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Modeling Metallic Precipitate Subcontract Report 

NREL/SR-520-37991 Dissolution in Silicon Under May 2005 

Point Defect Injection 

Final Subcontract Report 

20 January 2004 – 19 January 2005 

T.Y. Tan

Duke University 

Durham, North Carolina 

NREL is operated by Midwest Research Institute ● Battelle Contract No. DE-AC36-99-GO10337 
Modeling Metallic Precipitate Subcontract Report 

NREL/SR-520-37991 Dissolution in Silicon Under May 2005 

Point Defect Injection 

Final Subcontract Report 

20 January 2004 – 19 January 2005 

T.Y. Tan

Duke University 

Durham, North Carolina 

NREL Technical Monitor: Richard Matson 

Prepared under Subcontract No. XAD-4-33653-01 

National Renewable Energy Laboratory 

1617 Cole Boulevard, Golden, Colorado 80401-3393 

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Table of Contents 

Introduction.................................................................................................................................... 1 

Formulation of the Problem ........................................................................................................... 1 

Numerical Simulation Results ....................................................................................................... 4 

Conclusions.................................................................................................................................... 5 

References...................................................................................................................................... 5 

List of Figures 

Figure 1. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from

the Si matrix by an Al-Si liquid layer at the wafer backside (200 μm position).............6 

Figure 2. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates 

from Si by an Al-Si liquid layer at the wafer backside (200 μm position) ....................6 

Figure 3. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from

Si by an Al-Si liquid layer at the wafer backside (200 μm position) .............................7 

Figure 4. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from

Si by an Al-Si liquid layer at the wafer backside (200 μm position) .............................7 

iii 
INTRODUCTION 

Gettering of dissolved and precipitated metallic contaminants is important in Si solar cell 

fabrication for improving the cell efficiency. Gettering is needed because Si is an indirect bandgap semiconductor, with its carrier lifetime controlled by electrically active impurities (metals) 

and defects. Such impurity and defect electrical activities are responsible for the desirable fast 

action of switching devices, but also for the detrimental result of device/circuit failures due to 

excessively large pn junction leakage current and extremely short storage capacitor charge holding times. In the case of solar cells, these impurities and defects lead to low cell efficiencies. 

Point defects must be present to their thermal equilibrium concentrations. The presence of impurities is, however, an extraneous factor that can be controlled by cleaning and by gettering. 

We have recently devoted an effort to identify the physics and to model the metal electrical 

activity,1-3 and the metal gettering processes.4-7 The latter includes metal dissolution from the 

precipitates, diffusion of metal atoms to, and their stabilization at the gettering sites. In the present study, we further address the role of point defect in affecting the precipitate dissolution 

process, which originates from the fact that almost all metal silicide precipitate species are associated with a volume misfit with the Si matrix. We have formulated the problem using an Al-Si 

liquid layer as the gettering substance and interstitial Fe and FeSi2 as the metal species and precipitates that are gettered. In this problem, the misfit of the precipitate is assumed to be accommodated by the point defect species vacancies (V) and Si self-interstitials (I). The point defects 

are either generated or absorbed during gettering and are hence rendered non-equilibrium for 

which equilibrium restoring process is assumed to be in- or out-diffusion from or to the Si exterior surfaces. Based on the physical modeling, numerical simulation results showed that injection 

of the appropriate point defect species from the Si exterior surfaces can be an effective means of 

speeding up the gettering process. 

FORMULATION OF THE PROBLEM 

Gettering of M by the Al-Si liquid involves the segregation processes taking place at the interface of Si and the liquid layer, which occurs simultaneously with impurity diffusion. Accounting for M diffusion, segregation, and dissolution of the precipitates, the time change rate of M is 

given by 

⎡ ⎛ ⎞⎤ 4πr 2 ∂CM 

= 

∂ 

∂t ∂x 

⎜

⎝

∂CM CM ⎜ ∂mM dr ⎢

⎢

⎣

⎥

⎥

⎦

⎟

⎠ DM − ⎟ − ρ (1) dt , ∂x ∂x ΩMSix mM 

where DM is the M diffusivity, CM is the M concentration in Si, mM is the M segregation coefficient, ρ is the precipitate density, r is the radius of the precipitate, ΩMSix is the volume of one 

MSix molecule, and dr/dt is the precipitate radius change rate given by 

* dr 

dt 

= ΩMSix 

CM −

r 

CM , (2) 

* where CM is the dynamical equilibrium concentration of M at the precipitate-Si interface which 

maintains the precipitate to neither grow nor shrink. 

1
Irrespective of whether the impurity is a substitutional or an interstitial species, upon precipitation, there is usually a volume change between the precipitate and the involved metal and Si 

atoms, i.e., the misfit. For the silicide MSix with its molecules formed by one M atom and x Si 

atoms, mass conservation condition requires the reaction 

M + xSi ⇔ MSix (3) 

to hold. On the other hand, if the precipitate is strain-free, volume conservation condition requires the relation 

ΩMSix = ΩM + (x + y)ΩSi (4) 

to hold. In Eq. (4) ΩM and ΩSi are respectively the atomic volumes of the M and Si atoms, and 

y is the misfit, with its value being either positive or negative. For M being a substitutional species ΩM = ΩSi , and for M being an interstitial species ΩM =0. Upon continued growth or shrinkage of the precipitate, the misfit cannot be cumulative, since then the strain energy will be too 

large. In the following discussion, we let such misfits be relieved by point defects. 

For the case assuming that the dominant point defect species is I, the combined mass and 

volume conservation condition requires 

M + xSi ⇔ MSix + yI (5) 

to hold. A consideration of the Gibbs free energy of the system in accordance with reaction (5) 

yields that, at the precipitate-matrix interface 

⎛ ⎞ y ⎛2Ω ⎞ MSix C σ * eq I CM = CM

⎜

⎜

⎝

⎟

⎟

⎠

⎟

⎠ ⎜⎝

⎜ ⎟

holds. In Eq. (6) CM

eq is the thermal equilibrium concentration of M, CI and CIeq are the actual 

and thermal equilibrium concentrations of I in the Si matrix, σ is the surface energy density of 

the precipitate-Si interface, r is the radius of the precipitate, kB is Boltzmann’s constant, and T is 

the absolute temperature. Now, with the CI change rate given by 

exp (6) eq CI rkBT 

4πr 2 ∂CI 

= 

∂ 

∂t ∂x 

⎛

⎜

⎝

DI

∂CI

∂x 

⎞ dr ⎟

⎠ + yρ (7) dt , ΩMSix 

where DI is the I diffusivity in Si, the I dominated case is modeled by Eqs. (1), (2), (6), and (7). 

For the case assuming that the dominant point defect species is V, the combined mass and 

volume conservation condition requires 

M + xSi + yV ⇔ MSix (8) 

to hold. A consideration of the Gibbs free energy of the system in accordance with reaction (8) 

yields 

⎛ ⎞ y ⎛ ⎞ eq CV ⎟

⎠

⎜

⎝

2 ΩMSix σ 

⎜ ⎟ , ⎜

⎜

⎝

⎟

⎟

⎠

* eq CM = CM exp (9) CV rkBT 

2 
where CV and CV

eq are the actual and thermal equilibrium concentrations of V in the Si matrix. 

With the CV change rate given by 

4πr

2 ∂CV 

= 

∂ 

∂t ∂x 

⎛

⎜

⎝

DV

∂CV

∂x 

⎞ dr ⎟

⎠ − yρ (10) dt , ΩMSix 

where DV is the V diffusivity in Si, the V dominated case is modeled by Eqs. (1), (2), (9), and 

(10). 

The most accurate approach is to assume that both I and V contribute to the relieve of the 

misfit, for with the combined mass and volume conservation condition requires the reaction 

M + xSi + zyV ⇔ MSix + (1 − z )yI (11) 

to hold. In reaction (11) z (with a value less than 1) is the fractional contribution of V to the relief 

of the misfit and (1-z) is that of the I contribution. Using reaction (11) in the Gibbs free energy 

considertion of the system, we obtrain 

(1− z)y ⎛ ⎞zy ⎛ ⎞ ⎛ ⎞ eq CV ⎜

⎝

2ΩMSix σ 

exp⎜ CI ⎜

⎜

⎝

⎟

⎟

⎠

* eq CM = CM

⎜

⎜

⎝

⎟

⎟

⎠

⎟

⎠

⎟ .

For this case, the change rate of both CI and CV need to be considered, and they are given by 

(12) eq CI C rkBT V 

4πr 2 ⎛

⎜

⎝

∂C ⎞ I

= 

∂ 

∂t ∂x 

DI

∂CI

∂x 

dr 

+ 4πrIV 

eqCVeq − CICV (DI + DV )(C ⎟ I ) ⎠ +(1− z)yρ , (13) ΩMSi dt x 

2 ⎛

⎜

⎝

∂C ⎞ V

= 

∂ 

∂t ∂x 

DV

∂CV

∂x 

4πr dr eqCV

eq − CICV (DI + DV )(C ⎟ I ) ⎠ − zyρ + 4πrIV , (14) ΩMSi dt x 

where rIV is the reaction radius of I and V. Via the last terms on the right-hand-sides, Eqs. (13) 

and (14) have been written down assuming that the I and V recombination/generation reaction 

I + V ⇔ φ , (15) 

where φ is a Si lattice atom, is operative. The I and V dynamical equilibrium condition according 

to reaction (15) is 

eqCV CICV = CI

eq . (16) 

The assumption that Eq. (16) holds allow to continue the analysis of this case. First, Eq. (12) is 

now reduced to 

⎛ ⎞ y ⎛ y ⎛ ⎞ ⎛ ⎞ ⎞ eq CV 2ΩMSix 

σ 2ΩMSix C σ I ⎜

⎜

⎝

⎟

⎟

⎠

* eq = CM

⎜

⎜

⎝

⎟

⎟

⎠

⎟

⎠

eq ⎟ = CM ⎟

⎠ ⎜

⎝ ⎜⎝

⎜ ⎜ ⎟

Second, note that in Eqs. (12)-(14) z is not yet known, and we can now use Eqs. (13), (14), and 

(16) to obtain the value of z. Equation (16) yields 

CV 

∂CI 

∂t 

+ CI 

∂CV 

∂t = 0 . (18) 

3 

CM exp exp (17) eq CI rkBT C rkBT V 
Assuming that Eq. (16) holds, then Eqs. (13) and (14) simplify to 

2 ∂C dr I 

= 

∂ ⎛ ∂CI ⎞ 4 πr

DI ⎜ (19) ⎝ ∂x 

⎟

⎠ + (1− z) yρ dt , ∂t ∂x ΩMSi x 

2 ∂C ⎛ ⎞ V 

= 

∂ 

∂t ∂x 

∂CV 

∂x 

4 πr dr ⎜DV ⎝ ⎟

⎠ − zyρ . (20) ΩMSi dt x 

Using Eqs. (18) and (19) in Eq. (17), we obtain 

⎛ ⎞

⎟

⎠

∂C ⎛ ⎞ I 

∂x 

∂CV 

∂x 

CV 

∂

∂x 

+ CI 

∂

∂x 

⎜DI DV ⎝ ⎜

⎝ ⎟⎠

z = 

CV + , (21) 2 CI + CV dr 

yρ 

4πr 

ΩMSix 

(CI + CV )

dt 

which is a function of t and x. The case can now be solved using Eqs. (1), (2), (17), (19), (20), 

and (21). 

NUMERICAL SIMULATION RESULTS 

The case of gettering precipitated Fe is of particular interest. Multicrystalline solar Si contain 

regions with high densities of dislocations and metal silicide precipitates, for which the minority 

carrier diffusion lengths are extremely low and cannot be improved by a normal gettering treatment using the Al-Si liquid layer at, e.g., 700o

C for less than 2 h. Our results showed that the 

reason is that precipitate dissolution is involved which requires more extensive treatment. 

As a test of the soundness of the model, numerical simulations were carried out for gettering 

Fe in Si wafers 200 μm in thickness by a 2 μm thick Al-Si liquid layer at the wafer backsurface 

(at the 200 μm position). It is assumed that Fe was introduced at 900oC to its solubility of 

~4.3x1013 cm-3 and then precipitated out to steady state at 700oC to the density of _=1011 cm-3. 

For the interstitial Fe atoms ΩM =0 and for the FeSi2 precipitates ΩMSix =1.85 ΩSi ,

8

 and hence 

y=-0.15 obtains from Eq. (4). The negative y value means that growth of FeSi2 precipitates is associated with a volume contraction while their dissolution with a volume expansion. Thus, generation of V or consumption of I or both will result from FeSi2 precipitate growth, while generation of I or consumption of V or both will result from FeSi2 precipitate dissolution. Consequently, gettering will be faster if a V-supersaturation or I-undersaturation exists. Figure 1 shows 

two hypothetical cases for which CV =10CV

eq holds for case (a) and CV =CVeq for case (b), and it 

is clear that gettering case (a) is more effective becasue of the V supersaturation. In Fig. 1 the Fe 

atom concentration in Si is plotted as a function of position in Si, and gettering time is used as a 

parameter. Another ways of showing the results is to plot the total gettered Fe in the Al gettering 

layer and also to examine the precipitate size in Si, both as functions of gettering treatment time. 

First, Fig. 2 shows the result of the no volume misfit case, to be used to compare with results including the volume misfit. Fig. 3 shows the case assuming, at both Si surfaces, I concentration is 

either at thermal equilibrium value or undersaturated to 0.2 times of the thermal equilibrium 

value. As a function of gettering time, Fig. 3(a) shows the percentage of Fe depleted from the Si 

wafer by gettering into Al, while Fig. 3(b) shows the precipitate size change at the center of the 

wafer. Compared to the case of fig. 2, it is seen that the process of gettering is slowed down by 

4
the volume misfit, but the gettering rate improves with V supersaturation at the Si wafer boundaries. Figure 4 shows a corresponding case for which V supersaturation is assumed instead, for 

which the V concentration at both boundaries are assumed to be either at thermal equilibrium or 

at 4 times of the thermal equilibrium values. Compared to Fig. 2, it is seen that the presence of 

the volume misfit slowed down the gettering process (“equilibrium” case, solid line), but with 

the presence of the Assumed V supersaturation, the gettering rate has essentially became that 

given by Fig. 2. Thus, gettering are more effective when the appropriate point defect supersaturation (V case, Fig. 4) or undersaturation (I case, Fig. 3) are present. For these calculations, the 

point defect thermal equilibrium concentration and diffusivity values are those given in the lit-

-3 erature,9

 as follows. For the I case, we used CI

eq = 5x1030

exp(−4.4eV / kBT ) cm and 

2 DI = 10−5

exp(−0.4eV / kBT ) cm s

-1, which are believed to be accurate. For the V case, we used 

-3 2 CV

eq = 2 x1023

exp(−2 eV / kBT ) cm and DV = 0.1exp (−2eV / kTB ) cm s

-1. It is noted that for 

while the product DVCV

eq is fairly well known, individual values of DV and CVeq are not, and the 

values used in obtaining Fig. 4 represent one extreme choice,

9

 while in the other extreme the V 

formation energy is ~3.5 eV and the V migration activation ion energy is ~0.5 eV, for which our 

results showed that the gettering process will be much slower than the case shown in Fig. 4. 

CONCLUSIONS 

In this study the effect of point defect injection on the dissolution process of metal silicide 

precipitate in Si has been modeled. Practically all metal silicide precipitate species posses volume misfits with the Si matrix material, and this is the basic reason why point defects are involved in the precipitate dissolution processes (as well as in the precipitate formation process). 

Fundamental equations describing such processes have been written. Using these equations, numerical simulations has been carried out for the case of FeSi2, in which the role of point defects 

have been demonstrated: under the appropriate point defect super- or undersaturation conditions, 

precipitate will dissolve faster and hence speeding up the gettering process. This effect may be 

an important contributor to the highly effective optical gettering results obtained at NREL.10 Further works are needed for obtaining a more complete understanding on this subject. 

References 

1. P. S. Plekhanov and T. Y. Tan, Appl. Phys. Lett. 76, 3777 (2000). 

2. M. D. Negoita and T.Y. Tan, J. Appl. Phys. 94, 5064 (2003). 

3. M. D. Negoita and T.Y. Tan, J. Appl. Phys. 95, 191 (2004). 

4. P. S. Plekhanov, R. Gafiteanu, U. M. Gösele, and T. Y. Tan, J. Appl. Phys. 86, 2453­

2458 (1999). 

5. S. M. Joshi, U. M. Gösele, and T. Y. Tan, Solar Energy Materials and Solar Cells 70, 

231 (2001). 

6. T. Y. Tan, "Impurity Gettering in Silicon" in Encyclopedia of Materials: Science and 

Technology, ed. Subhash Mahajan (Elesevier, Amsterdam, 2001) pp. 4031-4042. 

7. P. S. Plekhanov, M. D. Negoita, and T.Y. Tan, J. Appl. Phys. 90, 5388 (2001). 

8. U. Starke, W. Weiss, M. Kutschera, R. Bandorf, and K. Heinz, J. Appl. Phys. 91, 6154 

(2002). 

9. T.Y. Tan and U. Gösele, Appl. Phys. A37, 1 (1985). 

10. Bhushan Sopori, Teh Tan, and David Carlson, “Precipitate Dissolution and Gettering in 

Si Wafers using Optical processing”, abstract submitted to 2005 GADEST. 

5
0 

12 

52,5 

53 

53.5 

54 

0 

12 

38.5 

39 

39.5 

(a) (b) 

Fig. 1. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from the Si matrix by an AlSi liquid layer at the wafer backside (200 μm position). See text for initial and other experimental conditions. Plots 

(a) and (b) show the dissolved Fe concentrations at various gettering times in hrs at 700o

C. It is seen that gettering is 

more effective in (a) than in (b), because in (a) a V supersaturation is assumed. 

Fig. 2. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from the Si matrix by an AlSi liquid layer at the wafer backside (200 μm position). See text for initial and other experimental conditions. The 

plot shows the total % of Fe gettered into the Al layer as a function of time. In this case, it is assumed that the FeSi2 

precipitate is not associated with a volume misfit with the Si matrix. 

6
(a) (b) 

Fig. 3. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from Si by an Al-Si liquid 

layer at the wafer backside (200 μm position). See text for initial and other experimental conditions. Plot (a) shows 

the percentage of Fe gettered to the Al layer at various gettering times in hrs at 700o

C, under either the I thermal 

equilibrium assumption or I undersaturation assumption. Plot (b) shows the corresponding wafer center precipitate 

radius changes. 

(a) (b) 

Fig. 4. Simulation results of gettering dissolved interstitial Fe and FeSi2 precipitates from Si by an Al-Si liquid 

layer at the wafer backside (200 μm position). See text for initial and other experimental conditions. Plot (a) shows 

the percentage of Fe gettered to the Al layer at various gettering times in hrs at 700o

C, under either the V thermal 

equilibrium assumption or V supersaturation assumption. Plot (b) shows the corresponding wafer center precipitate 

radius changes. 

7
ABSTRACT 

We have formulated the problem of gettering of metallic precipitates in Si for which 

there exists a volume misfit between the precipitate and the Si matrix material. The gettering process is modeled using an Al-Si liquid layer, and the volume misfit associated 

with dissolving the precipitates is assumed as accommodated by point defects, which 

may be dominated by vacancies (V), self-interstitials (I), or both contributing. Under the 

condition that V and I attained dynamical equilibrium, we found that for analytic purpose 

the problem reduces closely to either the V or the I alone case, with the fast diffusing one 

dominating. Our initial simulation results on gettering of FeSi2 (with a misfit of –0.15) 

showed that the process can be sped up by the injection of V. 

8
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1. REPORT DATE (DD-MM-YYYY) 

May 2005 

2. REPORT TYPE 

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3. DATES COVERED (From - To) 

20 January 2004–19 January 2005 

4. TITLE AND SUBTITLE 

Modeling Metallic Precipitate Dissolution in Silicon Under 

Point Defect Injection, Final Subcontract Report: 

20 January 2004 – 19 January 2005 

5a. CONTRACT NUMBER 

DE-AC36-99-GO10337 

5b. GRANT NUMBER 

5c. PROGRAM ELEMENT NUMBER 

6. AUTHOR(S) 

T.Y. Tan 

5d. PROJECT NUMBER 

NREL/SR-520-37991 

5e. TASK NUMBER 

PVA52201 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Department of Mechanical Engineering and Materials Science 

Duke University 

Durham, North Carolina 27708-0300 

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REPORT NUMBER 

XAD-4-33653-01 

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NREL Technical Monitor: Richard Matson 

14. ABSTRACT (Maximum 200 Words) 

We have formulated the problem of gettering of metallic precipitates in Si for which there exists a volume misfit 

between the precipitate and the Si matrix material. The gettering process is modeled using an Al-Si liquid layer, 

and the volume misfit associated with dissolving the precipitates is assumed as accommodated by point defects, 

which may be dominated by vacancies (V), self-interstitials (I), or both contributing. Under the condition that V 

and I attained dynamical equilibrium, we found that for analytic purposes, the problem reduces to either the V or 

the I alone case, with the fast-diffusing case dominating. Our initial simulation results on gettering of FeSi2 (with 

a misfit of –0.15) showed that the process can be sped up by the injection of V. 

15. SUBJECT TERMS 

PV; gettering process; metallic precipitates; vacancies (V); self-interstitials (I); Al-Si liquid layer; thermal equilibrium 

value 

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