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Numerical Modeling of Stress and Dislocations in Si Ingots Grown by Seed-Directional Solidification, Comparison to Experimental Data Olga Smirnova, Vasif Mamedov, and Vladimir Kalaev Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/cg500736j • Publication Date (Web): 08 Oct 2014 Downloaded from http://pubs.acs.org on October 10, 2014

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Crystal Growth & Design

Numerical Modeling of Stress and Dislocations in Si Ingots Grown by Seed-Directional Solidification, Comparison to Experimental Data Olga V. Smirnova*, Vasif M. Mamedov, Vladimir V. Kalaev STR Group, Inc, Engels av. 27, P.O. Box 89, 194156,St.-Petersburg, Russia Unsteady modeling of the thermal stress and dislocations in silicon ingots grown by the seeddirectional solidification have been done for three cooling regimes with studying the effect of model parameters. To track the history of dislocation multiplication, our computations consider continuously the growth process from the beginning of crystallization on the seed to the end of cooling of grown crystal. It is discussed how a cooling regime may affect the residual stress and dislocation distribution in the crystal. Computational results show a good agreement with experimental data for both residual stress and dislocation density. Our results have confirmed that conjugated unsteady modeling is a promising way to investigate the effect of various growth conditions as well as growth system design on residual stress distributions, which is an important parameter of crystal quality.

a

b

c

Figure. Computed residual stress distributions (top) and experimental SIRP images22 (bottom) obtained for the fast-fast cooling (a), the fast-slow cooling (b) and the slow-fast cooling (c). *Olga V. Smirnova, STR Group, Inc, Engels av. 27, P.O. Box 89, 194156,St.-Petersburg, Russia, Phone: +7 812 320 4390, Fax: +7 812 326 6194, [email protected]

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Numerical Modeling of Stress and Dislocations in Si Ingots Grown by Seed-Directional Solidification, Comparison to Experimental Data Olga V. Smirnova*, Vasif M. Mamedov, Vladimir V. Kalaev STR Group, Inc, Engels av. 27, P.O. Box 89, 194156,St.-Petersburg, Russia *Phone: +7 812 320 4390, Fax: +7 812 326 6194, [email protected] ABSTRACT Unsteady modeling of the thermal stress and dislocations in silicon ingots grown by the seeddirectional solidification have been done for three cooling regimes with studying the effect of model parameters. To track the history of dislocation multiplication, our computations consider continuously the growth process from the beginning of crystallization on the seed to the end of cooling of grown crystal. It is discussed how a cooling regime may affect the residual stress and dislocation distribution in the crystal. Computational results show a good agreement with experimental data for both residual stress and dislocation density. Our results have confirmed that conjugated unsteady modeling is a promising way to investigate the effect of various growth conditions as well as growth system design on residual stress distributions, which is an important parameter of crystal quality.


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INTRODUCTION The seed-directional solidification is a cost-effective way of producing quasi mono-Si ingots for the photovoltaic (PV) industry. Numerical simulation is widely used to study different aspects of solidification growth. Coupled 2D models accounting for conductive heat transfer, radiative heat exchange and convection are successfully applied for optimization of solidification growth process

1-7

. Advanced 3D approaches are as well used for detailed analysis of melt flow

and crystallization shape geometry8-10. Dislocation density is an important parameter characterizing crystal quality and responsible for the efficiency of solar cells. It is known that one of the major factors affecting generation and multiplication of dislocations inside the crystal bulk is the thermal stress. The thermal stress is provided by thermal deformation due to spatial temperature variation during crystal growth and cooling processes. A 2D or 3D numerical approach can be successfully used to study the thermal stress and dislocation density in directional solidification11-13. Cooling rate dependence on dislocation density in multicrystalline silicon for solar cells is considered by Nakano et al11 for three cooling regimes within the Haasen-Alexander-Sumino model. However, the authors did not present modeling of growth process, starting from the seed, while, coupling the history of dislocation formation during the growth and cooling seems to be important to get adequate values of residual stress and dislocation density at the end of cooling process. Three-dimensional analysis has been recently applied to clarify details of dislocation formation in the crystal12-13 for Si Directional Solidification. These approaches allow one to consider three-dimensional effects of stress distribution in industrial crystallization12 or the effect of the cooling rate on activation on slip system13. However, Fang et al12 are using the critical resolved shear stress model, which does not account for the history of dislocation multiplication and not provide the analysis of the


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residual stress after cooling the crystal. Gao et al13 have studied in details the effect of the cooling rate on the activation of slip systems and other refined effects of dislocation interaction, assuming one single crystal grown in the crucible without alternative grains and crystallographic orientations. However, a real crystal grown by Seed-Directional Solidification usually has several grains with different orientations, which is not accounted in the model. Moreover, the industrial standard of Si Directional Solidification12 implies the growth of multi-crystalline structure with grain scale much lower that the ingot size, which should be accounted in a computer model. As well, to our knowledge, still there are no publications related to Si (Seed-) Directional Solidification with a comparison of experimental and modeling results on the residual stress and dislocation density. In this paper, we present results of conjugated unsteady modeling of seed-directional solidification process, simulated from the crystallization beginning till the end of cooling to reproduce the thermal stress and stress-related dislocations in Si ingots. The melting of seed during the melt down stage, melt convection effect on the crystallization front dynamics and the transition from crystallization to the cooling stage are simulated within a coupled algorithm using CGSim software. Within the Alexander-Haasen

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(AH) model, we have accounted for

resistance of dislocation gliding on grain boundaries. Also we discuss the effect of dislocation generation during the crystal growth on the dislocation multiplication during cooling stage. To verify the model, we present a detailed comparison with experimental data presented in the papers of Jiptner et al 20-22.


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NUMERICAL MODEL A special crystal growth simulation software, CGSim, developed by STR Group, is used for computations in this paper. The software has been verified by using a significant number of experiments

14–18

. Unsteady modeling accounts for conductive and radiative heat transfer in the

seed-directional solidification growth system. The melt flow is here considered as laminar within the Navier-Stokes equations. The melt-crystal interface shape is computed at each time step, starting from the seed, including the melt down stage. Stress and dislocation distributions are modeled within the Alexander-Haasen model 19, which gives the relationship between the plastic deformation and dislocation density. In the model, the total strain ε i j is assumed to be subdivided by components

ε i j = ε ie j + ε iT j + ε ic j

, (1)

where ε ie j , ε iTj and ε ic j are elastic strain, thermal strain and creep strain, respectively. Thermal strain is produced by isotropic thermal expansion

ε iTj =

δ ij 

ρ (T )  1 − , 3  ρ (Tref ) 

(2)

where δ ij is the Kroneckerdelta-function, ρ is the crystal density depending of absolute temperature T. Reference temperature Tref corresponds to non-deformed material. In case of crystal growth simulation the melting temperature is used as reference one. The total strain ε i j is related to the displacement ui of thermoelastic solid body as follows

1  ∂u

∂uj  .  

ε i j =  i + 2  ∂ x j ∂ xi

(3)


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In AH model, the creep strain rate and the multiplication rate of the mobile dislocation density N m can be expressed as follows

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:

dε icj

1 1 = bNm Si j v , dt 2 J2

(4)

dN m N v  Q p +l = K k0 (τ eff ) exp −  N m − m , dt L  kT 

(5)

τ eff = J 2 − D N m ,

(6)

D=R J2 =

Eb , 4π 1 −ν 2

(

)

1 Si2j , ∑ 2 i, j

Si j = σ i j − δ i j

(7)

(8) 1 ∑σ kk , 3 k

 Q p v = k0 (τ eff ) exp −  ,  kT 

(9)

(10)

where b is the magnitude of Burgers vector, τeff is the effective stress, Q is the Peierls potential, k is Boltzmann’s constant, Si j is the deviatoric stress, J 2 is the second invariant of deviatoric stress, D and R are strain hardening factor and relative strain hardening factor, respectively, E is the Young’s modulus and ν is the Poisson's ratio, k0, K, p, l are material constants, σi j denotes the stress tensor components, L is the average grain size, v is the average velocity of mobile dislocation. The last term in equation (5) is formulated in our approach to account for the fact that grain boundaries are expected to be effective barriers to dislocation glide23. The average grain size L should be specified as an additional model parameter, available from experimental data or theoretical estimations.


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In case of

J 2 − D N m < 0 the value of τeff is set to zero i.e. creep strain rate and dislocation

density rate are set equal to zero. The elastic strain ε ie j and total stress σi j can be expressed by the momentum balance equations and Hook's law:

∂σ i j

∑ ∂x j

σij =

=0

j

,

∑с

i j

(11)

ε ie j

j

,

(12)

where сi j is the coefficient of elastic stiffness matrix. The numerical mesh is special designed to be suitable for moving crystallization front up during unsteady computations of the solidification growth. The total number of cells in the whole computational domain including all parts of the growth facility is about 9000. Particular attention is paid to the grid quality in the crystal, because the crystal block increases during the unsteady computation and its cells extend upward. The number of crystal block cells is about 2000.

RESULTS and DISCUSSION Three unsteady computations of seed-cast Si crystal growth have been made with the same growth parameters but different cooling regimes. Fig.1 illustrates the instant temperature distribution, melt convection contributing to a “w-shaped” crystallization front, and 2D computational grid. The growth parameters and cooling regimes (Fig.2) were selected according to experimental data described in the papers of K. Jiptner et al

20-22

. The computations consider

the growth process from the crystallization beginning till the end of cooling, allowing to study evolution of dislocations from the seed crystals, through growth process, till grown ingot reaches the room temperature. The seed shape was calculated assuming zero crystallization rate before


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the start of crystal growth. The initial dislocation density in the seed was set as 1 cm-2 to represent the monocrystalline material.

Monitoring points

Figure 1. Temperature distribution (left), melt flow and computational grid (right).


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100

1500 1000

C1 (fast-fast) C2 (fast-slow) C3 (slow-fast)

60 40

500

20 0

2

4

6

8

10

Time,h

-500

0 -20 -40 -60

-1000

Crucible position, mm

80

Heater temperature, C

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-80 -1500

-100

Figure 2. Growth conditions including heater temperature (upper part) and crucible position (lower part) Grown crystals were analyzed by a Scanning Infra Red Polariscope (SIRP) to get residual stress distribution as published in 22

20-22

. Fig.3 shows a comparison of the measured SIRP images

and the Von Mises stress in the crystals computed with CGSim software. All simulated stress

distributions have been obtained in the end of cooling when the ingots reach the room temperature.


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a

b

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c

Figure 3. Computed residual stress distributions (top) and experimental SIRP images22 (bottom) obtained for the fast-fast cooling (a), the fast-slow cooling (b) and the slow-fast cooling (c). One can see a good qualitative agreement of the computations and experiment. There are areas with maximums at the crystal top and bottom center due to high temperature gradients during the beginning of crystallization (at the bottom) and during the start of cooling (at the top). Smaller maximum areas are placed at the middle of the side boundary as an effect of cooling stage. In the crystal bulk, central low stress area has smoothly resolved four local minima for the fast-fast cooling case (C1), which is very well reproduced by modeling. For slow-fast cooling experiment (C3), there is a larger central area of low residual stresses, which is just qualitatively reproduced by modeling. It is worth to note that, maybe, due to grain boundaries revealed by discontinuous changes in the experimental stress distributions, experimental results for the fast-slow and slowfast cooling regimes have not axisymmetric residual stress distributions. Looking at SIRP data for slow-fast cooling, represented by small separate areas (maybe representing separate grains)


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of continuous stresses, we can conclude only that the residual stress generally reduced in comparison to the fast-fast regime because lowering the temperature slowly at the high temperature level provides a longer time for active multiplication of dislocations reducing the stress. Also it looks strange that there are no local maxima in the experimental residual stress in the bottom and along the right side of the crystal in the case of slow-fast cooling. Even the experimental stress is lower in those areas in comparison to the center of the crystal, which looks to be unphysical because cooling firstly affects the temperature gradients along the external boundary of the crystal. 5.0E+06

C1 (fast-fast) C2 (fast-slow) C3 (slow-fast)

4.0E+06

Stress, Pa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.0E+06 2.0E+06 1.0E+06 0.0E+00

0

20

40

60

80

Distance from the crystal bottom, mm Figure 4. Computed residual stress distribution along the crystal symmetry axis Figure 4 shows distribution of computed residual stress as a function of the crystal height along the crystal symmetry axis. One can see that regimes with fast cooling at the beginning (C1 and C2) give close stress distributions, as supported by experimental data. The slow–fast regime (C3) results in a lower stress level near the crystal top and bottom, as confirmed in the experimental data shown in Fig.3 and in other similar modeling results 11,13.


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6.0E+04

EDP experimental Standard AH AH + GB resistance

Dislocatin density, cm-2

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Dislocations during cooling 4.0E+04

2.0E+04

0.0E+00

0

20

40

60

80

distance from bottom, mm Figure 5. Computed dislocation density distribution obtained for the fast-fast cooling case (C1) Analyzing experimental SIRP results (Fig.3), one can see that only crystal C1 (fast-fast cooling) has manly single monocrystalline structure along the whole symmetry axis, while the crystals C2 (fast-slow cooling) and C3 (slow-fast cooling) have rapid changes of the residual stress along internal lines or plains, which correspond to different grains of different crystallinity. K. Jiptner et al

20-21

have presented results of Etch Pit Density measurements for these crystals

and speculated that only measurements in the center of the crystal are relative to cooling conditions. So, we have selected the center line of the crystal C1 for a comparison of our computational results to experimental data on EPD. The computed dislocation density distribution along the central symmetry axis of crystal C1 and the respective experimental results are shown in Fig.5. Experimental EPD in the crystal core was found to be about 104 cm-2 with the maximal density on the crystal bottom. The same order of the dislocation density is found to be


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in our computations. The standard AH model neglects the effect of the resistance to dislocation glide at a grain boundary because of plastic and elastic incompatibility. One can see that modeling within the standard AH model overestimates the dislocation density in about 3 times, especially in the crystal top. Introducing the effect of the grain boundary resistance via the additional term in eq. (5) has helped to get much better predictions of EPD: the density is maximal on the crystal bottom and there is a general increase of the density in the crystal center to the crystal top, like in the experimental data. The average grain size L was specified here as the half crystal height because there are 2 grains from the bottom to the top well seen in the experimental data. Finally, we have examined the importance of modeling crystal growth dynamics coupled to dislocation evolution. Modeling results entitled as “Dislocations during cooling” in Fig. 5 are obtained without modeling of dislocations till the beginning of crystal cooling: the AH model was activated exactly at the start of cooling with the initial dislocation density of 1 cm-2. One can see that the density is strongly underestimated in the bottom of the crystal in this case. So we can conclude that dislocations in the bottom part of the crystal are mainly generated during crystal growth.

CONCLUSION Verification of computed stress distributions in Si crystals grown by the seed-directional solidification process has shown a good agreement with respective experimental data. It confirms that the stress density depends on cooling regime parameters at the temperature higher than 900C, which was already reported by other authors. Predictions within the standard AH model have provided overestimated dislocation density in the crystal in comparison to the experimental data. For multi crystalline Si ingots and for quasi


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mono crystals grown with the seed, there is a need to account for the resistance to dislocation glide at a grain boundary. We have introduced such a correction into AH model and obtained a reasonable correlation with the respective experimental EPD: the density is maximal near the crystal bottom and generally increases from the center to the crystal top. As well it was shown that neglecting the evolution of dislocations during crystallization results in a significant underestimation of EPD in the crystal bottom, which shows the importance of modeling crystallization with melt convection from seeding to the end of growth, coupled to dislocation and stress analysis. So, the coupled unsteady numerical modeling is an efficient tool allowed one to get adequate stress distributions during crystal growth and cooling process. Thus, it is possible to use modeling as a tool to investigate the effect of different growth and cooling parameters and, also, furnace design on residual stress distributions and the dislocation density in the crystal.

ACKNOWLEDGMENT The authors acknowledge Dr. Karolin Jiptner from National Institute for Materials Science (Japan) for interesting discussion of measurements of the residual stress and dislocation density. As well we would like to thank Dr. Mikhail Rudinsky from STR Group for useful ideas on dislocation evolution in multi crystalline silicon.


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REFERENCES (1) Maa, X.; Zheng, L.; Zhang, H.; Zhao, B.; Wang, C.; Xu, F. J. Cryst. Growth 2011, 318, 288–292. (2) Li, Z.; Liu, L.; Liu, X.; Zhang, Y.; Xiong, J. J. Cryst. Growth 2012, 360, 87–91. (3) Li, Z.; Liu L.; Nan, X.; Kakimoto K. J. Cryst. Growth 2012, 346, 40–44. (4) Chen, L.; Dai, B. J. Cryst. Growth 2012, 354, 86–92. (5) Yu, Q;, Liu, L.; Ma, W.; Zhong, G.; Huang, X. J. Cryst. Growth 2012, 358, 5–11. (6) Ma, W.; Zhong, G.; Sun, L.; Yu, Q.; Huang, X.; Liu, L. Sol. Energy Mater. Sol. Cells 2012, 100, 231–238. (7) Li, Z.; Liu, L.; Liu, X.; Zhang, Y.; Xiong, J. J. Cryst. Growth 2014, 385, 9–15. (8) Simons, P.; Lankhorst, A.; Habraken, A.; Fabera, A.-J.; Tiuleanu, D.; Pinge, R. J. Cryst. Growth 2012, 340, 102–111. (9) Kuliev, A.T.; Durnev, N.V.; Kalaev, V.V. J. Cryst. Growth 2007, 303, 236–240. (10) Vizman, D.; Tanasie, C. J. Cryst. Growth 2013, 372, 1–8. (11) Nakano, S.; Chen, X.J.; Gao, B.; Kakimoto, K. J. Cryst. Growth 2011, 318, 280–282. (12) Fang, H.S.; Wang, S.; Zhou, L.; Zhou, N. G.; Lin, M. H. J. Cryst. Growth 2012, 346, 5–11. (13) Gao, B.; Nakano S.; Harada H.; Miyamura Y.; Kakimoto, K. Cryst. Growth Des. 2013, 13, 2661-2669.


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(14) Kalaev, V.V.; Evstratov, I.Yu. J. Cryst. Growth 2003, 249, 87. (15) Smirnova, O.V.; Kalaev, V.V. J. Cryst. Growth 2007, 303, 141. (16) Sim, B.C.; Jung, Y.H.; Lee, J.E. J. Cryst. Growth 2007, 299, 152. (17) Jana, S.; Dost, S. ; Kumar, V. Int. J. Eng. Sci. 2006, 44, 554. (18) Dornberger, E.; Tomzig, E.; Seidl, A. J. Cryst. Growth 1997, 180, 461. (19) Alexander, H.; Haasen, P. Solid State Phys. 1968, 22, 27-158. (20) Jiptner, K.; Fukuzawa, M.; Miyamura, Y.; Harada, H.; Kakimoto, K.; Sekiguchi, T. 28th European Photovoltaic Solar Energy Conference and Exhibition, 1396-1399. (21) Jiptner, K.; Fukuzawa, M.; Miyamura, Y.; Harada, H.; Kakimoto, K.; Sekiguchi, T. Phys. Status Solidi C 10, 2013, 1, 141–145. (22) K.; Fukuzawa, M.; Miyamura, Y.; Harada, H.; Kakimoto, K.; Sekiguchi, T.; Solid State Phenom., 2014, 205-206, 94-99. (23) J. P. Hirth, Theory of Dislocations: 2nd (second) Edition, Krieger Publishing Company, 1983.


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For Table of Contents Use Only

Unsteady modeling of the thermal stress and dislocations in silicon ingots grown by the seeddirectional solidification have been done for three cooling regimes with studying the effect of model parameters. To track the history of dislocation multiplication, our computations consider continuously the growth process from the beginning of crystallization on the seed to the end of cooling of grown crystal. It is discussed how a cooling regime may affect the residual stress and dislocation distribution in the crystal. Computational results show a good agreement with experimental data for both residual stress and dislocation density. Our results have confirmed that conjugated unsteady modeling is a promising way to investigate the effect of various growth conditions as well as growth system design on residual stress distributions, which is an important parameter of crystal quality.

a

b

c

Figure. Computed residual stress distributions (top) and experimental SIRP images22 (bottom) obtained for the fast-fast cooling (a), the fast-slow cooling (b) and the slow-fast cooling (c).


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Numerical Modeling of Stress and Dislocations in ... - ACS Publications

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