A model for distribution of iron impurity during silicon purification by directional solidification

Accepted Manuscript A model for distribution of iron impurity during silicon purification by directional solidification Shutao Wen, Yi Tan, Tao Yuan, Pingting Li, Michele Forzan, Dachuan Jiang, Fabrizio Dughiero PII:

S0042-207X(17)30208-7

DOI:

10.1016/j.vacuum.2017.09.012

Reference:

VAC 7582

To appear in:

Vacuum

Received Date: 15 February 2017 Revised Date:

5 September 2017

Accepted Date: 6 September 2017

Please cite this article as: Wen S, Tan Y, Yuan T, Li P, Forzan M, Jiang D, Dughiero F, A model for distribution of iron impurity during silicon purification by directional solidification, Vacuum (2017), doi: 10.1016/j.vacuum.2017.09.012. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT A model for distribution of iron impurity during silicon purification by directional solidification Shutao Wena, b, c, Yi Tana, b*, Tao Yuand, Pingting Lia, b, Michele Forzan c, Dachuan Jianga, b, Fabrizio Dughiero c* School of Materials Science and Engineering, Dalian University of Technology, Dalian 116023, China

b

Key Laboratory for Solar Energy Photovoltaic System of Liaoning Province, Dalian 116023, China

c

Department of Industrial Engineering, University of Padova, Padova 35131, Italy

d

Qingdao Longsun Silicon Technology Co., Ltd, Qingdao 266234, China

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a

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Corresponding author at:

1. Yi Tan: School of Materials Science and Engineering, Dalian University of Technology, No. 2 Linggong Road, Ganjingzi District, Dalian 116023, China. E-mail address: [email protected] (Yi Tan)

Padova 35131, Italy.

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2. Fabrizio Dughiero: Department of Industrial Engineering, University of Padova, Via Gradenigo 6/a,

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E-mail address: [email protected]

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ACCEPTED MANUSCRIPT Abstract A theoretical model to determine distribution of iron impurity during silicon purification by directional solidification with fluctuant crystal growth rate is proposed in this paper. The crystal growth

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rate is fluctuant usually and it has profound effect on the distribution of iron impurity in practical production. The model validation by the distribution of iron impurity during silicon purification by directional solidification in industrial production and the results show that the calculation agrees with

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existing experimental results. The results also indicate that distribution of iron impurity is directly

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correlated with the instantaneous value of crystal growth rate. The high fluctuant distribution of iron impurity during silicon purification by directional solidification can be well explained. Many potential applications of the model in practical production are found, such as predicting the distribution of iron impurity with fluctuant crystal growth rate, evaluating the effect degree of production accident, design

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and optimization of the process parameters and evaluating of maximum yield for raw silicon with different impurity concentration. Silicon purification with low energy consumption is possible based on the research in this paper.

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Keywords: Silicon, Directional Solidification, Distribution of Impurity, Fluctuant Growth Rate

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ACCEPTED MANUSCRIPT 1. Introduction Driven by the dynamics of the photovoltaic industry, the demand for solar grade silicon is increasing greatly [1-4]. It is well known that the metal impurities content has a great impact on the

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photoelectric conversion efficiency of solar cells [5-7]. Iron impurity is one of the most common metal impurities in silicon [7, 8], which can affect the minority carrier lifetime of solar cells [7, 9, 10]. Recent studies have indicated that the photoelectric conversion efficiency of solar cells will decline

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when the content of iron higher than 1×1014/cm3 [11, 12]. The directional solidification is used for

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silicon purification as an effective way to remove the iron impurity from silicon [13]. The purifying effect is unsatisfactory in the experiment compared with the calculation by the Scheil equation. Thus, any factor must exist to affect the distribution of impurity during silicon purification by vacuum directional solidification. One of them is back diffusion behavior of impurity,

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which has proven by both of experimentation and calculation model [14]. However, either experimentation or theoretical model in the Ref. [14], the crystal growth rate is constant for a single directional solidification process. There will be a high fluctuant distribution of iron impurity in actual

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directional solidification if the crystal growth rate is fluctuant for a single directional solidification

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process [13], even after considering back diffusion behavior of impurity [14]. Actually, our previous research had proved the crystal growth rate is fluctuant usually in experiment [16], and this phenomenon maybe the most likely cause of the fluctuant distribution of iron impurity. The Scheil equation is inapplicable for calculating distribution of impurity during silicon purification by directional solidification when the crystal growth rate is fluctuant, however, Scheil equation was still used in practical production due to lack of mature theory, which resulting to the distribution of impurity is not clear. This triggers a series of problems such as instability product quality, long production cycle,

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ACCEPTED MANUSCRIPT lacking of theoretical support for designing or optimizing production parameters, low production yield and unreasonable cutting standard of silicon ingot. The crystal growth rate is an important parameter for directional solidification, which combines

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several factors, such as temperature gradient in molten silicon, heat transfer in the solidification process and the content of impurity. It is widely investigated by many researchers and had proved the iron impurity can be removed efficient from silicon with a proper crystal growth rate [10, 15]. However,

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studies on distribution of impurity during silicon purification by vacuum directional solidification with

published data needed for our application.

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fluctuant crystal growth rate are limited. The present study was conducted because of the lack of

In this paper, a theoretical model is proposed for iron impurity distribution during silicon purification by directional solidification with fluctuant crystal growth rate. The fluctuant distribution of

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iron impurity during silicon purification by directional solidification is explained. And the distribution of iron impurity can be predicted with fluctuant crystal growth rate. In addition, production parameters can be optimized for reducing energy consumption.

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2. Theoretical Model

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The mathematical model is established based on the following assumptions: 1. Liquid phase composition is uniform in the solidification process. 2. Diffusion in the solid silicon is negligible. 3. The solid-liquid interface is flat. According to above assumptions, the mass concentration of impurity in solid silicon at solid-liquid interface, ρs (ppmw), can be expressed as [17]:

ρ s = ke ρ l

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(1)

ACCEPTED MANUSCRIPT where ρl (ppmw) is the mass concentration of impurity in molten silicon; ke is the effective segregation coefficient, which is can be obtained by Burton’s equation [18] or by the method proposed by Mei [19], For clarify the effect of crystal growth rate on the distribution of impurity in silicon ingot, Burton’s

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equation is chosen in this paper. To calculate ke, Burton et al. [18] assumed a stagnant liquid layer of thickness δ (m) in which solute transport occurred only by diffusion. Outside this layer, the liquid was

ke =

k0 k 0 + (1 − k 0 )e −δv / Dl

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mixed completely by convection and had a uniform composition.

(2)

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where Dl (m2·s-1) is the diffusion coefficient of impurity in molten silicon at melting point. The value of Dl can be considered as constant, Dl =2.95×10-9 (m2·s-1), under certain conditions for iron impurity [20]. As an important parameter in Eq. (2), the stagnant layer is investigated by many researchers. Kodera’s research shows that the stagnant layer thickness is a function of rotation rate as shown in Eq. (3) [21].

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δ = 1.6 Dl1 / 3γ 1 / 6ω −1 / 2

(3)

where γ (m2/s) is the kinematic viscosity of the melt; ω (rad /s) is the angular velocity of rotation.

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Wilson proposed the modified stagnant layer, as defined as:

δ = ( ρ0 − ρl ) /(−∂ρ / ∂x) x = 0

(4)

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ρ0 (ppmw) is the original mass concentration of impurity in raw silicon; x=0 means the liquid-solid interface location. Martorano’s research shows that apparently the change in solidification conditions did not modify the fluid flow pattern significantly, and a constant value of δ=0.004m could be used for silicon solidification [22]. Huang consider the stagnant layer thickness as a value of 0.004m in their model and the calculation results agree well with the experiments [23]. Ren explained their experiment results with a constant value of δ=0.006m [24]. There was no rotation of silicon and crucible in the current experiments, a constant value of δ=0.005m is used in this paper. 5

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The only variable left in Eq. (2) is the crystal growth rate for iron impurity when the stagnant layer thickness is fixed, namely the ke is a function of crystal growth rate in directional solidification process.

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In order to investigate the impurity distribution during silicon purification by directional solidification with fluctuant crystal growth rate, the finite element idea is introduced into the directional solidification process. The solidified region is subdivided into N equal regions, thickness of each region

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in the x-direction is △x=L/N and the divisions between the regions are selected as the nodes. Therefore,

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there are N+1 nodes labeled 0, 1, 2,..., n-1, n, n+1,...,N, as shown in Fig. 1. The x-coordinate of any node n is simply xn=n· △x, and the impurity concentration at each point is ρs(xn)= ρs(n). Elements are formed by drawing vertical lines through the midpoints between the nodes. The elements are independent of each other for crystal growth rare, which means that each element can have its own

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crystal growth rare. Each of neighboring elements is linked by impurity concentration. For instance, when the solidification of the element represented by node n just finished, concentration of silicon melt at that moment will consider as the initial concentration for the element represented by node n+1.

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2.1. Relation for each interior node

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Note that all interior elements represented by interior nodes are full-size elements with a thickness of △x, whereas the two elements at the boundaries are half-sized with a thickness of △x/2. To obtain a general difference equation for the interior nodes, assuming the location of solidification at the element represented by node n-1. The total content of iron impurity in solid phase and liquid phase can be expressed as: n −1

M s ( n − 1) = A∑ ρ s (i )dx ,

M l ( n ) = M Total − M s ( n − 1)

and

1

,

where A (m2) is the sectional area of impurity diffusion; MTotal is the total content of iron impurity in 6

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ACCEPTED MANUSCRIPT raw silicon, which can be expressed as:

M Total = A ⋅ L ⋅ ρ0 ,

(6)

where L (m) is the total solidification length; ρ0 (ppmw) is initial concentration of iron impurity.

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Recall the assumptions at the beginning of the modeling: liquid phase composition is uniform in the solidification process. Before solidifying, the concentration of the element represented by node n can be expressed as:

M l (n) A ⋅ L ⋅ (1 −

n . ) N

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ρl (n) =

(7)

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After solidifying, the concentration of the element represented by node n can be expressed as:

ρ s ( n) = ke (n ) ⋅ ρ l ( n) 2.2. Boundary conditions

.

(8)

A general relation for each interior node is developed in Section 2.1. This relation is not

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applicable to the nodes on the boundaries, however, since it requires the presence of nodes on both sides of the node under consideration, and a boundary node does not have a neighboring node on at

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least one side. Therefore, the concentration of boundary nodes needs to obtain separately. This is best done by applying an mass conservation on the elements of boundary nodes. The concentration of

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boundary nodes before solidifying can be expressed as:

ρl (0) = ρ0 ,

ρl ( N ) =

and

N ⋅ [ M Total − M s ( N − 1)] . A⋅ L

(9)

The concentration of boundary nodes after solidifying can be expressed as:

ρs (0) = ke (1) ⋅ ρ0 ,

and

ρ s ( N ) = ke ( N ) ⋅ ρ l ( N ) .

(10)

It is noteworthy that when the crystal growth rate is constant; Eqs. (6)~(9) then turns into the Scheil equation:

ρ s = ke ρl [1 − f s ]( k 7

e

−1)

,

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ACCEPTED MANUSCRIPT where fs is the solidified fraction. 3. Experiment and Model Verification The experiment to validate the model is carried out with a casting furnace (JS-MCS480). The

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original mass concentration of iron, ρ0 (ppmw), in raw silicon is 694.9 ppmw. The charging mass of raw silicon in crucible is 404.9 kg. The total solidification length is 0.248 m. For the impurity detection method of the prepared silicon ingots as shown in Fig. 2: the silicon ingot as produced in experiments

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is shown in Fig. 2(left), and it was cut evenly into 2 pieces according to the red line direction, 1 of the 2

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silicon blocks was selected randomly. The crystal growth appearance of silicon ingot in experiments is shown in Fig. 2 (right) and the outlined in red line shows the sampling location. A silicon column sized 10mm×10mm×248mm was cute and removed from the center of the silicon block. The samples sized 10mm×10mm×10mm were cut from bottom to top of the silicon column every 20mm by diamond saw.

Spectrometer (ICP-MS).

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The impurity concentrations of each sample were determined by Inductively Coupled Plasma Mass

Figure 3 shows the crystal growth rates in experiment. The crystal growth rate is obtained by

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measuring the height of silicon ingot every once in a while with a quartz rod, therefore, Fig. 3 shows

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the average growth rate over a period of time rather than the instantaneous growth rate. The crystal growth rates fluctuate strongly with time as shown in Fig. 3. In addition, the ke calculating by Eq. (2) is inserted into Fig. 3 as a thumbnail Figure for showing the segregation of iron impurity in the experiment. It is obvious that the values of ke also fluctuate strongly with time. Figure 4 shows the computational and experimental data of the distribution of iron impurity in silicon ingot. The distribution of iron impurity, ρs (ppmw), can be calculated by Eq. (8). The computational data with the stagnant layer thickness value of 0.004m, 0.005m and 0.006m by the

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ACCEPTED MANUSCRIPT presented model are plotted in Fig. 4 using red dash line, solid line and black dash line, respectively. The experimental data is plotted using point. The computational data by Scheil equation with the average of crystal growth rate is also plotted in Fig. 4 using blue dash line for comparison.

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The results show that it is appropriate to the experimental conditions with the stagnant layer thickness value of 0.005m. The computational data by the presented model is more consistent with the experimental data compared with the computational data by Scheil equation, particularly those

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locations that deviation from the average of crystal growth rate, such as points A, B and C. However,

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the experimental data are considerably higher than the computational data at the end region of solidification. Therefore, other factors may also influence the distribution of iron impurity during silicon purification by directional solidification, for instance, Mei report that there exists impurity diffusion in silicon solid during directional solidification process [25]. Rational reason for this

4. Model Application

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phenomenon will be given in subsequent research.

The model has many potential applications in practical production. For demonstrate the

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applications of the model, any assumption of the parameters in model should be clear: initial

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concentration of iron impurity in the raw silicon is 694.9 ppmw, the total solidified length is 0.248m and the iron content standard is 0.1 ppmw. 4.1. Prediction of impurity distribution As discuss in Section 2, distribution of impurity in silicon ingot can be predicted with the

instantaneous value of crystal growth rate during silicon purification by directional solidification. It is very important for evaluating the product quality in practical production. Thus, this model can be used to evaluate the effect degree of production accident. Production accidents are very common in the

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ACCEPTED MANUSCRIPT practical production. It is necessary to analyze the effect degree of production accident on the quality of product. Fig. 5 shows the influence of production accidents occurred in practical productions on the distribution of iron impurity. Three times of power failures, t1, t2 and t3, were occurred in the production

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process, and each time of power failure can make the iron impurity concentration, ρ1, ρ2 and ρ3, relative higher in silicon ingot. It indicates that production accident has a significant affect on the distribution of iron impurity. And the effect degree of each production accident is different.

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To demonstrate the problem to further, assuming power failure appeared twice in the production

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process: when solidified fraction is equal to 0.2, the first time of power failure occurred, and the backup power is enabled when the solidified fraction is equal to 0.235; when solidified fraction is equal to 0.7, the second time of power failure occurred, and the backup power is enabled when the solidified fraction is equal to 0.735. The crystal growth rate during the process and the distribution of iron

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impurity in silicon ingot are as shown in Fig. 6. Shadow region represent the exceeding standard region for iron impurity content. The peaks value of iron impurity content in two time of power failure, C1 and C2, are different, and the exceeding standard region for iron impurity content, |x1x2| and |x3x4|, are

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different, even though the change of crystal growth rate, from 1×10-6 m·s-1 to 1×10-5 m·s-1, in two time

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of power failure is same. It indicates that the distribution of iron impurity is correlated with the instantaneous value of growth rate, the occurrence time and the duration of power failure. And the production yield, which is equivalent to |x5|-|x1x2|-|x3x4|, is directly correlated with the distribution of iron impurity.

4.2. Optimization of process parameters The model can be also used to optimize process parameters. Two different production processes have been compared in Fig. 7. The red dash line indicates the optimized crystal growth rate. The red

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ACCEPTED MANUSCRIPT solid line indicates the distribution of iron impurity in silicon ingot with the optimized crystal growth rate. The black dash line indicates a fixed value of crystal growth rate, which is used by majority of the researchers during silicon purification by directional solidification [13, 26]. The black solid line

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indicates the distribution of iron impurity in silicon ingot with the fixed value of crystal growth rate. Although the production yield of silicon ingot is equivalent (80%) in the two production processes, time consuming are different in solidified stage. The solidified times are 33.99h and 58.25h in the two

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production processes, respectively. Compared with the conventional method, the optimized crystal

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growth rate can make energy consumption reducing 41.65% in the solidified stage. It is significant in practical production.

4.3. Evaluation of maximum production yield

As is known, same production parameters and same furnace being used, but difference of

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production yield often occurred in production. We are frequently mistaken for stability of equipment or stability of production parameters cause of the production yield instability. However, each batch of the raw silicon used in silicon purification by directional solidification is different usually easily neglected

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in production. It may be the key factor for the different production yield.

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Assuming the crystal growth rate during silicon purification by directional solidification is the red line in Fig. 7. The distributions of iron impurity in the silicon ingot with different impurity concentration are as shown in Fig. 8. Maximum yield with initial concentration of 500 ppmw, 1000 ppmw, 1500 ppmw and 2000 ppmw are 0.93, 0.70, 0.64 and 0.61 respectively. The results show that same production parameters and same furnace being used, but difference of production yield can be obtained. 5. Conclusions

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ACCEPTED MANUSCRIPT A theoretical model to determine distribution of iron impurity during silicon purification by directional solidification with fluctuant crystal growth rate is proposed in this paper. Distribution of iron impurity can be calculated with fluctuant growth rate in practical production. The results show that

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distribution of iron impurity is directly correlated with the instantaneous value of crystal growth rate. The fluctuant distribution of iron impurity during silicon purification by directional solidification can be well explained. Many potential applications of the model in practical production are found, such as

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predicting the distribution of iron impurity with non-constant growth rate and evaluating the effect

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degree of production accident, design and optimization of the process parameters and evaluating of maximum yield for raw silicon with different impurity concentration. Silicon purification with low energy consumption is possible through the model proposed in this paper. Acknowledgments

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The authors gratefully acknowledge financial support from the Natural Science Foundation of China (Grant No. U1137601, 51304033 and 51404053), China Scholarship Council (File No. 201506060102) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant

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directional solidification of multicrystalline silicon ingots. Acta Mater., 2011; 59: 7703-7710.

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ACCEPTED MANUSCRIPT Figure captions Fig. 1. Sketch of the directional solidification process. Fig. 2. Crystal growth rate and ke in the experiment.

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Fig. 3. Computational and experimental data of the distribution of iron impurity in silicon ingot. Fig. 4. Influence of production accidents occur in practical productions on the distribution of iron impurity.

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Fig. 5. Distribution of iron impurity in silicon ingot with non-constant crystal growth rate.

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Fig. 6. Distribution of iron impurity in silicon ingot with different process parameters. Fig. 7. Production yield for raw silicon with different impurity concentration during silicon purification

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by directional solidification.

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ACCEPTED MANUSCRIPT Table Table 1: Parameters used in the theoretical model. Property

Value

Units

ρ0

Original mass concentration of impurity in raw silicon

ppmw

ρs

Mass concentration of impurity in solid phase

ppmw

ρl

Mass concentration of impurity in liquid phase

ppmw

ke

Effective segregation coefficient

k0

Equilibrium segregation coefficient

v

Growth rate

δ

The stagnant layer thickness

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Symbol

f(v)

8×10-6

m·s-1

0.005

m

2

m ·s-1

Diffusivity of impurity in molten at 1687K

A

Sectional area of impurity diffusion

L

Total solidification length

fs

Solidified fraction

△x

Thickness of the element

MTotal

Total content of iron impurity in raw silicon

kg

Ms

Total content of iron impurity in solid phase

kg

Ml

Total content of iron impurity in liquid phase

kg

γ

Kinematic viscosity of the melt

m2/s

ω

Angular velocity of rotation

rad /s

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Dl

17

m2 m m m

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Fig. 1

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Fig. 1: Sketch of the directional solidification process.

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Fig. 2

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Fig. 2 Photograph of silicon ingot as produced in experiments and sampling method.

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Fig. 3

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Fig. 3. Crystal growth rate and ke in the experiment.

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Fig. 4

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Fig. 4. Computational and experimental data of the distribution of iron impurity in silicon ingot.

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Fig. 5

Fig. 5. Influence of production accidents occur in practical productions on the distribution of iron

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impurity.

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Fig. 6

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Fig. 6. Distribution of iron impurity in silicon ingot with non-constant crystal growth rate.

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Fig. 7

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Fig. 7. Distribution of iron impurity in silicon ingot with different process parameters.

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Fig. 8

Fig. 8. Production yield for raw silicon with different impurity concentration during silicon

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purification by directional solidification.

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ACCEPTED MANUSCRIPT Highlights

Distribution of iron impurity is calculated with fluctuant growth rate. Fluctuant distribution of impurity in silicon ingot is explained.

Process parameters can be design and optimization by the model.

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Effect degree of production accident can be evaluated by the model.

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Maximum yield for different impurity concentration of raw silicon can be evaluated.