 5;10 cm\ where, again, the typical cellular structure appears. However, if the well-known cell structure occurs their dimensions are markedly larger (1}2 mm) than in typical LEC-material (ca. 0.2}0.5 mm). A further sensitive indicator of the interface curvature is the radial distribution of the shallow acceptor carbon [C ], in#uenced by segregation.
Fig. 6. Radial distributions of the etch pit density (EPD) along 11 0 02 and 11 1 02 directions in 4 crystals grown by the optimised VCz conditions according to Fig. 5. For comparison the EPD distribution of a 3 LEC crystal, grown in the authors laboratory, is added.
Fig. 7. Radial carbon distributions across 3 (䉬) and 4 (䊏) VCz-grown wafers analysed by the LVM IR-absorption method.
Precondition for a nearly uniform radial [C ] distribution like the two shown in Fig. 7 is a nearly #at interface shape during growth [10]. At a mean value of 7.6;10 cm\ a standard deviation of only 7% was measured along the wafer diameter. In general, it is possible to control the total carbon content in VCz GaAs crystals in the range between some 10 and 10 cm\ (note this is markedly below the limits being reported for VCz crystals so far [2,11]). More details will be given in Ref. [10]. Unfortunately, the resulting resistivity distribution does not only follow the homogeneous carbon distribution, but is also in#uenced by the EPD
16
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
Fig. 8. Radial distributions of the electrical resistivity o (a) and carrier mobility k (b) in 3 (䉬) and 4 (䊏) VCz crystals corresponding with those of Fig. 7.
distribution via EL2 [6,12]. This is illustrated in Fig. 8a. While the resistivity is roughly uniform in the middle part of the wafer it decreases towards the edges due to the increasing EPD (increasing EL2 and, thus, carrier concentration). However, this is only the case in as-grown crystals. The EL2 and carrier concentrations can be homogenised by subsequent bulk annealing very e!ectively [6]. Fig. 8b shows radial distributions of the electron mobility corresponding to Fig. 8a. In the 3 wafer it is quite high for as-grown crystals and corresponds to the maximum of the mobility}resistivity curve of SI GaAs at o+2;10 ) cm [13].
5. Conclusions Global modelling of the thermal "eld and the von Mises stress by using the code CrysVUN# # was successfully used for the tailoring of the growth arrangements of 3 and 4 VCz crystals. The gas convection in a VCz equipment was calculated for the "rst time by applying the code STHAMAS. Comparing the results of both programs no considerable di!erences were observed for the thermal
"elds inside the crystals. Therefore it can be concluded that the heat transport by gas convection plays a minor role in the VCz growth mode because of (i) the smaller volume of the inner chamber and (ii) the thermal shielding of the crystal from the water-cooled walls of the outer vessel. This makes the VCz arrangement completely di!erent from the conventional LEC where the gas convection is of marked in#uence on the thermal "eld inside the crystal. Contrary to this, melt convection cannot be neglected in the VCz growth mode. Best growth results were obtained if the melt convection is nearly suppressed by choosing appropriate crucible and crystal rotation rates. Under those preconditions, CrysVUN# # is justi"ed to be applied for the optimisation of the thermal "eld inside the apparatus, i.e. reduction of thermally induced stress. The "nally obtained conditions are characterised by low dislocation densities with a good radial homogeneity of its density distribution. The obtained #at growth interfaces are additionally indicated by the uniform radial carbon concentration distribution. Electrical parameters are in the expected range of typical SI GaAs. Their good radial homogeneity is promising to be again improved after bulk annealing.
Acknowledgements The authors are indebted to the following co-workers of the IKZ Berlin: M. Pietsch and M. Czupalla for growing the crystals, B. Lux for crystal preparation, S. Bergmann for EPD analysis, and K. Irmscher for Hall measurements. They are grateful to Prof. W. Ulrici (Paul-Drude Institut fuK r FestkoK rperelektronik, Berlin) for the analysis of the radial carbon distribution by LVM IR-absorption. This work was partially supported by the German Federal Ministry of Education, Science, Research & Technology under contract No. 5224001-01 BM 501/0.
Appendix Tables 1}3 show the data used for simulations with STHAMAS and CrysVUN# #.
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
17
Table 1 Global temperature "eld simulations [14,15] Material Graphites EK90 EK98 EK51 CFC (radial) CFC (axial) CBCF (radial) CBCF (axial) CBCF (rad.) CBCF (axial) Graphite felt Sigra#ex (foil) Others Non-corrosive steel CrNi Mo Al O PBN (radial) (axial) Fused silica B O GaAs(solid) GaAs(liquid) As(solid) Argon gas
Thermal conductivity [W/mK]
Emissivity
65(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 90(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 25(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 9.3175 # 2.5;10\ ¹ 20 0.2489 #4.166;10\ ¹ # 3.537;10\ ¹ !2.976;10\ ¹ 3 ) CBCF (radial) 3 ) CBCF (axial) !0.1011 #8.0301;10\ ¹ !5.386;10\ ¹ !1.7857;10\ ¹ 4.3;10\ # 3.24;10\ ¹ # 9.4;10\ ¹ 218.25 ! 0.25176 ¹ # 1.3165;10\ ¹ ! 0.23433;10\ ¹
0.8 0.8 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.7 0.58
15 25 105 2.15 3.135 62.7 0.94 # 0.001348 ¹ 0.237 # 0.0011 ¹ 7.12 17.8 50 1.8;10\ # 5.885;10\ ¹ ! 2.249;10\ ¹ # 4.921;10\ ¹
0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.75 0.55 0.55 0.5 0.5
Table 2 Data used for calculation of gas convection [16,17] k!e constants c l C e C e R R e Grashof number, Gr
0.09 1.44 1.92 1.0 1.3 6;10
Table 3 Data used for calculation of melt convection [14] Grashof number, Gr Kinematic viscosity, l Crystal rotation rates Crucible rotation rates
1.7;10 0.00488 cm/s 5}26 rpm 10}25 rpm
References [1] P. Rudolph, M. Neubert, S. Arulkumaran, M. Seifert, Cryst. Res. Technol. 2 (1997) 35. [2] M. Tatsumi, K. Fujita, in: M. Isshiki (Ed.), Recent Developments of Bulk Crystal Growth, Research Signpost, Trivandrum, 1998, p. 47. [3] M. Kurz, A. Pusztai, G. MuK ller, J. Crystal Growth 198/199 (1999) 101. [4] J. Fainberg, H.-J. Leister, G. MuK ller, J. Crystal Growth 180 (1997) 517. [5] K. BoK ttcher, P. Rudolph, M. Neubert, M. Kurz, A. Pusztai, G. MuK ller, J. Crystal Growth 198/199 (1999) 349. [6] M. Neubert, P. Rudolph, to be published. [7] W. Miller, U. Rehse, K. BoK ttcher, Solid-State Electron. 44 (2000) 825. [8] G.O. Meduoye, D.J. Bacon, K.E. Evans, J. Crystal Growth 108 (1991) 627. [9] A.S. Jordan, A.R. von Neida, R. Caruso, J. Crystal Growth 70 (1984) 555. [10] K. Jacob, Ch. Frank, M. Neubert, P. Rudolph, W. Ulrici, to be published.
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Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
[11] M. Tatsumi, T. Kawase, Y. Iguchi, K. Fujita, M. Yamada, in: M. Godlewski (Ed.), Semi-Insulating III}V Materials, World Scienti"c, Singapore, 1994, p. 11. [12] S. Miyazawa, T. Mizutani, H. Yamazaki, Jpn. J. Appl. Phys. 21 (1982) L542. [13] T. Flade, M. Jurisch, A. Kleinwechter, A. KoK hler, U. Kretzer, J. Prause, Th. Reinhold, B. Weinert, J. Crystal Growth 198/199 (1999) 336.
[14] http://www.ikz-berlin.de/&kb/daten/mat-1/mat.html [15] M. Kurz, Ph.D. Thesis, University of Erlangen-NuK rnberg, 1998. [16] B.E. Launder, D.B. Spalding, Appl. Mech. Eng. 3 (1974) 269. [17] R. Henkes, C.J. Hoogendorn, Int. J. Heat Transfer 32 (1989) 157.
Temperature "eld simulation and correlation to the structural quality of semi-insulating GaAs crystals grown by the vapour pressure controlled Czochralski method (VCz) Ch. Frank , K. Jacob , M. Neubert *, P. Rudolph , J. Fainberg, G. MuK ller Institut fu( r Kristallzu( chtung, Max-Born-Str. 2, D-12489 Berlin, Germany Institut fu( r Werkstowwissenschaften WW6, Universita( t Erlangen-Nu( rnberg, Martensstr. 7, D-91058 Erlangen, Germany Received 14 October 1999; accepted 21 January 2000 Communicated by K.W. Benz
Abstract Recent results of global thermal "eld modelling using the software programs CrysVUN# # and STHAMAS are presented for 3- and 4-inch VCz SI GaAs growth assemblies. For the "rst time global simulations including the gas and melt convection in a VCz arrangement are shown. In contrast to conventional LEC the impact of gas convection on the temperature distribution in the growing crystal is not dominant in the VCz case. Contrary to that melt convection cannot be neglected and needs to be controlled by crucible and crystal rotation rates. Simulations were used to optimise the temperature "elds in the VCz arrangements and thus, to reduce the von Mises stress within the growing crystal. Doing this, the EPD could be reduced by approximately one order of magnitude compared to conventional LEC. Minimum EPD's were found to be in the range of some 10 cm\ for 4 crystals. The observed nearly #at (slightly convex) interfaces are in agreement with the simulations. The achieved good radial uniformity of the carbon incorporation and electrical parameters is due to the improved thermal boundary conditions. 2000 Elsevier Science B.V. All rights reserved. PACS: 02.70.Fj; 02.60.Pn; 07.05.Tp; 44.40; 68.45.!v; 81.05.Ea; 81.10.Fq Keywords: GaAs; VCz; Global simulation; Interface; Homogeneity
1. Introduction The demand for semi-insulating (SI) GaAs crystals was rapidly increasing during the last decade because of the strongly growing market of microwave devices. Consequently, the rise of mass production is accompanied by the demand for higher
* Corresponding author. Fax: #49-30-6392-3003. E-mail address: [email protected] (M. Neubert).
productivity, i.e. longer crystals and transition from the well-established 4 production line towards wafer dimensions of 6 (150 mm). Key issues being connected with are the reduction of structural and electrical inhomogeneities in the as-grown crystals. Additionally, many e!orts are aimed to maintain the 4 crystal qualities, i.e. dislocation densities of some 10 cm\ passing to the 6 production line. This is not possible to be achieved by the conventional liquid encapsulation Czochralski (LEC) technique, the major industrial SI GaAs
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 2 0 8 - 6
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
production method with a share of nearly 90%. The way to achieve this goal is to reduce the thermally induced stress in order to minimise or avoid dislocation multiplication within the growing crystal. This again can be achieved by markedly lowering the temperature gradients in axial and radial directions. Additionally, to obtain a high uniformity of structural and, consequently, electrical properties, the temperature "eld in the crystal needs to be homogenized. Among others, the so-called vapour pressure controlled Czochralski (VCz) technique is a powerful method to provide such temperature "elds. The axial temperature gradients are in the range of 15}35 K/cm. General features of VCz can be found in Refs. [1,2]. Typical VCz pullers have a very sophisticated and complex set-up of the inner growth chamber which makes it di$cult to measure temperature distributions inside the high pressure vessels and within the growing crystal. Numerical simulations are therefore of high importance for the design and `tailoringa of such facilities. This work compiles global temperature "eld simulations of VCz arrangements carried out by using the "nite volume codes CrysVUN# # [3] and STHAMAS [4]. The present study is mainly concerned with the correlation between the temperature "eld and structural crystal properties, but also some electrical properties.
2. Experimental procedure The main constructive feature of a VCz arrangement is the presence of a gas-tight inner chamber shielding the growing crystal from the water-cooled walls of the outer high-pressure vessel. This is the precondition to grow GaAs under markedly reduced temperature gradients because of the tendency of arsenic to evaporate from the relatively hot surface of the crystals. A separate, temperature controlled, source of solid arsenic establishes a certain partial pressure within the inner chamber to keep the crystal in equilibrium with its gas phase. The inert gas pressure (Ar or N ) and boric oxide en capsulant are still employed like in the conventional LEC process.
11
Fig. 1. Typical as-grown 4 SI VCz GaAs crystal.
In order to avoid constitutional supercooling due to the low temperature gradients in the melt (2}10 K/cm) depending on the particular growth situation) the pulling rate has to be reduced compared to LEC down to a value of )5 mm/h. Fig. 1 shows a typical 4 VCz crystal grown in [0 0 1] direction applying an arsenic partial pressure of about 0.05 MPa. The mirror-like phenotype of the crystal gives evidence that its surface was in thermodynamic equilibrium with the gas phase throughout the whole run. The diameter variations are caused by the lack of su$ciently high radial temperature gradients and thus, complexity with diameter control. To analyse the dislocation density polished crystal wafers were etched in a KOH melt and the developed etch pits counted automatically. Growth striations were investigated from cuts parallel to the crystal axis using DSL etchant. The content of substitutional carbon [C ] was measured by local
12
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
vibrational mode (LVM) IR-absorption at 80 K. Electrical resistivity and free carrier concentration were obtained from Hall measurements.
3. Simulation of thermal and stress 5elds inside the growing crystal Simulations of the global temperature "eld were carried out with the two software programs CrysVUN# # and STHAMAS for both 3 and 4 VCz growth arrangements installed in commercial LPA Mark 3 and CI 358 pullers, respectively. Both programs are considering axisymmetric geometries (2D). CrysVUN# # is a "nite volume code with unstructured grid currently comprising heat transfer by conduction and radiation and the release of the latent heat. STHAMAS is also a "nite volume code, but it works with structured grids and can additionally simulate the heat and momentum transfer in #uid phases by convection. In both programs the melt temperature at the tri-junction crystal, melt and boric oxide is established by "tting the heating power. The thermal boundary conditions at the outer high-pressure vessel are a constant temperature of 300 K (see also Ref. [5]). The material properties applied in the simulations are shown in the appendix. The calculated temperature "elds within the crystals are used to calculate the thermo-elastic stress distributions in terms of the scalar von Mises stress. It is used to estimate the in#uence of the thermal "eld on the structural quality of the growing crystals. At "rst, calculations were made without considering convective heat transport with both codes (CrysVUN# # and STHAMAS) in order to analyse their comparability concerning the basic model of radiative and conductive transport of heat. As can be seen from Figs. 2a and b the results are quite similar and the calculated heater powers of 11.5 kW (CrysVUN) and 11.4 kW (STHAMAS) agree well. In comparison, the experimental value for the same growth situation was measured to be 12.5 kW. This is a fortunately low deviation of about 9% only. In order to study the impact of gas convection on the temperature distribution the LPA Mark 3 set-up was simulated using the STHAMAS code, includ-
ing turbulent gas convection (k}e model with wall functions; for more details see Ref. [4] and the appendix). This is to our knowledge the "rst numerical study of this kind of a VCz set-up (Fig. 2c). Comparing Figs. 2a or b with c the isotherms, i.e. the temperature "eld in the gas within the VCz chamber is altered considerably, although nearly the same heater power was calculated. Fig. 3 shows a line scan of the temperature distribution along the axis of the apparatus for both cases: with and without gas convection. Surprisingly, apart from little absolute temperature di!erences, nearly identical temperature gradients of about 37 K/cm are calculated within the growing crystal. This is a very important result because in the conventional LEC case calculations with and without gas convection di!er markedly each to the other [4]. This may be understood because the gas-"lled volume surrounding the crystal is much larger in the LEC case than in the VCz case and, as it is well known, the Grashof number (Gr) scales with the 3rd power of the length but only with the 1st power of the temperature di!erences. However, the temperature differences in both cases also di!er signi"cantly. They are in the range of 1200}1300 K under LEC conditions, much higher than those of VCz where the inner chamber exhibits temperature di!erences of about 400}500 K (Fig. 3) and in a "nally optimised system even only 200}300 K (CI 358 puller) [6]. All in all, the impact of heat transport due to gas convection to the temperature "eld inside the growing crystal should be very di!erent for LEC and VCz. The observed small di!erences in the VCz case are the major argument for carrying out the following optimisation of the thermal "eld using CrysVUN# #, i.e. without considering the convective transport of heat by the inert gas. Another question is of which e!ect is the melt convection on the shape of the phase boundary and the resulting temperature "eld inside the crystal. Generally, it was found that the ratio and absolute values of crucible and crystal rotation rates in#uence the interface curvature considerably. This was shown by experiments and by some local FIDAP calculations of another paper of the IKZ considering melt and crystal only [7]. In connection with this, Fig. 4a shows the stream lines and isotherms in the melt of a 4 VCz growth situation in the CI 358
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
13
Fig. 2. Calculated global temperature "elds of a VCz arrangement inside the puller LPA Mark 3 carried out with the codes CrysVUN# # (a) and STHAMAS (b), in both cases without consideration of convective transport and with STHAMAS (c) including gas convection of argon gas at 0.9 MPa. The calculated stream "eld is added on the left side of (c). P indicates the calculated heater powers, the measured power consumption was 12.5 kW. The distances of the isotherms are 50 K in all the three cases.
puller obtained from global simulations by STHAMAS. In the demonstrated case, the rotation rates were chosen close to the results in Ref. [7] with 26 and 21 rpm for crystal and crucible, respectively. This was guided by the intention to nearly suppress the buoyancy-driven convection, i.e. that heat transfer by conduction becomes dominant within the melt. In fact, the isotherm pattern of the CrysVUN# # simulation of the same situation in Fig. 4b shows a quite similar result like in Fig. 4a although convection is not considered in this
model. In other words: for the presented VCz snapshot one can "nd appropriate rotation rates of crystal and crucible where the melt #ow plays only a minor role for the heat transport in the melt. Choosing such a priori conditions, the application of CrysVUN## is justi"ed because the in#uence of melt convection can be nearly neglected. This, in fact, was done in all of the simulations presented in the following. Additionally, the use of CrysVUN# # is advisable because of its better practicability compared to STHAMAS.
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Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
After learning the above facts, the thermal "eld inside the growing crystals was optimised for the relevant VCz set-ups using the CrysVUN# #
Fig. 3. Calculated axial temperature distribution (left) taken from Figs. 2b and c and values of the local temperature gradients. The global temperature "eld is added on the right side to correlate the axial coordinate with the interior of the VCz arrangement. The isotherms are separated by 25 K.
program. The main objectives were (i) reduction of the thermal stress due to the #attening of the interface boundary and (ii) homogenisation of the heat #ow through the crystal by adjusting it to be nearly uniaxial. To achieve this, the design of the thermal insulation of the VCz chamber, the height of boric oxide, the crucible position etc. were varied in wide ranges during the simulation procedure. Finally, results like illustrated in Fig. 5a were obtained. As expected, the nearly uniaxial heat #ow (nearly equidistant and #at isotherms) through the crystal guarantees relatively low values of the von Mises stress as demonstrated in Fig. 5a. The main part of the interface area and the core region of the growing crystal are exposed to very low stress, often
Fig. 5. 4 crystal in "nally optimised thermal conditions: (a) calculated using CrysVUN# #, the "gure shows isotherms, *¹"10 K (left) and lines of equal von Mises stress, separated by 0.25 MPa (right). (b) striation patterns of a real 4 GaAs crystal, grown under those optimised conditions, cut parallel to its growth axis [0 0 1].
Fig. 4. Comparison of a typical 4 growth arrangement calculated with: (a) STHAMAS; crystal and crucible rotation rates are 26 and 21, respectively, the "gure shows convection stream lines (left) and isotherms (right). (b) CrysVUN# #; without convective transport of heat, isotherm are shown. In both (a) and (b) the isotherms are separated by 12 K. The stream lines in left side of (a) are normalised between 0 and 1.
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
15
below the critical resolved shear stress (CRSS [kPa]"22.72 exp(4334/¹ ) [8], i.e. approximately 0.5 MPa near the melting point of GaAs). However, near the crystal edge stress peaks of 1.8 and 3.7 MPa appear. They are mainly caused by the discontinuities of the material properties like conduction and emissivity (solid and liquid GaAs, boric oxide). These e!ects are well known from the conventional LEC process [9] where the absolute stress values are, however, higher by of about one order of magnitude than in VCz crystals.
4. Experimental veri5cation To verify the above results experimentally, the inner VCz constructions were manufactured according to this. As shown in Fig. 5b the real curvature of the interface, obtained by the striation technique, is close to the calculated one in Fig. 5a. Please note that several stages of growth were simulated while only one stage is shown in Fig. 5 exemplarly. Fig. 6 shows two typical radial EPD distributions of 4 VCz crystals (lower curves) pulled in the computer-tailored growth arrangement. For comparison the EPD distribution of a conventional 3 LEC crystal is added. Obviously, the optimisation in the VCz case leads to an EPD reduction of about one order of magnitude. Further, the radial distribution is found to be more U-shaped instead of the well-known W-shape in conventional LEC crystals. The EPDs in Fig. 6 show a good qualitative correlation with the calculated stress "eld of Fig. 5a, i.e. within the main area of the slices the EPD is rather low and homogeneously distributed. The lowest dislocation densities obtained are at about some 10 cm\. In these areas the most dislocations are randomly distributed and only rarely polygonised. Only near the crystal edge the dislocation density increases to maximum values of about 5;10 cm\ where, again, the typical cellular structure appears. However, if the well-known cell structure occurs their dimensions are markedly larger (1}2 mm) than in typical LEC-material (ca. 0.2}0.5 mm). A further sensitive indicator of the interface curvature is the radial distribution of the shallow acceptor carbon [C ], in#uenced by segregation.
Fig. 6. Radial distributions of the etch pit density (EPD) along 11 0 02 and 11 1 02 directions in 4 crystals grown by the optimised VCz conditions according to Fig. 5. For comparison the EPD distribution of a 3 LEC crystal, grown in the authors laboratory, is added.
Fig. 7. Radial carbon distributions across 3 (䉬) and 4 (䊏) VCz-grown wafers analysed by the LVM IR-absorption method.
Precondition for a nearly uniform radial [C ] distribution like the two shown in Fig. 7 is a nearly #at interface shape during growth [10]. At a mean value of 7.6;10 cm\ a standard deviation of only 7% was measured along the wafer diameter. In general, it is possible to control the total carbon content in VCz GaAs crystals in the range between some 10 and 10 cm\ (note this is markedly below the limits being reported for VCz crystals so far [2,11]). More details will be given in Ref. [10]. Unfortunately, the resulting resistivity distribution does not only follow the homogeneous carbon distribution, but is also in#uenced by the EPD
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Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
Fig. 8. Radial distributions of the electrical resistivity o (a) and carrier mobility k (b) in 3 (䉬) and 4 (䊏) VCz crystals corresponding with those of Fig. 7.
distribution via EL2 [6,12]. This is illustrated in Fig. 8a. While the resistivity is roughly uniform in the middle part of the wafer it decreases towards the edges due to the increasing EPD (increasing EL2 and, thus, carrier concentration). However, this is only the case in as-grown crystals. The EL2 and carrier concentrations can be homogenised by subsequent bulk annealing very e!ectively [6]. Fig. 8b shows radial distributions of the electron mobility corresponding to Fig. 8a. In the 3 wafer it is quite high for as-grown crystals and corresponds to the maximum of the mobility}resistivity curve of SI GaAs at o+2;10 ) cm [13].
5. Conclusions Global modelling of the thermal "eld and the von Mises stress by using the code CrysVUN# # was successfully used for the tailoring of the growth arrangements of 3 and 4 VCz crystals. The gas convection in a VCz equipment was calculated for the "rst time by applying the code STHAMAS. Comparing the results of both programs no considerable di!erences were observed for the thermal
"elds inside the crystals. Therefore it can be concluded that the heat transport by gas convection plays a minor role in the VCz growth mode because of (i) the smaller volume of the inner chamber and (ii) the thermal shielding of the crystal from the water-cooled walls of the outer vessel. This makes the VCz arrangement completely di!erent from the conventional LEC where the gas convection is of marked in#uence on the thermal "eld inside the crystal. Contrary to this, melt convection cannot be neglected in the VCz growth mode. Best growth results were obtained if the melt convection is nearly suppressed by choosing appropriate crucible and crystal rotation rates. Under those preconditions, CrysVUN# # is justi"ed to be applied for the optimisation of the thermal "eld inside the apparatus, i.e. reduction of thermally induced stress. The "nally obtained conditions are characterised by low dislocation densities with a good radial homogeneity of its density distribution. The obtained #at growth interfaces are additionally indicated by the uniform radial carbon concentration distribution. Electrical parameters are in the expected range of typical SI GaAs. Their good radial homogeneity is promising to be again improved after bulk annealing.
Acknowledgements The authors are indebted to the following co-workers of the IKZ Berlin: M. Pietsch and M. Czupalla for growing the crystals, B. Lux for crystal preparation, S. Bergmann for EPD analysis, and K. Irmscher for Hall measurements. They are grateful to Prof. W. Ulrici (Paul-Drude Institut fuK r FestkoK rperelektronik, Berlin) for the analysis of the radial carbon distribution by LVM IR-absorption. This work was partially supported by the German Federal Ministry of Education, Science, Research & Technology under contract No. 5224001-01 BM 501/0.
Appendix Tables 1}3 show the data used for simulations with STHAMAS and CrysVUN# #.
Ch. Frank et al. / Journal of Crystal Growth 213 (2000) 10}18
17
Table 1 Global temperature "eld simulations [14,15] Material Graphites EK90 EK98 EK51 CFC (radial) CFC (axial) CBCF (radial) CBCF (axial) CBCF (rad.) CBCF (axial) Graphite felt Sigra#ex (foil) Others Non-corrosive steel CrNi Mo Al O PBN (radial) (axial) Fused silica B O GaAs(solid) GaAs(liquid) As(solid) Argon gas
Thermal conductivity [W/mK]
Emissivity
65(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 90(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 25(1.286!0.10909;10\ ¹ # 0.40159;10\ ¹ !0.47697;10\ ¹) 9.3175 # 2.5;10\ ¹ 20 0.2489 #4.166;10\ ¹ # 3.537;10\ ¹ !2.976;10\ ¹ 3 ) CBCF (radial) 3 ) CBCF (axial) !0.1011 #8.0301;10\ ¹ !5.386;10\ ¹ !1.7857;10\ ¹ 4.3;10\ # 3.24;10\ ¹ # 9.4;10\ ¹ 218.25 ! 0.25176 ¹ # 1.3165;10\ ¹ ! 0.23433;10\ ¹
0.8 0.8 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.7 0.58
15 25 105 2.15 3.135 62.7 0.94 # 0.001348 ¹ 0.237 # 0.0011 ¹ 7.12 17.8 50 1.8;10\ # 5.885;10\ ¹ ! 2.249;10\ ¹ # 4.921;10\ ¹
0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.75 0.55 0.55 0.5 0.5
Table 2 Data used for calculation of gas convection [16,17] k!e constants c l C e C e R R e Grashof number, Gr
0.09 1.44 1.92 1.0 1.3 6;10
Table 3 Data used for calculation of melt convection [14] Grashof number, Gr Kinematic viscosity, l Crystal rotation rates Crucible rotation rates
1.7;10 0.00488 cm/s 5}26 rpm 10}25 rpm
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[11] M. Tatsumi, T. Kawase, Y. Iguchi, K. Fujita, M. Yamada, in: M. Godlewski (Ed.), Semi-Insulating III}V Materials, World Scienti"c, Singapore, 1994, p. 11. [12] S. Miyazawa, T. Mizutani, H. Yamazaki, Jpn. J. Appl. Phys. 21 (1982) L542. [13] T. Flade, M. Jurisch, A. Kleinwechter, A. KoK hler, U. Kretzer, J. Prause, Th. Reinhold, B. Weinert, J. Crystal Growth 198/199 (1999) 336.
[14] http://www.ikz-berlin.de/&kb/daten/mat-1/mat.html [15] M. Kurz, Ph.D. Thesis, University of Erlangen-NuK rnberg, 1998. [16] B.E. Launder, D.B. Spalding, Appl. Mech. Eng. 3 (1974) 269. [17] R. Henkes, C.J. Hoogendorn, Int. J. Heat Transfer 32 (1989) 157.