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Optimisation of the VGF growth process by inverse modelling
ARTICLE IN PRESS Journal of Crystal Growth 312 (2010) 2175–2178
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Optimisation of the VGF growth process by inverse modelling a b ¨ ¨ M.P. Bellmann a,b,, O. Patzold , M. Stelter a, H.J. Moller a b
Institut f¨ ur NE-Metallurgie und Reinststoffe, TU BAF, Leipziger Str. 34, 09596 Freiberg, Germany Institut f¨ ur Experimentelle Physik, TU BAF, Leipziger Str. 23, 09599 Freiberg, Germany
a r t i c l e in fo
abstract
Article history: Received 19 January 2010 Received in revised form 16 April 2010 Accepted 19 April 2010 Communicated by A.G. Ostrogorsky Available online 22 April 2010
Numerical and experimental results on the thermal optimisation of vertical gradient freeze crystal growth are presented. An inverse modelling approach is described aimed at solidification with a constant growth rate and planar solid–liquid interface. As a result of modelling an optimised growth process characterised by a modified ampoule configuration and thermal regime was established. For experimental confirmation Ga-doped germanium single crystals were grown with the optimised process. In good agreement with the numerical results, solidification with an almost constant growth rate was achieved with the interface deflection being significantly lower than in conventionally grown crystals. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Inverse problem A1. Numerical simulation A1. Optimization A2. Vertical gradient freeze technique B2. Semiconducting germanium
1. Introduction Modelling of crystal growth is often aimed at the computation of the temperature field in the crystal/melt at given thermal boundary conditions. In process optimisation, however, the reverse issue is usually of particular interest: which temperatures have to be given to achieve solidification under certain, favourable conditions? Such problems can be addressed with an inverse simulation procedure. In [1–3] the inverse method was applied to optimise the GaAs crystal growth in a vertical gradient freeze (VGF) furnace with focus on the reduction of thermoelastic stress at the solid–liquid interface and the related dislocation density in the grown crystal. Inverse numerical simulation of vertical Bridgman (VB) growth including crystal translation was presented in [4]. It was also used for optimising the thermal conditions in float zone (FZ) growth of SiGe/GeSi single crystals [5]. As a further example, the onset of remelting in the Czochralski (Cz) growth of Ge crystals under the constraints of a constant pulling rate and given crystal shape was investigated by inverse modelling [6–8]. This paper reports on the optimisation of VGF growth of single-crystalline germanium by means of an inverse modelling approach. The study is aimed at solidification with a constant growth rate and planar solid–liquid interface which is of high interest for crystal growers with respect to, e.g., (i) increasing of the growth rate and hence, the yield of the process, (ii) growth Corresponding author at: Alfred Getzv. 2, 7491-Trondheim, Norway. Tel.: +47 735 97093; fax: + 47 735 50203. E-mail address: [email protected] (M.P. Bellmann).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.04.036
under minimised thermoelastic stress as already mentioned, and (iii) influencing of the dopant transport in the melt and thus, the segregation of dopants in the crystal. Numerical results are presented and compared with growth experiments designed on the basis of inverse modelling. Details of the numerical model and the growth set-up are described in the Section 2. Crystal growth including the method of validating the results of modelling are briefly addressed in Section 3. Numerical and experimental results are discussed in Section 4 and concluding remarks are given in Section 5.
2. Numerical procedure The inverse modelling algorithm implemented in CrysVUN++ [1,9] has been used to obtain an optimised VGF process with a constant growth rate of vG ¼5 mm h 1. The growth ampoule and furnace set-up which were already described elsewhere in detail [10,11], are shown in Fig. 1a,b. Numerical modelling is characterised by computing the heating powers of each furnace heater necessary to reach certain target temperatures given at certain points in the melt or crystal. The thermal conditions in the melt were defined by the control points x1,y,x3 indicated in Fig. 1b. The position of the solid–liquid interface is fixed in the centre of the crucible at a certain height (x1) by the melting point Tm ¼1211.4 K. In order to achieve a planar solid–liquid interface a second control point (x2) is set to Tm at the same radial position at the peripherie. To avoid instabilities of the interface a fixed temperature gradient of
ARTICLE IN PRESS 2176
M.P. Bellmann et al. / Journal of Crystal Growth 312 (2010) 2175–2178
Fig. 1. (a): Global model of the VGF-furnace consisting of seven heaters Z1–Z7. On the left-hand side the unstructured grid and on the right-hand side the calculated temperature field at the beginning of the solidification is shown. Isolines spacing: 100 K. The heaters are controlled by thermocouples indicated by black squares. (b): Sketch of the growth ampoule with the positions of temperature control points of inverse modelling. (c): Sketch of the dummy ampoule used for validating the numerical results (see Section 3).
5 K cm 1 in the melt is set by the control point (x3) 1 cm above the solid–liquid interface. The control point arrangement was vertically shifted in steps of 1 cm to simulate the solidification. In each position (xn + 1) the heating power in the seven heating zones is calculated. In the seeding stage the growth rate is set to zero, whereas the release of latent heat was considered by a growth rate of 5 mm h 1 during growth. In the final stage where the melting isotherm has reached the graphite cap (see Fig. 1b), the growth rate was again set to zero. Linearly interpolated heater temperatures of several interface positions were taken as input data of the transient calculation just described. The corresponding process times were calculated from the positions via the constant growth rate. Thermophysical material properties used in the simulation are summarised in Table 1.
3. Experimental The numerical results were validated by means of dummy experiments using the ampoule shown in Fig. 1c. The dummy consists of a massive graphite body with the diameter of the crucible which is fixed in a sealed, evacuated silica ampoule. A centric channel allows a thermocouple to be shifted vertically to measure axial temperature profiles as a function of the furnace heater temperatures. Ga doped Ge single crystals with a diameter of 15 mm and a doping concentration of about 5 1017 cm 3 were grown under conventional and optimised thermal conditions, where conventional refers to typical VGF growth with a constant translation rate of the imposed temperature profile. The experimental growth rate was calculated from the positions of dopant striations induced during growth by means of repeated pulses of a rotating magnetic field. Such artificial interface demarcations can be easily detected on etched longitudinal slices of the crystals [11].
4. Results and discussion Fig. 2 shows experimental and numerical results of the growth rate in a conventional VGF process performed with a
Table 1 Material parameters used in the simulation. Germanium liquid Thermal conductivity
39 W (m 1 K 1)
l Specific heat cp Density r Melting point TS Heat of fusion z
393 J (kg 1 K 1) 5.51 103 kg m 3 1211.4 K 700 000 J Kg 1
Germanium solid Thermal conductivity
17 W (m 1 K 1)
l Specific heat cp Density r Emissivity
380 J (kg 1 K 1) 5.32 103 kg m 3 0.55
Graphite R6300 (EK90) Thermal conductivity 35 W (m 1 K 1)
l Emissivity
0.8
Pyrolitic boronnitrid Thermal conductivity
lJ ¼ 62:2, l? ¼ 2:1 W (m 1 K 1)
l Emissivity
Silica Thermal conductivity
l Emissivity Transparency
0.435
0.133 10 5 T2 0.000144579 T+ 1.350259 W (m 1 K 1) 0.1 260 nm22:6 mm
constant cooling rate of 6 mm h 1. The growth rate increases continuously but remains always below the imposed cooling rate. This can be mainly attributed to the latent heat released on solidification which is not fully transferred through the crystal leading to inhibited growth. In qualitative agreement with the experimental results the simulation gives crystallisation with an increasing growth rate. The differences between measured and calculated values in Fig. 2 indicate that inaccurate material data were used in process simulation. For example, heat aging of the furnace insulation and
ARTICLE IN PRESS M.P. Bellmann et al. / Journal of Crystal Growth 312 (2010) 2175–2178
recrystallisation of the silica ampoule surface can change the thermal properties of the set-up significantly. Attempts to optimise the growth with the standard ampoule set-up shown in Fig. 1b led to unrealistic thermal profiles because the maximum temperature differences between adjacent heating zones of the furnace set to 100 K for constructive reasons were exceeded. Therefore, the ampoule has been modified by including an additional graphite bar below the crystal as can be seen in Fig. 3. As a consequence of the increased heat flux in the modified ampoule the overall temperature gradient is reduced and the thermal constraints of the furnace could be fulfilled. Time-dependent heater temperatures of conventional and optimised growth are compared in Fig. 4. Discrete sets of heater temperatures from quasi-steady calculations of the optimised process are indicated by triangles. Each set corresponds to a certain position of the solid–liquid interface during growth. These values are taken as input data for the transient simulation resulting in the dashed profiles of Fig. 4. Compared to the conventional process, growth under thermally optimised conditions can be established by an initial increase of the temperatures of the heaters Z5–Z7 and decrease of the temperatures of Z2–Z4. Thus, the seeding stage of the optimised process
Fig. 2. Numerical and experimental obtained growth rate for the conventional growth set-up (see Fig. 1b) and thermal regime (see Fig. 4 below). Numerical values has been obtained from the transient simulation.
Fig. 3. (a) Modified growth ampoule configuration and (b) the related dummy ampoule.
2177
is characterised by a high axial temperature gradient in the furnace which is gradually decreased during further growth. The results of the dummy experiments are presented in Figs. 5 and 6. In Fig. 5 temperature profiles measured at different stages of the growth are compared with the profiles predicted by numerical simulation. For the experimentally relevant part indicated by the sketch of the ampoule in Fig. 5, the differences of the temperature profiles are smaller than 5 K. Good agreement between dummy experiments and numerical results were also obtained measuring axial temperature profiles as a function of process time which are shown in Fig. 6. In this case the differences are below 4 K. Results of Ge:Ga crystal growth using the modified ampoule set-up under optimised thermal conditions are shown in Fig. 7 and Table 2. The experimental growth rate is in a good qualitative agreement with the numerically calculated one. Due to the high axial temperature gradient established at the beginning of the process (see Fig. 4) a rapid increase and even a slight ‘overshooting’ is observed both in experiment and simulation. Apart from this initial effect the growth rate is nearly constant
Fig. 4. Heater temperatures versus process time of the optimised process (triangles: discrete temperature sets obtained from a quasi-steady calculation, dashed lines: temperature profiles from the inverse simulation—see text) in comparison with the temperature profiles of the conventional process (solid lines).
Fig. 5. Comparison of numerical and experimental temperature profiles at different stages of the optimised growth process.
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Table 2 Comparison of interface deflections Dz in Ge:Ga single crystals grown in a conventional and optimised VGF process. Position solidified fraction (g)
Conventional Dz (mm)
Optimised Dz (mm)
0 0.26 0.33 0.38 0.55 0.6
– – 0.69 – 0.71 –
0.14 0.21 – 0.19 – 0.19
The values of the conventionally grown crystals are taken from [10].
Fig. 6. Numerical and experimental temperature profiles vs. the process time of optimised growth at different axial positions which are indicated in the schema of the dummy ampoule on the right.
compared. At all positions denoted the interfaces are concavely bent as seen from the melt. Optimised growth, however, leads to a significant reduction of the interface deflection by about 70%, because the axial heat flux is much more efficient than in conventional growth.
5. Summary Inverse numerical simulation has been used to improve VGF growth of cylindrical germanium single crystals with a diameter of 15 mm with respect to solidification with a constant growth rate and as planar as possible solid–liquid interface. As a result an optimised process with modified ampoule set-up and thermal regime was established. It essentially bases on crystallisation under an increased axial heat flux which was achieved (i) by an additional graphite bar below the crystal seed, and (ii) by an increased axial temperature gradient established by the furnace heaters. Growth experiments under optimised conditions show that the target parameters of inverse modelling are approximately reached: Solidification proceeds with an almost constant growth rate and nearly planar melting isotherm. Fig. 7. Comparison of numerical and experimental growth rates for the optimised growth set-up and thermal regime.
References
throughout the process. The experimental growth rate, however, is found to be about 6.5 mm h 1 compared to 5 mm h 1 of the numerical modelling. This is a further indication of inaccurate material data available as already discussed above in the context of Fig. 2. A microscopical inspection of demarcated melting isotherms reveals that the numerically given target of a planar solid–liquid interface during growth is approximately reached in the optimised VGF process. In Table 2 interface deflections in Ge:Ga crystals grown under optimised and conventional conditions are
¨ [1] M. Kurz, G. Muller, J. Cryst. Growth 208 (2000) 341. ¨ [2] B. Birkmann, M. Rasp, J. Stenzenberger, G. Muller, J. Cryst. Growth 211 (2000) 157. [3] M. Metzger, J. Cryst. Growth 230 (2001) 210. [4] M. Margulies, P. Witomski, T. Duffar, J. Cryst. Growth 266 (2004) 175. [5] K. Lin, P. Dold, K.W. Benz, Cryst. Res. Technol. 38 (2003) 149. [6] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 166 (1996) 446. [7] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 171 (1997) 65. [8] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 171 (1997) 77. ¨ [9] M. Kurz, A. Pusztai, G. Muller, J. Cryst. Growth 198 (1999) 101. [10] U. Wunderwald, Ph.D. Thesis, TU Freiberg, 2006. ¨ ¨ ¨ [11] M.P. Bellmann, O. Patzold, G. Gartner, H.J. Moller, M. Stelter, J. Cryst. Growth 311 (2009) 1471.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Optimisation of the VGF growth process by inverse modelling a b ¨ ¨ M.P. Bellmann a,b,, O. Patzold , M. Stelter a, H.J. Moller a b
Institut f¨ ur NE-Metallurgie und Reinststoffe, TU BAF, Leipziger Str. 34, 09596 Freiberg, Germany Institut f¨ ur Experimentelle Physik, TU BAF, Leipziger Str. 23, 09599 Freiberg, Germany
a r t i c l e in fo
abstract
Article history: Received 19 January 2010 Received in revised form 16 April 2010 Accepted 19 April 2010 Communicated by A.G. Ostrogorsky Available online 22 April 2010
Numerical and experimental results on the thermal optimisation of vertical gradient freeze crystal growth are presented. An inverse modelling approach is described aimed at solidification with a constant growth rate and planar solid–liquid interface. As a result of modelling an optimised growth process characterised by a modified ampoule configuration and thermal regime was established. For experimental confirmation Ga-doped germanium single crystals were grown with the optimised process. In good agreement with the numerical results, solidification with an almost constant growth rate was achieved with the interface deflection being significantly lower than in conventionally grown crystals. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Inverse problem A1. Numerical simulation A1. Optimization A2. Vertical gradient freeze technique B2. Semiconducting germanium
1. Introduction Modelling of crystal growth is often aimed at the computation of the temperature field in the crystal/melt at given thermal boundary conditions. In process optimisation, however, the reverse issue is usually of particular interest: which temperatures have to be given to achieve solidification under certain, favourable conditions? Such problems can be addressed with an inverse simulation procedure. In [1–3] the inverse method was applied to optimise the GaAs crystal growth in a vertical gradient freeze (VGF) furnace with focus on the reduction of thermoelastic stress at the solid–liquid interface and the related dislocation density in the grown crystal. Inverse numerical simulation of vertical Bridgman (VB) growth including crystal translation was presented in [4]. It was also used for optimising the thermal conditions in float zone (FZ) growth of SiGe/GeSi single crystals [5]. As a further example, the onset of remelting in the Czochralski (Cz) growth of Ge crystals under the constraints of a constant pulling rate and given crystal shape was investigated by inverse modelling [6–8]. This paper reports on the optimisation of VGF growth of single-crystalline germanium by means of an inverse modelling approach. The study is aimed at solidification with a constant growth rate and planar solid–liquid interface which is of high interest for crystal growers with respect to, e.g., (i) increasing of the growth rate and hence, the yield of the process, (ii) growth Corresponding author at: Alfred Getzv. 2, 7491-Trondheim, Norway. Tel.: +47 735 97093; fax: + 47 735 50203. E-mail address: [email protected] (M.P. Bellmann).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.04.036
under minimised thermoelastic stress as already mentioned, and (iii) influencing of the dopant transport in the melt and thus, the segregation of dopants in the crystal. Numerical results are presented and compared with growth experiments designed on the basis of inverse modelling. Details of the numerical model and the growth set-up are described in the Section 2. Crystal growth including the method of validating the results of modelling are briefly addressed in Section 3. Numerical and experimental results are discussed in Section 4 and concluding remarks are given in Section 5.
2. Numerical procedure The inverse modelling algorithm implemented in CrysVUN++ [1,9] has been used to obtain an optimised VGF process with a constant growth rate of vG ¼5 mm h 1. The growth ampoule and furnace set-up which were already described elsewhere in detail [10,11], are shown in Fig. 1a,b. Numerical modelling is characterised by computing the heating powers of each furnace heater necessary to reach certain target temperatures given at certain points in the melt or crystal. The thermal conditions in the melt were defined by the control points x1,y,x3 indicated in Fig. 1b. The position of the solid–liquid interface is fixed in the centre of the crucible at a certain height (x1) by the melting point Tm ¼1211.4 K. In order to achieve a planar solid–liquid interface a second control point (x2) is set to Tm at the same radial position at the peripherie. To avoid instabilities of the interface a fixed temperature gradient of
ARTICLE IN PRESS 2176
M.P. Bellmann et al. / Journal of Crystal Growth 312 (2010) 2175–2178
Fig. 1. (a): Global model of the VGF-furnace consisting of seven heaters Z1–Z7. On the left-hand side the unstructured grid and on the right-hand side the calculated temperature field at the beginning of the solidification is shown. Isolines spacing: 100 K. The heaters are controlled by thermocouples indicated by black squares. (b): Sketch of the growth ampoule with the positions of temperature control points of inverse modelling. (c): Sketch of the dummy ampoule used for validating the numerical results (see Section 3).
5 K cm 1 in the melt is set by the control point (x3) 1 cm above the solid–liquid interface. The control point arrangement was vertically shifted in steps of 1 cm to simulate the solidification. In each position (xn + 1) the heating power in the seven heating zones is calculated. In the seeding stage the growth rate is set to zero, whereas the release of latent heat was considered by a growth rate of 5 mm h 1 during growth. In the final stage where the melting isotherm has reached the graphite cap (see Fig. 1b), the growth rate was again set to zero. Linearly interpolated heater temperatures of several interface positions were taken as input data of the transient calculation just described. The corresponding process times were calculated from the positions via the constant growth rate. Thermophysical material properties used in the simulation are summarised in Table 1.
3. Experimental The numerical results were validated by means of dummy experiments using the ampoule shown in Fig. 1c. The dummy consists of a massive graphite body with the diameter of the crucible which is fixed in a sealed, evacuated silica ampoule. A centric channel allows a thermocouple to be shifted vertically to measure axial temperature profiles as a function of the furnace heater temperatures. Ga doped Ge single crystals with a diameter of 15 mm and a doping concentration of about 5 1017 cm 3 were grown under conventional and optimised thermal conditions, where conventional refers to typical VGF growth with a constant translation rate of the imposed temperature profile. The experimental growth rate was calculated from the positions of dopant striations induced during growth by means of repeated pulses of a rotating magnetic field. Such artificial interface demarcations can be easily detected on etched longitudinal slices of the crystals [11].
4. Results and discussion Fig. 2 shows experimental and numerical results of the growth rate in a conventional VGF process performed with a
Table 1 Material parameters used in the simulation. Germanium liquid Thermal conductivity
39 W (m 1 K 1)
l Specific heat cp Density r Melting point TS Heat of fusion z
393 J (kg 1 K 1) 5.51 103 kg m 3 1211.4 K 700 000 J Kg 1
Germanium solid Thermal conductivity
17 W (m 1 K 1)
l Specific heat cp Density r Emissivity
380 J (kg 1 K 1) 5.32 103 kg m 3 0.55
Graphite R6300 (EK90) Thermal conductivity 35 W (m 1 K 1)
l Emissivity
0.8
Pyrolitic boronnitrid Thermal conductivity
lJ ¼ 62:2, l? ¼ 2:1 W (m 1 K 1)
l Emissivity
Silica Thermal conductivity
l Emissivity Transparency
0.435
0.133 10 5 T2 0.000144579 T+ 1.350259 W (m 1 K 1) 0.1 260 nm22:6 mm
constant cooling rate of 6 mm h 1. The growth rate increases continuously but remains always below the imposed cooling rate. This can be mainly attributed to the latent heat released on solidification which is not fully transferred through the crystal leading to inhibited growth. In qualitative agreement with the experimental results the simulation gives crystallisation with an increasing growth rate. The differences between measured and calculated values in Fig. 2 indicate that inaccurate material data were used in process simulation. For example, heat aging of the furnace insulation and
ARTICLE IN PRESS M.P. Bellmann et al. / Journal of Crystal Growth 312 (2010) 2175–2178
recrystallisation of the silica ampoule surface can change the thermal properties of the set-up significantly. Attempts to optimise the growth with the standard ampoule set-up shown in Fig. 1b led to unrealistic thermal profiles because the maximum temperature differences between adjacent heating zones of the furnace set to 100 K for constructive reasons were exceeded. Therefore, the ampoule has been modified by including an additional graphite bar below the crystal as can be seen in Fig. 3. As a consequence of the increased heat flux in the modified ampoule the overall temperature gradient is reduced and the thermal constraints of the furnace could be fulfilled. Time-dependent heater temperatures of conventional and optimised growth are compared in Fig. 4. Discrete sets of heater temperatures from quasi-steady calculations of the optimised process are indicated by triangles. Each set corresponds to a certain position of the solid–liquid interface during growth. These values are taken as input data for the transient simulation resulting in the dashed profiles of Fig. 4. Compared to the conventional process, growth under thermally optimised conditions can be established by an initial increase of the temperatures of the heaters Z5–Z7 and decrease of the temperatures of Z2–Z4. Thus, the seeding stage of the optimised process
Fig. 2. Numerical and experimental obtained growth rate for the conventional growth set-up (see Fig. 1b) and thermal regime (see Fig. 4 below). Numerical values has been obtained from the transient simulation.
Fig. 3. (a) Modified growth ampoule configuration and (b) the related dummy ampoule.
2177
is characterised by a high axial temperature gradient in the furnace which is gradually decreased during further growth. The results of the dummy experiments are presented in Figs. 5 and 6. In Fig. 5 temperature profiles measured at different stages of the growth are compared with the profiles predicted by numerical simulation. For the experimentally relevant part indicated by the sketch of the ampoule in Fig. 5, the differences of the temperature profiles are smaller than 5 K. Good agreement between dummy experiments and numerical results were also obtained measuring axial temperature profiles as a function of process time which are shown in Fig. 6. In this case the differences are below 4 K. Results of Ge:Ga crystal growth using the modified ampoule set-up under optimised thermal conditions are shown in Fig. 7 and Table 2. The experimental growth rate is in a good qualitative agreement with the numerically calculated one. Due to the high axial temperature gradient established at the beginning of the process (see Fig. 4) a rapid increase and even a slight ‘overshooting’ is observed both in experiment and simulation. Apart from this initial effect the growth rate is nearly constant
Fig. 4. Heater temperatures versus process time of the optimised process (triangles: discrete temperature sets obtained from a quasi-steady calculation, dashed lines: temperature profiles from the inverse simulation—see text) in comparison with the temperature profiles of the conventional process (solid lines).
Fig. 5. Comparison of numerical and experimental temperature profiles at different stages of the optimised growth process.
ARTICLE IN PRESS 2178
M.P. Bellmann et al. / Journal of Crystal Growth 312 (2010) 2175–2178
Table 2 Comparison of interface deflections Dz in Ge:Ga single crystals grown in a conventional and optimised VGF process. Position solidified fraction (g)
Conventional Dz (mm)
Optimised Dz (mm)
0 0.26 0.33 0.38 0.55 0.6
– – 0.69 – 0.71 –
0.14 0.21 – 0.19 – 0.19
The values of the conventionally grown crystals are taken from [10].
Fig. 6. Numerical and experimental temperature profiles vs. the process time of optimised growth at different axial positions which are indicated in the schema of the dummy ampoule on the right.
compared. At all positions denoted the interfaces are concavely bent as seen from the melt. Optimised growth, however, leads to a significant reduction of the interface deflection by about 70%, because the axial heat flux is much more efficient than in conventional growth.
5. Summary Inverse numerical simulation has been used to improve VGF growth of cylindrical germanium single crystals with a diameter of 15 mm with respect to solidification with a constant growth rate and as planar as possible solid–liquid interface. As a result an optimised process with modified ampoule set-up and thermal regime was established. It essentially bases on crystallisation under an increased axial heat flux which was achieved (i) by an additional graphite bar below the crystal seed, and (ii) by an increased axial temperature gradient established by the furnace heaters. Growth experiments under optimised conditions show that the target parameters of inverse modelling are approximately reached: Solidification proceeds with an almost constant growth rate and nearly planar melting isotherm. Fig. 7. Comparison of numerical and experimental growth rates for the optimised growth set-up and thermal regime.
References
throughout the process. The experimental growth rate, however, is found to be about 6.5 mm h 1 compared to 5 mm h 1 of the numerical modelling. This is a further indication of inaccurate material data available as already discussed above in the context of Fig. 2. A microscopical inspection of demarcated melting isotherms reveals that the numerically given target of a planar solid–liquid interface during growth is approximately reached in the optimised VGF process. In Table 2 interface deflections in Ge:Ga crystals grown under optimised and conventional conditions are
¨ [1] M. Kurz, G. Muller, J. Cryst. Growth 208 (2000) 341. ¨ [2] B. Birkmann, M. Rasp, J. Stenzenberger, G. Muller, J. Cryst. Growth 211 (2000) 157. [3] M. Metzger, J. Cryst. Growth 230 (2001) 210. [4] M. Margulies, P. Witomski, T. Duffar, J. Cryst. Growth 266 (2004) 175. [5] K. Lin, P. Dold, K.W. Benz, Cryst. Res. Technol. 38 (2003) 149. [6] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 166 (1996) 446. [7] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 171 (1997) 65. [8] N. Van den Bogaert, F. Dupret, J. Cryst. Growth 171 (1997) 77. ¨ [9] M. Kurz, A. Pusztai, G. Muller, J. Cryst. Growth 198 (1999) 101. [10] U. Wunderwald, Ph.D. Thesis, TU Freiberg, 2006. ¨ ¨ ¨ [11] M.P. Bellmann, O. Patzold, G. Gartner, H.J. Moller, M. Stelter, J. Cryst. Growth 311 (2009) 1471.