Use of computer modeling for optimization of
    Cz Si growth: strategy and examples




                 animated visuali...
Prehistory of STR:
1984:    Start of the MOCVD modeling activities
         at Ioffe Institute, St. Petersburg, Russia;

1...
STR Today:
modeling of crystal growth, epitaxy, and devices

- STR Group, HQ and R&D, St.Petersburg, Russia
- STR Inc., Ri...
Content
Introduction

- Description of Computer Model

- Verification Results

- Strategy of Cz Si Growth Optimization

- ...
INTRODUCTION
                                        There are 3 basic techniques for silicon
                            ...
What is computer modeling?
Computer modeling is a comprehensive reconstruction or reproduction of
crystal growth process b...
Published CGSim successful stories

- Development of perfect 300 mm silicon crystal growth by the Czochralski technique in...
Content
Introduction

- Description of Computer Model

- Verification Results

- Strategy of Cz Si Growth Optimization

- ...
Navier-Stokes equations, heat transfer, electric current,
                                                               s...
Original approach for turbulence modeling: combined
                                                                     L...
Specific boundary conditions
(i) fixing velocity on solid               rr                                                ...
Strategy of coupled 2D/3D approach
 2D global HT model produces thermal boundary conditions for 3D/2D flow computations
  ...
Study of crystal growth using 2D models
              Temperature analysis coupled
                                       ...
Study of melt flow structure and crystallization using 2D
                                             models
            ...
Analysis of unsteady features of crystal cooling
      Crystal cooling at 450 mm Cz Si growth (animated visualization)
   ...
Content
Introduction

- Description of Computer Model

- Verification Results

- Strategy of Cz Si Growth Optimization

- ...
Temperature measurements in a Cz furnace
                                                                                 ...
Temperature measurements in the melt
T [K]
                                        no magnetic field (animated visualizati...
Temperature measurements in the melt by an optical sensor
                                                                ...
3D Unsteady Calculations of the Crystallization Front
                                                                    ...
Content
Introduction

- Description of Computer Model

- Verification Results

- Strategy of Cz Si Growth Optimization

- ...
Increasing the crystallization (pulling) rate

- by modifications of the heat shields and other elements surrounding
the c...
Effect of the heat shield modifications




    Initial heat shield                Optimization of HSh shape and compositi...
Reducing the heater power

- to find weak points in the insulation design by
analyzing the heat flux vectors and integral ...
Content
Introduction

- Description of Computer Model

- Verification Results

- Strategy of Cz Si Growth Optimization

- ...
Growth of Cz Si crystals the with
diameter of 400 mm for electronics:
effect of DC magnetic fields on the
   melt and crys...
II.II. 3D Unsteady Parametric Analysis of
                                                              400 mm Si Crystal ...
Modeling Results for Case 1: without MF
                                        all plots here are animated to visualize u...
Modeling Results for Case 2: with Cusp MF of 30 mT
                                        all plots here are animated to ...
Modeling Rresults for Case 3: with Horizontal MF of 30 mT
                                        all plots here are anima...
Modeling Results for Case 4: with Horizontal MF of 300 mT
             all plots here are animated to visualize unsteady e...
Modeling results for Case 4: with horizontal MF of 300 mT
                            all plots here are animated to visua...
Results for Case 5: Horizontal MF of 300 mT and with the
                                          Reduced Argon Flow Rate...
Time-Averaged Temperature Distributions over the Melt Free
          Surface and in Vertical Cross Sections for Cases 3, 4...
Comparison of the Computed Crystallization Front
                                                                       Ge...
I.VI. Defect incorporation and recombination
                              Initial defect kinetics:
dCm
    = ∇ ⋅(Dm∇Cm) +...
2D calculation of defect incorporation and recombination
                                                 (crystal diamete...
Defect clusterization
      Free energy gain for void and oxygen precipitate, volume and surface terms:
                  ...
Radial distribution of point defect density and average size
                                             6
              ...
Summary on modeling of Cz single Si growth
We have presented a combined 2D-3D model of Cz Si growth. The model
considers m...
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Modeling of Cz Si with CGSim

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Cz Si Growth Optimization
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Modeling of Cz Si with CGSim

  1. 1. Use of computer modeling for optimization of Cz Si growth: strategy and examples animated visualization of unsteady effects is not available in this edition of the presentation STR Group www.str-soft.com 2009
  2. 2. Prehistory of STR: 1984: Start of the MOCVD modeling activities at Ioffe Institute, St. Petersburg, Russia; 1993-96: Group for modeling of crystal growth and epitaxy at University of Erlangen-Nuernberg, Germany; History of R&D in STR on crystal growth from the melt: 1998: First cooperation project on Cz Si growth 2000: First cooperation project on encapsulated GaAs and InP growth 2001: Start of the development of CGSim package 2002: First cooperation project on Cz growth of oxides 2003: First cooperation project on Ge growth from the melt 2004: First industrial customers of CGSim package 2005: First cooperation projects on DS Si and Kyropoulos sapphire growth 2006: First cooperation projects on YAG and GGG (garnet) growth by Cz
  3. 3. STR Today: modeling of crystal growth, epitaxy, and devices - STR Group, HQ and R&D, St.Petersburg, Russia - STR Inc., Richmond VA, USA - STR GmbH, Erlangen, Germany - More than 30 scientists and software engineers in STR, local representatives in China, Korea, Taiwan and Japan. Software & consulting services : - Czochralski, LEC, and Bridgman crystal growth from the melt: CGSim - Polysilicon deposition by Siemens process: PolySim - PVT bulk crystal growth of SiC, AlN, GaN: Virtual Reactor - Modeling of epitaxy, MOVPE of Nitrides and III-Vs: CVDSim - Modeling of opto- and electronic devices: SimuLED, FETIS, BESST, SELES
  4. 4. Content Introduction - Description of Computer Model - Verification Results - Strategy of Cz Si Growth Optimization - Cz Growth with diameter of 400 mm for electronics
  5. 5. INTRODUCTION There are 3 basic techniques for silicon crystal growth from melt. DSS (Casting, multicrystal) Czochralski m. (monocrystal) A.T. Kuliev, N.V. Durnev, V.V. Kalaev, Y. Shiraishi et al., J. of Crystal J. of Crystal Growth (2007) 236-240 Growth 229 (2001) p.17
  6. 6. What is computer modeling? Computer modeling is a comprehensive reconstruction or reproduction of crystal growth process by a computer. Numerical model is used for technology optimization to lower operation costs. Advantages of computer modeling in comparison to experimental work: - fast and low cost (from few minutes to several days; only special software and computer are required) - provide information in all points of a furnace, including data which are impossible to measure quantitatively (temperature gradients, thermal stresses, crystallization behavior, fluid dynamics, species transport) -practically, no extra cost for changes of furnace design and growth parameters (several minutes of the work of the user): fast and efficient technology development
  7. 7. Published CGSim successful stories - Development of perfect 300 mm silicon crystal growth by the Czochralski technique in Siltronic AG, Germany - Optimization of impurity transport during EFG growth of dodecagonally shaped silicon tubes of 0.5 m diameter in SCHOTT Solar GmbH, Germany -Reduction of the macrodislocation generation probability in 3 times and 30% increase of the pulling speed during 200 mm Si Cz growth for solar cells in JSC PCMP, Russia -First technology for 300 mm sapphire growth by the Kyropoulos technique and reduction of dislocation density in Monocrystal Inc., Russia - Komatsu Metals Co., Japan (now a part of SUMCO) uses CGSim for optimization of large scale Cz Si growth for electronics
  8. 8. Content Introduction - Description of Computer Model - Verification Results - Strategy of Cz Si Growth Optimization - Cz Growth with diameter of 400 mm for electronics
  9. 9. Navier-Stokes equations, heat transfer, electric current, scalar transport p0 m ∂ρ r or ρ = f (T ) ρ= + ∇ ⋅ (ρ u) = 0 (1) Rg T ∂t r ∂( ρ u ) r r r + (u ⋅ ∇) ρ u = −∇p + ∇ ⋅τ + ( ρ − ρ 0 ) g + j × B + S u (2) ∂t ⎛ ∂ui ∂u j ⎞ 2 r ⎜ ⎟ − µ eff δ ij ∇ ⋅ u τ ij = µ eff ⎜ + ∂x j ∂xi ⎟ 3 ⎝ ⎠ ∂ (ρ c p T ) + ∇ ⋅ (ρ c p u T ) = ∇ ⋅ (λ eff ∇ T ) + S T r (3) ∂t ( ) ( ) j = σ − ∇Φ + u × B ∆Φ = B ⋅ ∇ × u (4) Electric potential equation ∂ ( ρϕ i ) ( ) r + ∇ ⋅ ( ρ u ϕ i ) = ∇ ⋅ ρ Dϕ i eff ∇ ϕ i + S ϕ i scalar (5) ∂t
  10. 10. Original approach for turbulence modeling: combined LES/RANS ⎡⎛ ν t ⎞ ∂k ⎤ σm 2 σm 2 k ∂ . dk ⎢⎜ν ⎟ ⎟ ∂x ⎥ + 2ν t S − ε − ρ B k ⋅ exp(−0.025 ρ B ε ) = + 2 ⎜ σk ⎠ j ⎥ dt ∂x j ⎢⎝ ⎣ ⎦ 3 ε = max(ε ), Ck 2 ,ε RANS LES Cε = 0.75 ε ∆ = 3 ∆ x∆ y∆ z , =ε LES , ∆ 3 k2 ky −3 ε l = κCµ 4 y, = Fε = f (Re y ), Re y = RANS , ν l ⋅ Fε ⎧ lt = min ⎨l ⋅ Fε , ∆ ⎫ ν t = C µ f µ lt k Cε ⎬ ⎭ ⎩ Wolfshtein model, 1969: ⎛ Re y ⎞ ⎛ Re y ⎞ f µ = 1 − exp⎜ − ⎟, Fε = 1 − exp⎜ − ⎜ A ⎟, Aε = 5.1 Aµ = 10.0 ⎟ ⎜ A⎟ ⎝ ε⎠ ⎝ µ⎠ N.G. Ivanov, A.B. Korsakov, E.M. Smirnov, K.V. Khodosevitch, V.V. Kalaev, Yu.N. Makarov, E. Dornberger, J. Virbulis, W. von Ammon, J. Crystal Growth, 250/1-2 (2003) pp. 183-188
  11. 11. Specific boundary conditions (i) fixing velocity on solid rr 15 m/s in the gas u = Uw walls and inlet boundaries: 0.015 m/s in the melt (ii) coupling of stresses along gas/liquid interfaces: W [m/s] 0.2 r r 0.15 (u )1 = ( u ) 2 0.1 0.05 0 -0.05 -0.1 ∂ uτ i ∂ uτ i ∂σ -0.15 -0.2 ( µ eff )1 = ( µ eff )2 + gradτ i (T ) ∂n ∂n ∂T ∂T ∂T (iii) heat transfer along the case of (λ )1 = (λ )2 solid/solid interfaces, ∂n ∂n opaque blocks liquid/liquid and (T )1 = ( T ) 2 + ∆Tgap liquid/solid interfaces : (iv) along opaque/transparent interface radiative heat transfer is accounted for: ∂T ∂T (λ ) opaque = (λ ) transparent + σ b ε rad Tw4 − Qrad in ∂n ∂n
  12. 12. Strategy of coupled 2D/3D approach 2D global HT model produces thermal boundary conditions for 3D/2D flow computations ⎛ ∂T ⎞ = Tmelting T and ⎜ λ in ⎟ Q melt / crystal rad ⎝ ∂n ⎠ gas 2D model of global heat transport T [K] 1600 1400 1200 1000 40 0 800 500 600 600 400 argon 1300 crystal 1500 1700 melt λeff melt crucible 3D unsteady model heater ⎛ ∂T ⎞ ⎛ ∂T ⎞ graphite 1700 λ + Qrad = ⎜ λ ⎟ + σεTwall in 4 ⎜ ⎟ 1600 ⎝ ∂n ⎠3 D ⎝ ∂n ⎠ gas insulation CZ growth of 400 mm Si crystal: V.V. Kalaev et al., J. Crystal Growth, 250/1-2 (2003) p.203 Y. Shiraishi et al., J. of Crystal V.V. Kalaev et al., Mat. Sci. in Semiconductor Processing, 5/4-5 (2003) p.369 Growth 229 (2001) p.17
  13. 13. Study of crystal growth using 2D models Temperature analysis coupled 450 mm Cz Si growth with melt and gas flow (animated visualization) Temperature Flow Velocity Crystallization front animation Global heat transfer simulation coupled to melt and gas convection is a powerful tool for day-by-day engineering calculations to improve hot zone design and growth parameters.
  14. 14. Study of melt flow structure and crystallization using 2D models 450 mm Cz Si growth animated visualization Temperature Flow Velocity In large scale crystal growth, the effect of melt convection on crystallization front formation is significant. Melt convection features often govern defects dynamics and macro-dislocation generation in the crystal.
  15. 15. Analysis of unsteady features of crystal cooling Crystal cooling at 450 mm Cz Si growth (animated visualization) Temperature Temperature gradient in the crystal Si charge meltdown and crystal cooling are optimized using 2D unsteady modeling. This helps to speed up melting and cooling stages and to prevent beginning of melt crystallization from the gas/melt interface.
  16. 16. Content Introduction - Description of Computer Model - Verification Results - Strategy of Cz Si Growth Optimization - Cz Growth with diameter of 400 mm for electronics
  17. 17. Temperature measurements in a Cz furnace 1400 calculation Temperature [ C ] experiment 0 1200 1000 800 0 100 200 300 Crystal Length [ mm ] a) calc. ins/ins exp. ins/ins calc. ins/graph 2000 Temperature [ 0C ] exp. ins/graph 1750 1500 1250 1000 0 100 200 300 400 b) Insulation Length [ mm ] calc. upper exp. upper 2000 Tempearture [ 0C ] calc. lower 1800 exp. lower 1600 1400 1200 1000 25 50 75 100 Insulation Length [ mm ] c) (b) (c) (a) The furnace geometry and experimental data are published in J. Crystal Growth 180 (1997) pp. 461- 467. The predicted temperature distribution (b) is compared to the experimental data (c) obtained in the points shown in (a). The comparison shows that CGSim can adequately predict temperature distribution in the industrial growth setup and inside the growing silicon crystal if material properties are well known.
  18. 18. Temperature measurements in the melt T [K] no magnetic field (animated visualization) 1750 experimental data 1745 90 grad 1740 3D computation 1735 2D computation 1730 1725 1720 1715 1710 1705 0 grad 1700 1695 1690 0 grad 1685 0 20 40 60 80 T [K] 1750 cusp magnetic field (animated visualization) 1745 experimental data 1740 3D computaion 2D computation 1735 1730 1725 1720 1715 1710 1705 1700 Angle from the axis of rotation [grad] 1695 1690 1685 0 20 40 60 80 The measurements by thermocouples along the melt/crucible interface are published in Microelectronic Engineering 56 (2001) 83–88. 3D calculations with CGSim predicts the experimental distributions quantitatively.
  19. 19. Temperature measurements in the melt by an optical sensor 10 0 T [K] power s pectral dens ity experiment 1700 calculations 10 -1 1695 -2 10 1690 experiment s imulations 10 -3 -3 1685 10 -2 10 -1 10 0 0 20 40 60 80 10 frequency [ Hz ] Time [s ] Power spectral density of temperature Temperature fluctuations in the melt, 1 fluctuations. cm lower the free surface. Growth parameters: The crystal diameter is 100 mm. The crucible/crystal rotations are 5/(-20) rpm. Argon flow rate and pressure are 750 slh and 30 mbar. The average crystallization rate is 2 mm/min. The experimental data are taken from the following paper: Mat. Sci. and Eng. B73 (2000) p.130. CGSim predicted the average amplitude and spectral characteristics of the temperature fluctuations in the melt near the triple point.
  20. 20. 3D Unsteady Calculations of the Crystallization Front Geometry animated visualization ∂T ⎛ ∂T ⎞ nx ⎜ λcrys crys − λmelt melt ⎟ Vcrys = ρ crys ∆H ⎜ ∂n ⎟ ∂n ⎝ ⎠ = Vcrys − Vcrys relative * Vcrys ∆X = Vcrys * TimeStep relative The experiments provided in Siltronic AG are published in: [1] Mat. Sci. in Semiconductor Processing 5/4-5, 2003, p.369-373; [2] J. Crystal Growth 250/1-2, 2003, p.203-208. [3] J. Crystal Growth, 266/1-3 ,2004, pp. 20 - 27 The crystal diameter is 100 mm The crystal diameter is 300 mm Computation, H=240mm 20 Computation, H=300mm Experiment, H=240mm Experiment, H=300mm 40 Computation, H=300mm Interface deflection, [mm] Computation, H=700mm Experiment, H=300mm 35 15 Experiment, H=700mm Interface deflection, [mm] 30 25 10 20 15 5 10 5 0 0 0 50 100 0 100 200 300 Radial position, [mm] Radial position, [mm]
  21. 21. Content Introduction - Description of Computer Model - Verification Results - Strategy of Cz Si Growth Optimization - Cz Growth with diameter of 400 mm for electronics
  22. 22. Increasing the crystallization (pulling) rate - by modifications of the heat shields and other elements surrounding the crystal - changes in the crystal and crucible rotation rate - optimization of the crucible position with respect to the heaters - changes in the crucible design General idea: One needs to increase the heat flux (temperature gradient) in the crystal and to decrease the heat flux in the melt. But it is necessary to avoid (i) crystal twisting, (ii) macrodislocations, and (iii) melt supercooling.
  23. 23. Effect of the heat shield modifications Initial heat shield Optimization of HSh shape and composition design up to +50% increase of pulling rate Modifications of heat shield design can be efficiently used to increase crystal growth rate
  24. 24. Reducing the heater power - to find weak points in the insulation design by analyzing the heat flux vectors and integral heat fluxes along surfaces - to improve insulations and repeat the analysis by computer model Usually, only 8-25% of the total heater power is used for the maintenance of the temperature in the melt !
  25. 25. Content Introduction - Description of Computer Model - Verification Results - Strategy of Cz Si Growth Optimization - Cz Growth with diameter of 400 mm for electronics
  26. 26. Growth of Cz Si crystals the with diameter of 400 mm for electronics: effect of DC magnetic fields on the melt and crystallization front
  27. 27. II.II. 3D Unsteady Parametric Analysis of 400 mm Si Crystal MCz Growth Ωcrystal = 10 rpm T [K] 1600 1400 Ωcrucible = -5 rpm 1200 1000 400 800 500 600 600 400 Vg = 0.45 mm/min argon 1300 crystal 1500 1700 P = 10 mbar melt crucible heater Flow rate is 5200 slh graphite 1700 1600 insulation Case 1: without MF 2D – 3D Case 2: with cusp MF of 30 mT model cusp MF Case 3: with horizontal MF of 30 mT Case 4: with horizontal MF of 300 mT Case 5: with horizontal MF of 300 mT and with the reduced argon flow rate of 520 slh horizontal MF Y. Shiraishi et al., J. of Crystal Growth 229 (2001) p.17 N. Machida et al., J. of Crystal Growth 186 (1998) p.362 K. Takano et al., Mat. Sci. and Eng. B73 (2000) p.30
  28. 28. Modeling Results for Case 1: without MF all plots here are animated to visualize unsteady effects 2 cm/s T [K] 1716 1711 1706 1701 1696 1691 1686 velocity and temperature evolution time-averaged velocity and temperature T ' [K] 2.6 0.8 2.2 1.8 0.6 1.4 1 0.6 0.2 time-averaged temperature fluctuation 2 2 k [m /s ] 0.002 0.0016 0.0004 0.0012 2 00 0.0008 0 .0 0.0004 0 evolution of the crystallization rate time-averaged turbulence kinetic energy
  29. 29. Modeling Results for Case 2: with Cusp MF of 30 mT all plots here are animated to visualize unsteady effects 2 cm/s T [K] 1716 1711 1706 1701 1696 1691 1686 time-averaged velocity and temperature velocity and temperature evolution T ' [K] 0.4 2.6 2.2 0.2 1.8 1.4 1 0.6 0.2 time-averaged temperature fluctuation 2 2 k [m /s ] 0.002 0.0004 0.0016 002 0.0012 0.0 0.0008 0.0004 0 evolution of the crystallization rate time-averaged turbulence kinetic energy
  30. 30. Modeling Rresults for Case 3: with Horizontal MF of 30 mT all plots here are animated to visualize unsteady effects 2 cm/s T [K] 1716 1711 1706 1701 1696 1691 r r 1686 B B time-averaged velocity and temperature velocity and temperature evolution T ' [K] 2.6 2.2 0.4 6 0. 1.8 1.4 1 0.6 0.2 time-averaged temperature fluctuation 2 2 k [m /s ] 0.0002 0.002 0.0016 0.0012 0.0008 0.0004 0 evolution of the crystallization rate time-averaged turbulence kinetic energy
  31. 31. Modeling Results for Case 4: with Horizontal MF of 300 mT all plots here are animated to visualize unsteady effects r r B B velocity and temperature evolution r evolution of the velocity and B temperature over the melt free surface velocity and temperature evolution k [m2/s2] 0.002 0.0016 0.0012 0.0008 0.0004 0 time-averaged turbulence kinetic energy evolution of the crystallization rate
  32. 32. Modeling results for Case 4: with horizontal MF of 300 mT all plots here are animated to visualize unsteady effects r Oxygen transport B crystal 3 XSiO[at/cm ] The results are obtained using a coupled gas model of oxygen transport in the melt and 3.2E+15 3E+15 region SiO in the gas. It has been found that SiO 2.8E+15 2.6E+15 distribution in the gas is extremely 2.4E+15 2.2E+15 asymmetric due to the horizontal MF, and 2E+15 melt 1.8E+15 there is significant oxygen concentration 1.6E+15 gradient over the melt/crystal interface. 1.4E+15 1.2E+15 1E+15 8E+14 evolution of oxygen concentration Evolution of SiO concentration in argon r over the crystallization front: B and oxygen content in the melt 3 crystal Xoxygen[at/cm ] 1.6E+18 1.5E+18 gas 1.4E+18 1.3E+18 region 1.2E+18 1.1E+18 1E+18 9E+17 r 8E+17 7E+17 melt B 6E+17 5E+17 4E+17
  33. 33. Results for Case 5: Horizontal MF of 300 mT and with the Reduced Argon Flow Rate of 520 slh r r B B velocity and temperature evolution r B evolution of the velocity and temperature over the melt free surface velocity and temperature evolution The strong decrease of the argon flow rate has resulted in a radical change of the flow structure and melt/crystal interface geometry K. Kakimoto and H. Ozoe, J. of Crystal Growth 212 (2000) p.429 evolution of the crystallization rate all plots here are animated to visualize unsteady effects
  34. 34. Time-Averaged Temperature Distributions over the Melt Free Surface and in Vertical Cross Sections for Cases 3, 4, and 5 r B 20 cm/s 40 cm/s 40 cm/s T [K] T [K] T [K] 1712 1702 1710 1702 1700 1708 1700 1698 1706 1698 1696 1704 1696 1694 1702 1694 1692 1700 1692 1690 1698 1690 1688 1696 1688 1686 1694 1686 1692 1690 1688 case 4 case 3 case 5 r r B B 2 cm/s T [K] 1716 1711 1706 1701 1696 1691 1686
  35. 35. Comparison of the Computed Crystallization Front Geometries and V/G Distributions no MF no MF Cusp DC MF, 30 mT Cusp DC MF, 30 mT Horizontal DC MF, 30 mT Horizontal DC MF, 30 mT V/G parameter [cm /Kmin] Interface deflection [mm] Horizontal DC MF, 300 mT Horizontal DC MF, 300 mT Horizontal DC MF, 300 mT; weak Gas Flow Horizontal DC MF, 300 mT; weak Gas Flow -03 2.5x10 20 2 10 -03 2.0x10 0 -03 1.5x10 -10 -03 1.0x10 -20 -100 0 100 -100 0 100 Radial position [mm] Radial position [mm] V. V. Voronkov, J. Crystal Growth, 59 (1982) p.625 R.A. Brown et al., Journal of Crystal Growth 225 (2001) p.97 M. Kulkarni, Ind. Eng. Chem. Res. 2005, 44, p.6246
  36. 36. I.VI. Defect incorporation and recombination Initial defect kinetics: dCm = ∇ ⋅(Dm∇Cm) + 4πar (D + Dv ) exp(− ∆G )(C Cve − C Cv ) m=v, i : interstitial and i kT ie i dt vacancy respectively. Initial defect equilibrium concentration and diffusion coefficient m⎞ 0 exp⎜ − H m ⎟ ⎛ 0 exp⎜ − H m − SmT ⎛ ⎞ Dm = Dm ⎟ Cme = C ⎜ kT ⎟ ⎜ ⎟ m ⎜ ⎟ kT ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Formation energy: Hv=2.48+2.33e-4*T (eV); Hi=3.46+3.08e-4*T (eV) Sv=(-3.7+3.53e-3*T)k (eV/K); Si=(1.4+3.85e-3*T)*k (eV/K) Concentration coefficient: Cv0=4.97*1022 (cm-3); Ci0=2.97*1023 (cm-3) Migration energy: Hv=0.457 (eV); Hi=0.937 (eV) Diffusion coefficient: Dv0=1.3*10-3 (cm2/s); Di0=0.242 (cm2/s) This values set has taken from [1] T. Sinno, et al., J. Electrochem. Soc. 145-1 302 (1998) Boundary conditions: melt/crystal interface – equilibrium concentration; side crystal surface – zero flux; top crystal surface – extrapolation from the inner crystal part.
  37. 37. 2D calculation of defect incorporation and recombination (crystal diameter is 300 mm) Interface Interface computed with a computed using a 2D approximation 3D approximation The effect of melt turbulent flow on Cv - Ci [cm-3] the formation of the crystallization 2.1E+14 1.8E+14 front is much 1.4E+14 more significant 1.1E+14 7.8E+13 for 300 mm 4.5E+13 crystal growth. 1.2E+13 -2.1E+13 -5.4E+13 -8.7E+13 2D 3D Technological parameters. Crystal height is 300 mm, pulling rate is 0.7 mm/min, cruciblecrystal rotation are 6-12 rpm, Ar flowratepressure are 1000 slh 15 mbar.
  38. 38. Defect clusterization Free energy gain for void and oxygen precipitate, volume and surface terms: ⎛ ⎞ ⎜C ⎟ G (n ) = − nf + λ ⋅ n 2 / 3 = − nkT ⋅ 4π R 2 V ⎟ +σ ⋅ ln ⎜ eq V V V V ⎜C ⎟ ⎜ ⎟ ⎝V ⎠ ⎛V ⎞ ⎜ ⎟ 1 ⎜ SiO2 ⎛ ⎞ −1⎟ ⎜C ⎟ γ= ⎜ ⎛ ⎞ 2 / 3 = −γnkT ⋅ ln⎜ V ⎟ − nkT ⋅ ln⎜ Co ⎟ + σ ⋅ 4πR2 ⎟ G (n) = −nf + λ ⋅ n 2⎜ V ⎜ eq ⎟ ⎜ C eq ⎟ ⎟ Si P PP P ⎜C ⎜ ⎟ ⎟ ⎝o⎠ ⎜ ⎟ ⎝V⎠ ⎝ ⎠ Interstitial injection mechanism of stress energy relaxation for oxygen precipitate is neglected, assuming high Frenkel pair formation energy and high vacancy flux. ∂H m Evolution of point defect size distribution function = − ∂ ⎛ Jm ⎞ ⎜ ⎟ ∂t ∂R ⎝ ⎠ describes using Fokker-Planck equation ⋅ ∂ ⎡(g m (R) + d m (R))H m ⎤ 1 m=V,P: voids and J m = ⎡ gm (R) − dm (R)⎤ H m − 8πR2 ρm ∂R ⎣ ⎢ ⎥ ⎢ ⎥ oxygen precipitate ⎣ ⎦ ⎦ respectively gm(R) and dm(R) are point defect growth and dissolution rate respectively. The initial defect loss equation takes into account the formation of new nucleus, precipitating to existing point defects and binding vacancy (V02) formation. This model based on the assumptions introduced in [1] V. V. Voronkov, R. Falster, J. Cryst. Growth 204, p.462 (1999) and [2] T. Sinno, R. Brown, J. Electrochem. Soc. 146 -6, p.2300 (1999)
  39. 39. Radial distribution of point defect density and average size 6 4.0x10 9 4x10 2D interface 2D interface 3D interface 3D interface -3 -3 Void density, cm Particle density, cm 9 3x10 6 3.5x10 9 2x10 6 3.0x10 9 1x10 6 2.5x10 0 0 1 2 3 4 5 0 1 2 3 4 5 7 Radial position, cm Radial position, cm 60 Oxygen precipitate average size, nm 55 Void average size, nm 6 50 5 45 2D interface 40 2D interface 3D interface 4 3D interface 35 3 30 0 1 2 3 4 5 0 1 2 3 4 5 Radial position, cm Radial position, cm Technological parameters: Crystal diameter is 100 mm. Crystal height is 300 mm, pulling rate is 2 mm/min, cruciblecrystal rotation are 5-20 rpm, Ar flowratepressure are 675 slh 25 mbar. This values correlate well with data presented in [1] T. Sinno et al., Mat. Sci. Eng. 28 p.149 (2000)
  40. 40. Summary on modeling of Cz single Si growth We have presented a combined 2D-3D model of Cz Si growth. The model considers melt and gas convection, heat transfer by radiation and conduction, calculations of crystallization front shape, oxygen transport in the melt and SiO transport in the gas, including deposition, and defects dynamics. 2D unsteady modeling of melt down and cooling stages provides information on crucible and melt temperature evolution, thermal gradients in the cooled crystal and freezing of the melt remainders Multiple verification examples show high predictive capability of CGSim software. Good correspondence between results of computations and experimental data is obtained for temperature distributions, thermal fluctuations in the melt, crystallization front shape, oxygen transport Using CGSim software it is possible to make considerable optimization of hot zone design for Cz Si growth and: increase the pulling rate, to decrease probability of twisting phenomena, to decrease probability of macrodislocation generation, to decrease power consumptions, to reduce operation time including meltdown and cooling stages, etc.

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