Czochralski silicon pull rate limits

Czochralski silicon pull rate limits

Journal of Crystal Growth 54 (1981) 267—274 North-Holland Publishing Company 267 CZOCHRALSKJ SILICON PULL RATE LIMITS Samuel N. REA * Texas Instru...

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Journal of Crystal Growth 54 (1981) 267—274 North-Holland Publishing Company

267

CZOCHRALSKJ SILICON PULL RATE LIMITS Samuel N. REA

*

Texas Instruments Incorporated, Dallas, Texas 75265, USA Received 22 July 1980; manuscript received in final form 10 February 1981

A detailed thermal model of the Czochralski silicon process was developed to predict maximum crystal pull rates as a function of crystal size and various puller parameters. Experimental data are presented which confirm the model validity. Extrapolations of the model are shown to provide guidance for possible tradeoffs in equipment design or operational mode.

1. Introduction

gradient at the interface is given by [3,4]:

Silicon crystal pulled by the Czochralski technique (Cz silicon) provides over 90% of the single crystal wafers utilized by the worldwide semiconductor

Vm~=

industry. This large and growing market base has resulted in continual improvements in Cz silicon technology with 100 mm diameter crystals routinely grown today and 125 mm and larger wafers available in evaluation quantities. Large diameter Cz silicon is playing a key role in meeting near-term goals of the US photovoltaic effort [1]. Lower cost polysiicon starting material [21 combined with higher productivity pulling techniques offer promise of substantially reducing the cost of Cz silicon used in solar cells. One element of the productivity matrix is the crystal pull rate since, obviously, higher pull rates imply greater throughput of the Cz process. This paper presents an improved Cz pull rate model which is useful in assessing pull rate limits for a variety of operational parameters.

(~),

~

(1)

ifP

in which k is the crystal conductivity, h1f the heat of fusion, p is crystal density, and dT/dx the temperature gradient in the crystal at the melt interface. All properties are evaluated at the crystal melting point. The minus sign in eq. (1) accounts for the fact that dT/dx is a negative quantity for the usual coordinate system in which x is zero at the interface and positively increases along the crystal length. For a one-dimensional analysis, Vmax depends only on the crystal temperature gradient at the interface which, however, is a very complex function of puller geometry and ambient conditions. Fig. 1 depicts the thermal model assumed in this analysis. Under the assumptions: (1) one-dimensionality, (2) constant crystal diameter, (3) constant pull rate, (4) constant crystal density, (5) constant specific heat, and (6) steady state conditions, the temperature distribution in a growing crystal is governed by:

2. Analysis It is well known thatthemaximumCzpullrate,

dIdT\ dx’~th)

dT

2h

2

Vpc~-——(T—T~)——qR0,

(2)

Vmax, assuming a flat growth interface, no radial

temperature gradients, and zero melt temperature

*

Present address: ARCO Solar, Inc., Chatsworth, California 91311, USA.

0 022-0248/81/0000—0000/$02 .50 © North-Holland

in which T is crystal temperature at x, V is pull rate, c is specific heat, h the thermal convective coefficient, r the crystal radius, Ta the average furnace interior ambient temperature, and q~is the radiation heat flux from the crystal at positionx.

S.N. Rea / Czochralski silicon pull rate limits

268

Ta

I

Thermal radiation is the dominant heat transfer mode in the Cz silicon process. A crystal element (fig. 1) exchanges thermal radiation with the melt surface, crucible wall, and puller interior surfaces, all of which are at different- temperatures and exhibit different surface emittances. Thus, the net radiative flux, q~,is a very complex function of geometry, temperatures, and surface emittances. With the surfaces and nomenclature indicated in fig. 1 and by use of radiation enclosure theory [5], a reasonable approximation of q~is given by:

-

1~R~ [A]]~2~

__

CONVECTION

fl~ ~

___

___

j __

I q4

ITil T~ F23T

F24T~ Fdl_2T~— Fdl_3T~— Fdl_4T

-

=UI I

MOITENS~JCON—



(l—F4_4)T~—F42T~—F4_3T

LF!_2T~+F1_3~ +F1_4T~ Fig. 1. Czochralski crystal thermal model.

(5)

The coefficients of the matrix A in eq. (5) are: a11 =

Boundary conditions on eq. (2) are: x0:

(3)

TTm;

x~L: ~ dx



k

T:)~(T_Ta);

where Tm silicon melting point (1685 K), e = crystal emissivity at seed end, and a = Stefan—boltzmann 2 K4). In eq.(5.729 (2) theX 1012 first term constant W/cmrepresents axial conduction up the crystal; the second term describes the rate at which heat is transported due to the crystal motion; the third term represents thermal convection from the crystal surface; and the fourth term involves the thermal radiation from the crystal surface. Boundary condition (3) states that the crystal temperature at the growth interface is the melting point and (4) relates the temperature gradient at the seed end to the thermal radiation and convective heat transfer from the crystal top. The radiative term q~is proportional to temperatures to the fourth power and can have a variety of values depending upon the physical parameters and assumptions employed,

a12=—(1—62)Fd12/e2,

(

~

e4)Fdl4/e4,

a14

=

0,

a21

=

0,

=

l/E2

=

—(1

=

aF2_1

a22 a 23 a24

a3 1 = a32 = —(1 a33





4)

F24/e4

2)

F4_2/e2

1/E4



=

0 (I



2)

a43

=

(1



a44

=

a.

a34

a41 a42

(1



~)F4_4/e4

=

=

F1_2/e2

(6)

S.N. Rea / Czochralski silicon pull rate limits

The F’s in eqs. (5) and (6) are radiation view factors and the es are emittances of the surfaces shown in fig. 1. The radiation flux qp~determined from eq. (5) is a function of both x and T since the Fdi factors are functions of x. Rea [6] presents the expression for Fd12 and illustrates how the other view factors can be determined. In order to solve eq. (5) for q~,the various surface temperatures must be specified. In all modeling work the silicon melt temperature (T2) was assumed 1698 K which is a nominal value experimentally observed for a wide variety of crystal pulls. The crucible liner temperature (7’4) was determined experimentally and its temperature as a function of H is shown in fig. 2. The data in fig. 2 were derived from temperatures obtained by manually scanning the side of a crucible with a PYRO Micro-Optical Pyrometer during 100 mm crystal growth. The measured crucible liner temperature distributions were averaged over the distance H and these averaged temperatures are shown in fig. 2. Generally, the liner temperature decreases exponentially with distance from the melt level and, therefore, the use of an average liner temperature in the modeling work is an approximation. The puller interior ambient temperature (T3 or T~)varies widely so that use of a single value for this temperature is a gross approximation. Fortunately, as will be shown later, this temperature has a minor influence on pull rate so the value used in modeling is not critical, The silicon total emissivity as a function of ternperature was estimated from Runyan [7] combined with measurements made by Texas Instruments on

I

100C

I

I

I

~~~169R

1060’

0

I

I

I

I

I

I I

oo

0.8

I

I

I

-

\ —

02

-

-

I ~

~

1200

.

1400

I

I

1800

18=

TEMPERATURE.K

Fig. 3. Estimated total emissivity of silicon.

molten silicon and at room temperature. The idealized emissivity curve is shown in fig. 3. A value of 0.3 was used for molten silicon (2). Crucible emittance, 4, was estimated at 0.59 based on data in Katzoff [8]. With q~now defined by eqs. (5) and (6) it is possible to solve the differential equation (2). Data from Glassbrenner and Slack [9] shown in fig. 4 were used for silicon thermal conductivity as a function of temperature. The convective heat transfer coefficients, h, were computed using natural convection relations [10] and are shown in fig. 5. Conventional pullers operate under reduced pressures of around 20—30 Torr with argon purge. Fig. 5 indicates a vanishingly small heat transfer coefficient at those pressures so that setting h = 0 in eq. (2) is justified and this value was generally used in the analyses. An iterative procedure was employed to solve eq. (2) for the maximum pull rate. A pull rate, V, was assumed and eq. (1) was usec to compute dT/dx at x = 0. This dT/dx, along with the assumed V, were

I 1.8

12=

269

I

I

400

600

I

I

I

I

I

1000

1200

1400

1600

K

I

I

20 40 00 80 100 DISTANCE FROM MELT SURFACE TOCRUCIBLE LIP. H. ,rn,

I

120

Fig. 2. Experimentally determined crucible wall temperature as a function of melt level for a 12 kg crucible.

0 800

Fig. 4. Silicon thermal conductivity.

270

S.N. Rea / Czochralski silicon pull rate limits 1O~

I

I 111111

I

I 11111

I

I

I

Constant silicon density of 2.3 1 g/cm3 was assumed and a specific heat of 920 J/kg K was used in eq. (2). In eq. (1) p = 2.29 g/cm3, k = 0.2 16 W/cm K, and a value of 0.502 W ‘h/g was used for the heat of fusion.

III:

.

.

~

i io2 a,



-

HELIUM

if

F

3. Experimental

II

a

Fig. 6 contains two data sets of experimental pull rate limits. The Digges [11] data were obtained in a commercial puller employing a 6 kg crucible with helium heat transfer enhancement. Strictly, these

15

2

:

T,,~~8S5K

• • I

1

I

I I II

I

il

I I

I I

HEAT TRANSFER COEFFICIENT. N. W/0,,2K

~~-2

Fig. 5. Crystal heat transfer coefficients.

then utilized to integrate eq. (2) and the boundary condition (4) was computed. If this boundary condition was not satisfied, a new V was chosen and the process repeated until adequate closure on the boundary condition was obtained. Solution convergence was slow in many cases due to the strong nonlinearities in q~coupled with the rapid temperature variation in the crystal over relatively short distances. Fig. 6 shows computed results of the maximum pull rate as a function of crystal diameter. The band in the figure is due primarily to the crystal length variation since, surprisingly, a 500 K swing in ambient temperature produces a negligible variation in maximum pull rate. Fig. 6 is based on a 12 kg crucible having R = 123 mm. Crucible melt level H varied with the crystal diameter and length from 67 to 160 mm.

I 40

I

I

-



1

I I

~

I -

• 5~50S 11 o n~

-

-



-

20

••

-

data should not agree with the theoretical curves since the puller geometry and operational mode were quite different from that assumed in the model. However, the helium enhancement compensated somewhat for the geometric differences and the data tend to support the pull rate model. The Rea data were obtained with a Varian Model 2848A puller modified for 12 kg charges. All runs were made under a low-pressure argon ambient similar to that assumed in the model. Once top growth was completed and the crystal brought out to the desired diameter, the average pull rate was systematically increased in 1.0 cm/h increments every 30 mm. The crucible lift was slaved to the pull rate so that melt level remained fixed relative to the diameter sensor and minor diameter corrections were made by fluctuating the pull rate around the average set point. Melt temperatures were maintained by the puller control system at a level slightly above freezing. This procedure was continued until freezing was just visible on the silicon melt surface and the pull rate at these conditions was considered the maximum. Obviously, this maximum pull rate was somewhat imprecise since melt freeze does not occur instantaneously so a qualitative judgement was required. Nevertheless, the data were repeatable and the overall agreement with theory is acceptable in view of the numerous model simplifications and ~~~~: 5ntul uncertainties in pulling Cz silicon at 4. Discussion

I

5 0

20

I

I

I

CHVS14~0I4HETER

I

I

I

100

120

100

Fig. 6. Theoretical and experimental Cz silicon pull rates,

Several pull rate models have been proposed over the past two decades and all have proven to be opti-

S.N. Rea / Czochralski silicon pull rate limits

mistic in comparison to actual experience. There are several reasons for this with a primary one being the underestimation in analyses of thermal radiation exchange between the crystal, melt, and crucible. For instance, Billig [3], with modifications by Ciszek [12], neglected the second and third terms in eq. (2) and assumed radiation from the crystal surface to a black body environment at zero temperature. This permits obtaining a closed-form expression for pull rate which is high by about a factor of two over the present model. The results are instructive, however, since they do predict the inverse diameter square root dependency of pull rate proven by experiment. Kuo and Wilcox [13] combined the radiation and convection terms in eq. (2) and were able to obtain a pull rate solution approximately 30% higher than the present model. Their model, however, overstates the ambient temperature influence which is a result of linearizing the radiation term, q~. Arizumi and Kobayashi [14] recognized the importance of the radiative exchanges but assumed a crucible emittance of unity and did not predict pull rates. Thus, a direct comparison cannot be made with the present model. Other factors enter into actual Cz silicon growth which tend to lower maximum pull rates. For instance, a truly zero liquid-side temperature gradient, assumed in eq. (1), is difficult to achieve and maintain for a sustained period. The melt surface continually loses heat to the puller ambient which automatically creates a liquid gradient. This can be overcome to some extent by dropping the melt level in the crucible and varying the crucible rotational rate

40

I

I

I

I

I

I

271

I

I

I

0

-

niHc~~ 1400

v=~33,~A

— “.‘..~

1200

-

~



,,,,

-



-



-

I I

60

100

200

I

I

1010

406

Fig. 8. Theoretical Cz silicon axial temperature profiles.

to achieve better mixing. The net effect will still lower the pull rate due to the insulation effect of the crucible side. Figure 7 illustrates this effect. The dashed curve assumes a constant H 60 mm which might be achieved by continuous melt replenishment to the crucible. The solid line is for a batch pull. At L = 400 mm the melt replenishment case shows a 4 cm/h improvement over batch pulling. This improvement is due solely to the lower crucible height possible with the melt replenishment. Another limiting factor on pull rates is the thermal response in actual pullers. The melt temperatures cannot be changed instantaneously due to thermal inertia. During rapid crystal growth it is possible for the puller temperature control system to sufficiently lag actual crucible melt temperatures so that melt freeze occurs. This is prevented by pulling slow enough to maintain thermal equilibrium throughout the puller system. I

I I

I

I

20



BATCH PULL

i~

I ‘6056

1

4116 HELIUM

OR~I~MEN1~H601616

100

200

006

CRYSTAL LENGTH.

40~

600

600

1616

Fig. 7. Theoreticai comparison os batch and melt replenishment pull rates.

12 0

I

I

I

I

I

200

460

000

000

1000

1200

AMBIENT TEMPERATURE. K

Fig. 9. Effects or puller ambient conditions on pull rate.

S.N. Rea / Czochralski silicon pull rate limits

272

In spite of possible optimistic results, the model described here is useful in predicting trends and pinpointing advantageous puller design/operating avenues. Already mentioned in fig. 7 is the pull rate advantage continuous melt replenishment has over batch pulls. Not only is the pull rate headroom opened but a significant productivity improvement can result due to greater crystal length per run. Axial temperature gradients for two batch pull crystal lengths at maximum pull rates are depicted in fig. 8. The pronounced hump in the larger crystal is due to thermal radiation interactions with the crucible which become more significant as the melt is depleted near the end of a pull. All the model results presented assume vacuum conditions in the puller. Fig. 9 illustrates the effect of ambient conditions on maximum pull rate. If 1 atm argon were utilized, the pull rate could be increased only 0.5 cm/h over a wide ambient temperature range. Fig. 9 emphasizes again the slight effect ambient temperature has on pull rate. Under low pressure conditions, the pull rate is virtually independent of ambient temperature up to 600 K. The influence of crucible size on pull rate is shown in fig. 10. Larger batch sizes imply larger crucibles which have a slight effect on maximum pull rate. Heat losses from larger crucibles increase dramatically, however, so that there is a practical upper limit on crucible size. The solution of eq. (5) yields the heat loss from the melt surface, q 2, and crucible wall, q4. The sum of these quantities is the total heat loss from the crucible. For 100 mm crystal growth, increasing the crucible size from 10 to 25 kg increases the heat loss from 9 to 17 kW. Thus, it appears preferable to increase crucible capacity by increasing

crucible depth rather than diameter although a small penalty will be paid in maximum pull rate with deeper crucibles. Perhaps of academic interest only is the modification of crucible emittance by use of heat shielding. Fig. 11 shows the effect on pull rate of varying cmcible emittance. Conventional fused silica liners used in Cz silicon have an emittance around 0.59 at operating temperatures. As much as a 20% pull rate increase could be gained through low emittance crucibles or heat shields. Nevertheless, the logistics and economics of employing such shields probably preclude their use in a high-production environment. Validity of the one-dimensional model developed here to describe large-diameter Cz silicon growth merits additional discussion. Under the assumption of rotational symmetry, Czochralski crystal temperatures can be determined more accurately from the two-dimensional partial differential equation:

I

I

k aT

~



a

=

0:

-s-—

a

=

r

—k —

=

0

(8) 9

=

aa ___________________________________________



I

24

I

20

2T

aT



Vpc— + k—~—+-— = 0, (7) 3x ax aa a where a is crystal radial position and, for simplicity, thermal conductivity is assumed constant. In this formulation q~ and the convective cooling term appear as a boundary condition at the crystal surface. Since convection is negligible in commercial silicon pullers only qR is of interest. Then, radial boundary conditions become: k—i-

I

I

I

I 22

I

I

1201616 CRYSTAL 400o,n H=60~~





~

SILICA 10 12 RO

I

I

100

120

I

I

140 100 CRUCIBLE RADIUS. ‘I,,



I 180

Fig. 10. Effect of crucible radius on pull rate.



I 200

0

0.1

I 0.2

I

I

0.3 0.4 CRUCIBLE EMITTANCE

I 0.5

I 0.0

Fig. 11. Influence of crucible wall emittance on pull rate.

S.N. Rea / Czochralski silicon pull rate limits

Axial boundary conditions remain as before in eqs. (3) and (4) although, technically, eq. (3) should be modified to account for interface curvature which is not known a priori and creates a sizeable complica-

273

two-dimensional eq. (7), the ordinary differential equation results: dT 2 d2T k ~j~j- Vpc 4~j~ = 0, (12) —



tion necessitating numerical solution [14—16]. Brice [4] obtained a solution to eq. (7) with V = 0 by assuming a flat growth interface and a simplified radiative boundary condition rather than eq. (9). The one-dimensional analysis of eq. (2) will exhibit maximum error at the melt interface because q~is maximum at that location. At crystal positions some distance from the interface the radial temperature variations flatten out due to both smaller q~’s as well as higher silicon thermal conductivity at the lower average crystal temperature. Thus, the onedimensional assumption is reasonably accurate within one to two crystal diameters from the growth interface. The numerical analyses cited previously, supported by Wilcox and Duty [17], show that the growth interface under normal circumstances is concave toward the melt and that radial temperature gradients in the crystal can be approximated by parabolic equations of the 2form: ~1o~ T T = Ca / —

where T~is the crystal centerline temperature and C is a constant. Eq. (10) automatically satisfies boundary condition (8) and from eq. (9) the constant C can be evaluated with the result: 2

T (rqR/2k)(a/r) (11) At the melt interface q~is typically around 11—12 W/cm2. Thus, a 100 mm diameter silicon crystal can have a center-edge temperature variation as large as 130°C in the immediate vicinity of the interface. However, calculated radiation fluxes within one diam= eter from the interface show q~to be near zero implying that the radial temperature profile is much flatter at that location. In fact, q~goes negative (heat into the crystal surface) in long crystals in the vicinity of the crucible lip which results in the pronounced temperature hump shown in fig. 8. Near the seed end q~will be fairly constant around 0.5— 0.8 W/cm2, depending on crystal and crucible dimensions and the center-edge temperature difference is only a few degrees. If eq. (11) is differentiated and inserted into the —

r

which is identical to the one-dimensional model, eq. (2), with constant conductivity and zero surface convection. This result implies that the analysis presented here is consistent with parabolic radial profiles and should not be in serious error In predicting average crystal axial temperatures, hence pull rates, even at substantial crystal diameters. This conclusion is supported by the agreement between theory and experiment shown in fig. 6.

5. Conclusions The Cz silicon modeling effort presented here indicates that very little can be done to increase pull rates. The pull rate is determined primarily by the crystal diameter and the crucible/melt level geometry. Thus, the pull rate curves in fig. 6 pretty wellheat out set the transfer limitsaugmentation. achievable in practice with or withA potential problem enhanced by high pull rates in Cz silicon is that of structure loss due to excessive impurity concentration at high growth rates [2]. Digges and Shima [11] suggest that the maximum practical pull rate is only 80% of theoretical maximum due to this structural breakdown phenomenon.

=

Acknowledgements The work presented in this paper was funded in part by the Low-Cost Silicon Solar Array Project, Jet Propulsion Laboratory, California Institute of Technology, sponsored under an interagency agreement between the Department of Energy and the National Aeronautics and Space Administration. Mr. Robert B. Morgan of Texas Instruments aided in computer-programming the equations presented.

References [1] The Low-Cost Siison Solar Array Project, administered by the Jet Propulsion Laboratory, California Institute

274

[2]

[3] [4] [5] [6] [7] [8] [9]

S.N. Rea / Czochralski silicon pull rate limits of Technology, and sponsored by the National Aeronautics and Space Administration under an interagency agreement with the Department of Energy. R.H. Hopkins, R.G. Seidensticker, J.R. Davis, P. RaiChoudhury, P.D. Blais and J.R. McCormick, J. Crystal Growth 42 (1977) 493. E. Billig, Proc. Roy. Soc. (London) 229 (1955) 346. J. Brice, J. Crystal Growth 2 (1968) 395. R. Siegel and J.R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972) ch. 8. S.N. Rea, AIAA J. 13 (1975) 1122. W.R. Runyan, Silicon Semiconductor Technology (McGraw-Hill, New York, 1965) p. 213. S. Katzoff, NASA SP-55 (1965) 251. C.J. Glassbrenner and G.A. Slack, Phys. Rev. 134 (1964) A1058.

[10] W.M. Rohsenow and H. Choi, Heat, Mass, and Momenturn Transfer (Prentice-Hall, Englewood Cliffs, NJ 1961) p. 159. [ill T.G. Digges, Jr. and R. Shima, J. Crystal Growth 50 (1980) 865. [12] T.F. Ciszek, J. Appl. Phys. 47 (1976) 440. [13] V.H.S. Kuo and W.R. Wilcox, J. Crystal Growth 12 (1972) 191. [14] T. Arizumi and N. Kobayashi, J. Crystal Growth 13/14 (1972) 615. [1511. Arizurni and N. Kobayashi, Japan. J. Appi. Phys. 8 (1969) 1091. [16] N. Kobayashi and T. Arizurni, Japan, J. Appl. Phys. 9 (1970) 361. [17] W.R. Wilcox and R.L. Duty, J. Heat Transfer 88 (1966) 45.