The weighing method of automatic Czochralski crystal growth

Journal of Crystal Growth 40 (1977) 13-20 0 North-Holland Publishing Company

THE WEIGHING METHOD OF AUTOMATIC

CZOCHRALSKI

CRYSTAL

GROWTH

I. Basic theory

W. BARDSLEY, D.T.J. HURLE and G.C. JOYCE Royal Signals and Radar Establishment,

Received

17 January

Great Malvern, Worm. WR14 3PS, England

1977; revised manuscript

received

5 April 1977

Theory relating crystal radius to the time-variation of the force exerted on the pull-rod by the weight of the growing crystal and surface tension forces is developed. The nature of the relationship is investigated for different classes of materials and an anomalous dependence is demonstrated for the case of materials which expand on solidification and/or not completely wetted by their own melts. The consequences for automatic diameter control are considered.

1. Introduction

understanding of the physical processes involved. It is the purpose of this paper to provide a physical model of the Czochralski weighing technique. Some of the main results of this paper have been originally summarised in refs. [9] and [lo]. A control system based on these ideas is described in a companion paper

Automatic control of crystal diameter during Czochralski growth using either crystal weighing [ 1,9] or crucible weighing [l-3,9] has recently been described. The method is very versatile and has significant advantages. To understand and exploit the process fully an understanding of what the sensor (the weighing cell) is measuring and the way in which it changes with growth conditions must be acquired. The force experienced by the weighing cell during crystal weighing consists not only of the static weight of pull rod and crystal but also has contributions arising from the surface tension of the melt and the hydrostatic head of liquid supported by the growing crystal. The implicit assumption heretofore has been that the rate of change of this force is a measure of the rate of accretion of crystal and hence, for a constant linear growth rate, a measure of the square of the crystal radius. However this will not, in general, be true if changes in the process induce changes in the height of the liquid meniscus or of the direction along which the surface tension vector is acting. It turns out that such effects are particularly important for materials for which ps
1111. Before we can write down an expression for the force experienced by the weighing cell we must first consider the properties of the meniscus.

2. The meniscus With reference to fig. 1, the first question to be settled is the magnitude of the angle 0: which the

CRYSTAL

I

Fig. 1. Schematic 13

representation

1

I

of the meniscus.

14

W. Bards/ev eta!.

/ Weighing method of automatic

ineniscus makes with the vertical at the point of contact with the crystal when the crystal is growing as a right cylinder. At first sight one might suppose that = 0 since, one might argue, if the crystal is to continue to grow at the same diameter any other angle would necessitate surface diffusion. However this conclusion is at variance with experiment for germanium, and the situation has been discussed recently by Bardsley et al. [4]. These authors have shown that *:

C:ochralski crystal growth. I

Table 1 Relevant properties of germanium Property

+ 0LG

USL)

2~SG~LG

(I)

where ULG is the surface tension and USG and cJSL are respectively the average values of the surface free energy of the solid and the solid—liquid interfacial free energy. The component of surface tension acting vertically downward is then: Yst

=

(2)

cos 0L’

0LG

where (3)

OL=O~+Os

(positive Os signifies increasing radius), O~being the inclination of the surface of the crystal to the vertical at the line junction of the three phases. Since the objective is to obtain conditions for the growth of a crystal of constant diameter we will assume that Os is small and expand eqs. (2) and (3) to first power in Os only. Thus:

YoOs,

YstYo

Ref. —

[13]

3

~

5.26 g cm 5.51 g cni~ 616 crgcln 2 216 erg cm 2 40 erg cm 2 595 crgcm 2 159 ergcm2 15 dega)

PL °LG

GSL GSG 00

~

Value

1131 [4]

141

0.228 a) This value from ref. [4]. After this paper was prepared an experimental value of = 13 1° was published 1121.

tance is its height. Tsivinski [5] and Gaulé and Pastore [61have shown that this is given approximately by:

ii

[~

=

Sill °L) +

(~

~ cos 4r O~ (5)

C05 4r o)2]1/2

where ~ = 2aL~/p~gis the Laplace constant, g the acceleration due to gravity and r the crystal radius. ___________________________________________ 0 4

—

ho

03

(4)

where: Yo=uL(;cos0~ =

Ye

= =

02

(a~

0+ ULG

ULG

USL)/2t7SG,

sin O~. 0LG

—

(‘~~G+ 0LG

OSL)1’/2SJSG.

[4ã~G

0I

The above relations are valid only if the liquid does not completely wet the solid (i.e., only if O°L> 0). If wetting is complete and O°L = 0, then 70 = 0LG and Ye

0. The other parameter of the meniscus of impor=

0

G 5

10

5

2 0

r 0 *

Cf. Glossary of symbols, section 7.

cm)

Fig. 2. Dependence of meniscus parameters on crystal radius.

W. Bardsley eta!.

Table 2 Magnitude of various parameters

/ Weighing method of automatic Czochralski crystal grosvth. I

15

(defined in the text) for germanium crystals of the radii indicated

Crystal radius, r

Parameter

0 (cm)

ha — pL)rOhe 2y~/g (PS PL)ha 2pLhO 2yo/r~ig

~wo Xu~

0.25

0.5

1.0

1.5

2.0

0.246 0.116 0.205 0.013 0.324 0.029 2.711 4.852 0.722 0.032

0.315 0.082 0.237 0.030 0.324 0.021 3.471 2.426 0.563 0.034

0.360 0.048 0.253 0.063 0.324 0.012 3.967 1.213 0.494 0.037

0.376 0.033 0.258 0.097 0.324 0.008 4.144 0.809 0.472 0.040

0.384 0.026 0.260 0.130 0.324 0.007 4.232 0.607 0.461 0.043

Again, because we are interested only in small deviations, we can linearise in O~and in a = r r 0, where a is the error in the radius and r0 is its desired value. Thus we can write: h =h 0 + (halro)a hoes, (6)

(static weight at time

t

=

0)

—

f p~g~r2udt

+0

—

(weight 2p~hof crystal grown in time t) + ~r (excess pressure due to meniscus)

where we obtain from eq. (5): =A 4r 0 —

+ ha

=

—

4r0 =

—cos

and A =[13(l

(13 cos O°L 2 A ‘~

(resolved component of surface tension)

4r~ /

r13

oOLf~~.Pi

4r~

—

2irry5~

—

+ (13)2

[~

O~1A

~4r0~

j

J

sino~)÷~~01~)]/

where vis the growth velocity. Substituting eqs. (4) and (6) into (7) and retaining only first order terms:

(

ha FFref=27rp~gr0v0 ~—a0

rh0 —~

V0

4r0 Values for the various materials parameters are given for germanium in table 1 and values of h0, ha and h6 for various r0 are given in table 2 and displayed graphically in fIg. 2.

(7)

2v~

\

+fa dt + ritz

We are now in a position to write down the force experienced by the weighing cell as a function of time. For the case where the crystal is weighed: F(t)m0g

vo~~s)~

(8)

where we have written Fref =

3. The force experienced by the weighing cell

—

m0g + p5g1rr~v0t + 7rr~p~gh0 + 2lrr0Yo,

(9)

which is the expected value of F if the crystal grows at constant velocity v0 with the required radius r0. a0 and ~ are respectively the values of the error in the radius and the slope angle of the crystal at time t = 0; ~=—

2p5v0

[2(m..ho

+y0/r0g)—(p5

PL)hal,

(lOa)

16

W. Bardsley eta!.

Values of

1

1’O

[2~0/r0g

A=

(Ps

-

/ Weighing inel/mod 01 automatic (lOb)

PL)

likely to remain valid at very slow growth rates. In

(F~‘~‘ref)

(II)

2irgr0v0p~

as a re-normalised nieasure of the error signal yields:

r h

Ii H= ~La0

.~

~

0

~ +fadt + ~a

v0AO5.

this event we must expand eq. (15) to second order in

the small quantities to obtain: I ha (v0 ---—a+/1005105 =d. ~

r0

/

(19)

and must now seek to solve the coupled equations

(12)

0

or differentiating:

(13) and (19) simultaneously rather than the single equation (18). (To be consistent one should additionally introduce non-linear terms into eqs. (12) and

(13). However, it does appear that the most impor-

a + r~à— v0XO

(13)

tant terms are those in eq. solution, non-linear when obtained, yields a value for(19).) a theThe error in the radius in terms of the time variation of H(t) (and

where: 11Tg1~ V p 0 0 S ‘

H

naniely that thefluctuations fractional ischange in growth produced by the negligibly small, is rate un-

s~V

0and Av~as a function of r0 for gernianium are given in table 2. Writing

H

Czochralski crystal growth. I

(F

—

Fref),

(14)

therefore of F(t)). This error value may then, in principie, be used to servo-control the power to the melt in order to achieve radius control. Published methods

and Fref = p~g7rr~V from 0 eq. (9). For the case where the crucible is weighed one can write as error signal JJC =

these be arespectively. measure of the integrated and currentquantities errors in to radius Before seeking solutions, we first examine the

(Fe + Fret),

~

irgr0v0p~ where I~is the rate of change of force experienced by the crucible weighing cell (FC < 0). Then H + He = c where e is the rate of evaporation of melt and/or crucible and the He signal can be used for servo control provided e is negligibly small or at least constant. In what follows we explicitly consider crystal weighing only

~.] ~

—

. —

.

—

r—

a,

where v = v0

‘

h and V0 is equal to the pulling rate plus the rate of fall of the melt level in the crucible. —

-

.

.

.

-

To first order in the small quantities a, a, O~and O~, eq. (15) becomes:

O V0

~l6~

— —

a

‘

and substitution of eq. (16) into (1 2) and (1 3) yields: H=

h a~ 2v0 a+

properties of eqs. (17) and (18) and show tile importance of the terms in ~1and A.

4. Properties of the force equation

but all results carry over to the crucible

weighing case simply by writing H -+ _fjc + e. Finally we need to relate 0,, to a. Clearly: V tan

of automatic control by weighing use either H [1,2] or H [3] directly as error signal implicitly assuming

r Ii — ~

—

0~° +fa dt

+ na

—

Xá ,

r = r0 + ~ hr [1 + tanh (t/r)].

(20)

(If we wish to know the exact effect of a step function change of power in the time evolution of the crystal radius we must solve the heat transfer problem in the susceptor—crucible—melt—crystal assembly.

(17)

This is clearly a very complex task, which it is not realistic to tackle for the present purpose.) Substituting eqs. (16) and (20) into (6):

(18)

h

~..V0

All.

Consider, for illustration, a situation where the crystal radius changes smoothly from an initial value of r0 to a final larger, value of r0 + hr (see fig. 3). Such a change could occur as the result of a small sudden decrease in heater power. Suppose that this change in radius can be analytically represented by:

However, the condition v0~’h implied by eq. (16)

=

~1(ha/2ro)[1 + tanh (tin)] 2(t/r)} br, (he/2VoT) sech

h0 —

+

(21)

W. Bardsley et al.

/ Weighing method of automatic Czochralski crystal growth. I

17

40

a lcn,I

0

a

30 -

0

do

20 ho

F 0

0

0

0-5 -...~

I 0

I 1-5

2.0

—,

r 0(cffl)

_________________________________________ t=o TIME Fig. 3. Schematic representation of the time evolution of various parameters following a small step change in heater power. The dashed curves represent possible behaviour for x>0.

Fig. 4. Dependence of stability parameter on crystal radius.

those methods of automatic control that involve imaging of the crystal [7,8]. An examination of the equations (10) shows that n is always positive, since for realistic values of the

parameters (see table 2)

(ps PL)haN~2 (pLho + ‘y0/r0g). (22) For materials which expand on solidification (such as —

which is also plotted in fig. 3. Note that as the crystal begins to increase its radius, the meniscus can first fall

in height before later rising to a steady, larger, value. This will occur if

Ge and the other Group IV and Ill—V semiconduc-

tors), (ps

is negative

PL) is negative and hence A is positive. A only if Ps > PL and

—

1 dh~ <0,

Ye <~r1~h0 (p~— pL)g.

(However it will be recalled that Ye i.e. if V0 T <

(2h0iha) r0.

=

(23) 0 if the melt

completely wets the solid.) Substituting eq. (20) into (17) and (18), evaluating 0=

The characteristic time for therelaxation change totime occurof cannot be less than the thermal the system Tth (which is typical of the order of 102 sec for Ge in an RSRE puller). Values of (2h 0/ha)ro are plotted versus r 0 in fig. 4 from which it can be seen that itexcept is greater than UOTth for all practical growth rates at very small crystal radii. Therefore h is expected to behave anomalously except for very slow changes having r ~ Tth. This result has implications for

the integral in (17) from _oo to t, with a0 yields: 2H/brr = (n + t)/r + ln[2 cosh(t/r)] 2) sech2(t/r) +

(n/n) tanh(t/i-) —(Air

=

0

O~

(24)

2(t/r) 2H/br = 1 + tanh(t/r) + (n/n) sech +(2X/r2) sech2(t/r) tanh(t/r).

(25)

Inspection of these equations shows that all the terms

W. Bardsley eta!. / Weighing met/mod of automatic Gzochralski crystal growth. I

18

contributions to H/br and H/br except the terms in A when A > 0. If A is sufficiently large then the situation can arise whereby, as the radius starts to increase, H and H first decrease before finally increasing to larger values. This is shown in fig. 3. Clearly this anomalous change is a potential cause of instability in a servo-loop controlling with H or H as error signal. Such instabilities have been observed in this laboratory [9,10]. Since the sign of the term in A is anomalous (for A > 0) for all t in the expression for H (eq. (24)) whereas it is anomalous only for t < 0 in the expression for H (eq. (25)), we can anticipate that control will be more difficult to achieve with the H signal than with the H signal. This result is in accord with experience for materials with A > 0. The physical cause of the instability is readily seen. The term in Ye represents the fact that as, for example, the crystal decreases in diameter, °L decreases, thus increasing the contribution of the surface tension force to the recorded “weight” so that the “weight” increases more rather than less rapidly. Similarly the term in (p~ PL) expresses the fact that the first result of the crystal “growing in” is an increase in the meniscus height h so that the cell is weighing an additional volume of melt in replacement for a similar volume of crystal and as the former is the denser for Ge-like materials again the weight increases more rapidly even though the crystal is “growing-in”. . The condition that the H or H signals should behave anomalously is: provide positive

(cml

2

s

2

—

1

d2H

hr

~.

—

=

or, from eqs. (24) or (25) that: I

+

2~/r 4Am2 <0, —

I

.0

5

2.0 ~o (em)

Fig. 5. Dependence of the stability parameter 9 on crystal radius.

might expect a spatial period on the crystal of the order of 21TVOTth 2iri~iwhich for the crystal shown in fig. 2a of ref. [11] is 5 mm compared to an observed value of 7 mm. This period does not depend on the growth rate again in agreement with experi-

(26)

i e that

I

0~5

It should be stressed that the above represents the response to a particular perturbation that given by eq. (20) and is presented to illustrate the principal characteristics of the response. Other perturbations will of course produce differing responses but the —

—

v

2 + 4Av~]1/2 (27) 0r < ~v0 + [(nv0) Hence the instability should not occur for growth rates in excess of t,l1/Tth. In fig. 5, the dependence of m,Li on r 0 is shown for Ge from which we see that small crystals are stable down to lower growth rates. For a Ge crystal with = 1 cm and r~ = 102 sec we see that anomalous 3 behaviour should occur only for v0 < 10 cm sec This is in accord with experimental findings. One —

.

above tics of isthethought system.to demonstrate the main characteris-

5. Solution of the force equation The Laplace transform of eq. (18) is 2ä a pa A[p 0] + n[pã a0] +ã = p11— H0,(28) —

—

—

—

W. Bardsley et al. / Weighing method of automatic Czochralski crystal growth. I

where a0 and a0 are initial values of the error in the radius tor. H and its time derivative; p is the Laplace opera0 is the initial value of H. Hence: - —

a—

[(p a) + 2 a]H c2] A[(p a) H 0 na0 + ~ + Apa0 + 2 2 A[(p a) C 1 —

—

—

—

—

[(p [(p where: +

— —

—

c2] a)2 +— cx]a0 a)

growing with H = 0 this can mean is growing 0s either ~0s= 0it or that its according to plan with a = a = net growth rate is zero (i.e., v0 h = 0) corresponding to the meniscus height increasing at a rate v0. The inclusion of otheralter nonlinear terms complicates but does not basically this picture. Reluctantly then we are forced to the conclusion —

that we cannot deduce the radius of a slowly growing Ge-like crystal in an “open-loop” manner by observ-

-

—

19

‘

‘~29~

closing the servo-loop applying corrections ing the time dependenceand of its “weight”. However for by the effects of the terms in n and A it has proved possible to achieve stable control of the radius and this is

(30) (31)

described in the following paper [11].

a= i 7/2A, 2=a2+l/A. c Re-transforming gives: a

=

—

1 ~

J. H(t

t’)

—

6. Summary The rate of change of the weight of a growing crystal (F(t)) can be used to servo-control for constant crystal radius for materials for which the melt corn-

e0t

o

r

X [cosh ct’

a

+

—

C

pletely wets the solid and is less dense than the solid. In other cases i.e. solid having lower density and/or incomplete wetting F(t) can be used directly for

sinh ct’]dt’

—

—

+H

0

+ a0

—

~a0 + Ac

e0t

r[c05h

~0

e0t

a

Ct +

—

sinh Ct

sinh

(32)

Ct].

C

In principle, given H(t), we can obtain a(t) by evaluating eq. (32). However, in practice, the growing exponential (a > 0) in eq. (32) prevents this. Thus any error in the initial value a0 is amplified so that repeated evaluation of (32) produces a strongly divergent value of a. Clearly then, for A > 0, the “weight” of the crystal is not a well conditioned measure of its radius. In practice the divergence of a will be arrested by non-linear effects not embodied in eq. (18). If such effects become important before the magnitude of a reaches significant proportions control may still be possible. Consequently we sought to numerically integrate the simultaneous eqs. (13) and (19). However inspection of these equations show that spurious behaviour is again possible. Specifically the equations have a solution corresponding to0sv =but v0 with h Os0 =which —vo/hoccurs for d = 0, with arbitrary 0 and a = H v~A/h0.Thus, for example, if a crystal is —

—

servo-control only for fast rates of growth. The minimum allowable growth rate decreases with decreasing crystal radius. At slow rates of growth the dependence of radius on F(t) becomes anomalous and some additional processing of the F(t) signal is necessary in order to achieve a stable control loop.

Acknowledgement The authors gratefully acknowledge the assistance of Dr. M. Healey in the numerical work.

7. Glossary of symbols A F, F~ FrefF~ef

H, He Href

a

See eq. (6) Force experienced by weighing cell during crystal and crucible weighing respectively Expected value of F or FC if crystal is growing to schedule See eq. (14) Expected value of H Error in radius ( r r0) —

W. Bardsley eta!.

20

a0 c e

g h

ha

/ Weighing method of automatic Czochralski crystal gross’th. I

Radius error at time t = 0 See eq. (31) rate Evaporation Gravitational acceleration Meniscus height Meniscus height when crystal growing to schedule

=(~‘~

asL 0LG 5SG’ USL T

Tth i,lj

A

Solid—liquid interfacial Surface tension of liquidfree energy Angular averaged values of 05G and See eq. (20) Thermal relaxation time of puller See eq. (27) See eq. (lOb)

0SL

References

/ 3h\

h 0

=

~~0~=0

[11 W. Bardsley, G.W. Green, (‘Fl. Flolliday and D.T.J. Hurle, J. Crystal Growth 16 (1972) 277.

rn0 p

r r0 hr t v

a 13Yst Yo Ye

Mass being weighed at time t = 0 Laplace operator Crystal radius Desired value of crystal radius See eq. (20) Time Growth rate of crystal Pull rate minus rate offal! of melt level n/2A (see eq. (30))2uLG/pLg) Laplace constant (= See eq. (2) See eq. (4) See eq. (4) See eq. (lOa) Meniscus angle at point of attachment to =

crystal

Os no

L

Ps. PL USC

121 T.R. Kyle and G. Zydsik, Mater. Res. Bull. 8 (1973)

131 141

A.E. Zinnes. BE. Nevis and CD. Brandle, J. Crystal Growth 19 (1973) 187. W. Bardsley, F.C. Frank, G.W. Green and D.T.J. liurle, J. Crystal Growth 23 (1974) 341.

151 161

S.V. Tsivinskii, Inzh. Fir. Zh. 5 (1962)9. G.K. Gaulé and JR. Pastorc, Met. Soc. Conf. 12 (1961) 201. 171 K.J. Gartner, K]. Rittinghaus and A. Seeger, J. Crystal 18] D.F. Growth O’Kane, 13/14 T.W. (1972)Kwap, 619. L. Gulitz and AL. Bednowitz, J. Crystal Growth 13/14 (1972) 624. 191 W. Bardsley, B. Cockayne, G.W. Green, D.T.J. Hurle, G.C. Joyce, J.M. Roslington, P.J. Tufton, H.C. Webber and M. Healey, J. Crystal Growth 24/25 (1974) 369. 1101 W. Bardsley, G.W. Green, C.H. Holliday, D.T.J. Hurle, G.C. Joyce, W.R. MacEwan and P.J. Tufton, Inst. Phys. Conf. Ser. 24 (1975).

Slope angle of crystal at growth interface

111] W. Bardsley, D.T.J. Uurle, G.C. Joyce and G.C. Wilson,

See eq. (3) Density of solid and liquid respectively Surface from energy of the solid

40 (1977) 21. 1121 J. T.Crystal Surek, Growth Scripta Met. 10(1976)425. 1131 M. Neuberger, Handbook of Electronic Materials, Vol.5 (IFl Plenum, New York, 1971) p. 14.