Thermally induced effects during initial stage of crystal growth from melts

ARTICLE IN PRESS

Journal of Crystal Growth 273 (2004) 320–328 www.elsevier.com/locate/jcrysgro

Thermally induced effects during initial stage of crystal growth from melts O.A. Louchev, S. Kumaragurubaran, S. Takekawa, K. Kitamura Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan Received 20 May 2004; accepted 19 August 2004 Communicated by D.T.J. Hurle

Abstract Thermal analysis of the initial stage of crystal growth from melt reveals the specific temperature distributions which lead to the onset of compressive stress inside the crystal shoulder formed from the initial seed. This compressive stress increases with increase in the shoulder angle. The specific temperature distribution is also characterized by the significantly decreased temperature gradient at the growth interface which may lead to morphological destabilization. The initial stage of growth, i.e. crystal diameter increase from the seed, should be performed at significantly lower pulling rates as compared with the pulling rates applied to the constant diameter crystal. r 2004 Elsevier B.V. All rights reserved. PACS: 81.10.h Keywords: A1. Heat transfer; A1. Morphological stability; A1. Stresses; A2. Czochralski method; A2. Floating zone technique

1. Introduction The theoretical simulation of crystal growth from melts and high-temperature solutions has drawn significant attention from the research community leading to the development of simplified analytical and comprehensive integrated models for all relevant techniques: Czochralski Corresponding author. Tel: +81-298-513351; fax: +(81)

–29-851-6159 E-mail address: [email protected] (O.A. Louchev).

[1–4], float zone [5–9], directional solidification [2,10,11]. Integrated computational models including hydrodynamics, heat and mass transfer [1–11], radiation transport [12–14], and electromagnetic field effects [15–18] allow a detailed parametric analysis of the crystal growth interface, of dopant and impurities distribution in crystals, of the temperature field along the crystal and associated stresses [19,20], of onset of constitutional supercooling and morphological destabilization [21,22], and of the formation of defect structures in the growing crystals [23]. Analytical and simplified

0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.08.023

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computational models being less comprehensive nevertheless have several advantages allowing fast and simple order of magnitude estimates that identify the main qualitative trends necessary for practical process optimization. Our previous models provide a set of various analytical and easily performed estimates for temperature distributions, radiation heat transfer, effects of natural and Marangoni convection, onset of constitutional and thermal supercooling, and related threshold pulling rates [24–28]. These theoretical models focus mostly on the steady-state stage of crystal growth when the crystal has a constant diameter and its length becomes sufficiently large. However, numerous experimental observations suggest that many crystals are, in fact, able to inherit defects generated at the initial stage when the crystal is extended from a small diameter seed into a full diameter crystal. This suggests also that processing parameters defined theoretically to be optimal for steady-state growth may in fact be detrimental for growth at the initial stage, leading to many initial defects which then can extend into the crystal body. In this paper we focus on a theoretical study of this initial stage considering the influence of initial thermal effects (i) on stress generation, and also (ii) on the constitutional supercooling which can lead to morphological destabilization of the growth interface.

2. Thermal model In this paper we consider a simplified onedimensional model (1D) of heat transfer through a growing crystal for which the cross-sectional area depends on the distance from the growth interface (Fig. 1). The steady-state temperature distribution along the growing crystal is defined by the 1D heat transfer equation  d
pRðxÞ2 J x ¼ 2pRðxÞ½As QðxÞ dx  
sT 4  hg ðT  T g Þ ;

321

melt

melt (a)

(b)

Fig. 1. Schematic of the initial stage of crystal growth by (a) Czochralski and (b) floating zone process.

temperature Tg, s is Stefan’s constant, As is the absorption coefficient, e is the emissivity, and Jx is the heat flux density through the crystal cross section defined by J x ¼ kðTÞ

dT þ f r cT; dx

(2)

where k(T) is the heat conductivity, f is the pulling rate, r is the crystal density and c is the specific heat. At the growth interface the temperature is the solidification temperature, i.e. T ¼ T sol at x ¼ 0: The radiation heat flux on the surface of the growing crystal includes radiation incoming on the particular crystal surface point from other surfaces, such as the melt surface and the insulation elements in the Czochralski growth system. The resulting radiation flux is defined by the following integral over the solid angle: ZZ s QðxÞ ¼ ðw; nÞ
ðwÞT 4 ðwÞd 2 O; (3) p O

ð1Þ

where R(x) is the cross-section radius, Q(x) is the radiation flux on the crystal surface, hg is the heat exchange coefficient with the surrounding gas at

where e(w) and T(w) are, respectively, emissivity and temperature of the radiation emitting elements in the direction of vector w, and n is the vector normal to the crystal surface. Details of estimations for Eq. (3) may be found in Ref. [29].

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In fact, the successful growth of the crystals has a narrow operational window restricted by different factors which may lead to the formation of various defects: cracks, dislocations, morphological destabilization of the growth interface causing grain, columnar and dendritic structures.

3. Results and discussion 3. 1. Thermal stress There are many papers devoted to thermal stress in crystal growth providing different formalisms and approximations (see review [19]). A detailed theoretical analysis performed in Refs. [30–32] and supported by detailed numerical simulations [19] suggests that thermal stress is mainly defined by the second-order derivative of the temperature and may be estimated as sst ffi aEW 2 d2 T=dx2 ;

(4)

where a is the linear thermal expansion coefficient, E is the elastic modulus, and W is a characteristic dimension parameter taken as 0.2 of the crystal diameter. The factor aE in Eq. (4) may be very sensitive to temperature increase, a fact which should be taken into account in estimation. This expression explicitly reveals that the thermal stress caused by the second-order derivative of the temperature depends strongly on the crystal geometry and specific heat transfer parameters. It should be noted that thermal stresses occurring during the growth may also be associated with other second derivatives of the temperature [32]. However, for the case of thermally thin crystals treated by the 1D approximation @2 T=@x2 is the most characteristic value. In effect, considering three-dimensional heat conductance equation and taking into account the radial symmetry of the problem one finds that @2 T=@y2 ¼ @2 T=@z2 ¼ 0:5@2 T=@x2 : Considering now a simple 1D model for a semi-infinite crystal one finds the exponential temperature approximation pTðxÞ ffi T g þ ðT s  T ng Þ expðx=lt Þ; where ffiffiffiffiffiffiffiffiffiffiffiffi lt ¼ Rk=hs and hs=hg+hr is the total heat exchange coefficient including linear approximation of the radiation heat exchange, hr, and T ng ¼

ðhg T ng þAs QÞ=hs is the effective temperature. On the other hand, one has @T=@r  hs ðT  T g Þ=k and, therefore, the exponential function for T(x) gives rffiffiffiffiffiffiffiffi @2 T hs @T h s R @2 T   : @r@x k @x2 k @x This expression shows that for low diameter crystals the stress associated with @2 T=@x2 dominates, and that @2 T=@x@r  @2 T=@x2 only for hsR/ k E1, i.e. for Si specimens of RE10 cm for which the 1D heat transfer approximation is not valid. Thus, this simple 1D model is sufficient to reveal different effects associated with specific temperature distributions. For relatively low growth rates   frcR=k 1 one finds from Eqs. (1) and (2) that:


d2 T 2
sT 4 þ hg ðT  T g Þ 1 dk dT 2   dx2 kðTÞ dT dx kðTÞRðxÞ 2As QðxÞ 2 dT dR  :  kðTÞRðxÞ RðxÞ dx dx (5) This expression reveals that thermal tensile stress (sst40) in the growing crystal is due to heat dissipation from its lateral surface (the first term on the right-hand side of Eq. (5)). In contrast to the first term, the second, third and fourth terms provide compressive stress components (ssto0) suggesting the possibility of the tensile stress inhibition during the initial stage. That is, the tensile effect of the first term is counteracted by the radiation flux incoming onto the crystal surface as well as by the fourth term of the right-hand side of Eq. (5) associated with a geometrical factor. This geometrical factor depends on the angle at which the crystal shoulder is formed, g , defining the value of dR/dx=tg(g/2). To demonstrate the above-described effects we provide here calculations done for a particular case of Si for which the thermophysical properties are well known, i.e. kðT Þ ¼ 0:9880  9:43  104 T þ 2:89  107 T 2 [4], As ¼
¼ 0:46 and a ¼ 2:225  106 þ 2:425  109 T approximating the experimental data [33]. The elastic modulus for Si is given by E ¼ 1:7  1011  2:771  104  T 2 (Pa) [34] which agrees well with recently performed high-temperature measurements [35].

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T (K)

1600 1400 1200 1000 0

2

4

6

8

10

12

14

10

12

14

10

12

14

x (cm)

(a)

gradT (K/cm)

0 -50 -100 -150 -200 0

2

4

6

8

x (cm)

(b) 100 80

stress (MPa)

It should be noted that the validity of Eq. (4) has been confirmed several times in the literature [19,32,36]. In particular, a direct comparison of Eq. (4) with 2D model calculations for InP crystals is made in Ref. [19] showing good agreement with the von Mises 2D stress distribution. That is, for the considered InP crystal Eq. (4) gives 1.5–2.5 MPa agreeing very well with 2D simulations. For a larger diameter InP crystal Eq. (4) gives a value of about 10 MPa, two times higher than that of related 2D simulations. Ref. [32] also provides 2D thermal stress calculations. For instance, for a Si crystal of D=3.6 cm and L=5 cm, 2D results give a stress distribution with maximal value of about 1.1 MPa. Eq. (4) gives a value up to sst  2 MPa for d2 T=dx2  10 K=cm2 a  4  106 K1 and E  100 GPa : The importance of the second-order temperature derivative in a thermal stress definition was also noted in a thermal stress study on Si ribbon growth [37]. To additionally verify the validity of Eq. (4) combined with the 1D heat transfer model proposed here, we have compared results given by Eqs. (4) and (5) with 2D calculations done for Si crystal growth by float zone process [38]. In particular, in Fig. 2 we show (a) temperature, (b) gradT, and (c) stress distributions along the x coordinate for a Si specimen with constant radius RE5 cm, main crystal body length E10 cm, and crystal shoulder angle g  70 : To provide compatibility with Ref. [38] we used in our calculation an effective temperature for the radiation heat  1=4 transfer T eff ¼ As Q=
s : 1D calculation is done for TeffE1000 K which provides a temperature gradient at the solidification interface   grad T   150 K=cm (see Fig. 2 b) similar to that of Ref. [38]. Our 1D stress calculations (see Fig. 2 c) give sstE95 MPa near the solidification interface decreasing by about an order of magnitude at x ¼ 10 cm. Additionally, for x410 cm the 1D model gives a sharp decrease in sst associated with the beginning of the conical part and related contribution of the last term in Eq. (5). For the same Si crystal dimensions 2D simulations [38] gives sstE80–140 MPa along the radius near the solidification interface and shows rapid stress decay towards the specimen end. Thus, this

323

60 40 20 0 0

(c)

2

4

6

8

x (cm)

Fig. 2. Temperature (a), temperature gradient (b) and thermal stress (c) distributions along Si crystal.

comparison suggests that notwithstanding possible significant stress variations along the radius the 1D approximation is quite reliable for making reasonable thermal stress estimates deviating from 2D results by no more than a factor of 2–3. In particular, the 1D model allows us to focus here on the stress formation mechanism at the initial stages of crystal growth neglected, in previous studies, which focused mostly on stress formation in the main bodies of the grown crystals. In Fig. 3 we show the distribution of the temperature (a) and related stress (b) along the crystal for different crystals extended from an initial seed of r0=2 mm towards a maximum

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324

Temperature (K)

1800

1600

1400

1 2 3

1200

1000 0.0

0.2

0.4

0.6

0.8

1.0

x/L

(a) 20

Stress (MPa)

10

0

1 2 3

-10

-20 0.0

(b)

0.2

0.4

0.6

0.8

1.0

x/L

Fig. 3. Temperature (a) and thermal stress (b) distribution along the crystal axis for different values of g: (1) g ¼ 40 ; (2) g ¼ 70 ; and (3) g ¼ 100 :

curve at the crystal shoulder vertex, providing a negative value in d2T/dx2o0 which in accordance with Eq. (4) induces the compression stress component at this point. Moreover, for g ¼ 401 (curve 1) d2T/dx2o0 and compression (ssto0) are shown to take place only in the vicinity of the vertex point whereas for g ¼ 701 the area of d2T/ dx2o0 and compression stress significantly extend, and for g ¼ 1001 is distributed over the whole conical part of the growing crystal. These figures also reveal tensile stress compensation by compressive stress. That is, the case of g ¼ 401 exhibits the maximum tensile stress of ffi14 MPa at the growth interface. In contrast, the cases of g ¼ 701 and g ¼ 1001 show a significant decrease of tensile stress at the growth interface, i.e. to E6 MPa and E1.5 MPa, respectively. However, significant compressive stress appears near the vertex point which increases in absolute value with increase in g. That is, for g ¼ 701 sst E MPa, whereas for g ¼ 1001 sstE-20 MPa. In Fig. 4 we show the contribution of different terms of the right hand side of Eq. (5) to total stress shown in Fig. 3b for the case of g ¼ 701 Term #1 (solid) provides tensile stress, whereas all the other terms give negative compressive components as discussed above. However, the contribution of the second (2) term, associated with the

Stress (MPa)

40

radius crystal of R=20 mm (x coordinate is directed in the upward direction from the solidification interface as shown in Fig. 1). The calculations are done for T g ¼ 1000 K; Q ¼ 1:6  105 W=m2 W/m2 (TeffE1300 K) hg ¼ 25 W=m2 K [24]. Crystal seed length is taken as L0 ¼ 4 cm: In Fig. 3 we show the results obtained for different angles at the crystal cone g ¼ 40 (curve 1), 701 (curve 2) and 1001 (curve 3). T and sst distributions are given as a function of coordinate along the crystal, dimensionalized to the total crystal length (seed plus conical part). These simulations show that with increase in g the temperature distribution along the crystal experiences a modification leading to a very significant change in stress distribution. In particular, the temperature distribution shows a specific

1 2 3 4

20 0

-20

-40 0.0

0.2

0.4

0.6

0.8

1.0

x/L Fig. 4. Contribution of different components of the right-hand side of Eq. (5) to stress distribution associated with : (1) the heat exchange by radiation and convection; (2) the temperature dependence of the heat conduction coefficient; (3) the radiative heating of crystal surface, and (4) the radius change in the conical part of the crystal.

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temperature dependence of the heat conductance (p dk/dT) is negligibly small. The most significant compressive contribution which counteracts the positive tensile stress is provided by the fourth (4) term, pdR/dz=tg(g/2). The effect of the radiation associated term (3), pAsQ, has a significantly lower effect. Fig 3 and 4 suggest that the various regimes may be practically realized in crystal growth by varying the value of g. For instance, the value g ¼ 701 significantly decreases the tensile stress at the growth interface leading at the same time to significant compressive stress occurring at the vertex point. The value g ¼ 1001 very significantly reduces tensile stress at the growth interface towards the level of tensile stress in the seed part. However, one should take into account that at the same time there is a very sharp increase of the compressive stress at the vertex point. There is also an additional effect revealed by more extensive simulation of different stages of crystal shouldering from the seed. Fig. 5 shows a sequence of temperature and stress distributions in the crystal

Temperature (K)

2000 crystal length 5.5 cm crystal length 4.5 cm

1800 1600 1400 1200 1000 0

1

2

4

4

5

6 10 5

crystal lenght 5.5 cm crystal length 4.5 cm

Stress (MPa)

2

Stress (MPa)

3

extended from the seed at g ¼ 701 for crystal lengths 4.5 and 5.5 cm (with the seed length 4 cm). These calculations reveal that at the initial stage of crystal shouldering there is a significant compressive stress in the entire cone section of the crystal. That is, for a crystal of the length 4.5 cm the 0.5 cm cone part of the crystal is under significant compressive stress E12 MPa near the vertex point and also E3 MPa at the interface. With extension in diameter to 1 cm and with total crystal length of E5.5 cm the compressive stress at the vertex point becomes lower (E10 MPa), whereas the stress at the growth interface becomes positive (tensile, E2 MPa). The evolution of stress with the increase in crystal length for g ¼ 701 at the vertex point and at the growth interface is shown in Fig. 6 by broken and solid lines, respectively. This figure shows that at the growth interface of the seed crystal (with L ¼ L0 ¼ 4cm) the stress is positive (E4 MPa), and at the very initial stage of crystal seed extension to a larger diameter a significant compressive stress (E15 MPa) appears at the growth interface (at the cone vertex). However, with time and with increase in crystal length the stress at the growth interface becomes positive and the compressive stress at the vertex point becomes more moderate ðjsst jo10 MPaÞ: It is worth noting here that in the case of constant diameter crystal growth stress may be decreased by additional heating of the crystal surface via increasing the temperature of growth chamber enclosure elements (and the value of Q in

x (cm)

(a)

325

0 -2 -4 -6 -8

interface vertex point

0 -5 -10 -15

-10 0

(b)

1

2

3

4

5

6

x (cm)

-20 4.0

4.5

5.0

5.5

6.0

6.5

crystal length (cm) Fig. 5. Temperature (a) and stress distributions (b) in the crystal grown at g ¼ 70 for different crystal lengths.

Fig. 6. Stress evolution with increase in crystal length.

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Eqs. (1) and (5)). For instance, for Si crystal growth thermal stress may be reduced below 10 MPa even for large diameter crystals. However, at the initial stage the crystal shoulder may be prone to significant compressive stresses leading to defect formation in the shoulder part of the crystal.

200

- gradTs (K/cm)

326

150 100

3. 2. Morphological destabilization

50

0 40

grad T m mC i1 ð1  k0 Þ ; o
V D k0 þ ð1  k0 Þ expðV d=D

γ (Ο )

120

160

700 600

γ=70o γ=100o

500 400 300 200 100 0

(6)

where grad Tm is the temperature gradient at the growth interface in melt, V is the pulling rate, m ¼ @Tm =@C is the liquidus line slope on the impurity concentration, CiN is the impurity concentration in the liquid bulk, D is the diffusion coefficient for impurity in liquid, d is the thickness of the diffusion boundary layer, and k0 is the segregation coefficient. The temperature gradient in the melt is coupled with the temperature gradient in the solid by ks grad T s ¼ km gradT m  V rs DH:

80

Fig. 7. Temperature gradient at the growth interface as a function of g.

- gradTs (K/cm)

This thermal model reveals an additional effect which may destabilize the growth interface: the increase in g causes a significant decrease of the heat flux density at the growth interface (i.e. the absolute value of dT/dx). This effect may produce morphological destabilization of the growth interface caused by constitutional supercooling leading to the formation of grain, columnar or dendritic structures in the growing crystal. In effect, the model of Burton, Prim and Slichter [39] leads to the following criterion for the onset of constitutional supercooling in front of the solidification interface [40]:

(7)

This expression shows that the in the  decrease  temperature gradient in solid, grad T s ; which is associated with the heat exchange conditions, leads also to a decrease in the temperature gradient grad T m  in melt, and to the onset of constitutional supercooling and morphological destabilization.   Fig. 7 shows the value of grad T s  at the growth interface of a crystal extended from r0=2 mm to R=20 mm as a function of g. An increase in g from

4

5

6

7

8

9

10

crystal length (cm) Fig. 8. Evolution of the temperature gradient at the growth interface occurring with increase in the crystal length.

  20 to 1501 leads to a sixfold decrease in grad T s : Thus, the increase in g allowing one to inhibit tensile stress at the growth interface may require a decrease in the applied pulling rate in order to inhibit the onset of the constitutional supercooling associated  with the significant decrease in grad T s :   Fig. 8 shows the dependence of grad T s  as a function of the current crystal length L for crystal growth from the seed towards maximum radius crystal of R ¼ 20 mm at g ¼ 70 (4 cmo Lo6.5 cm) and at g ¼ 100 (4 cmoLo5.5 cm), followed by growth at a constant radius until the length corresponding to the stabilization of the heat flux at the growth interface is achieved. This figure shows that during crystal shouldering   there is a very significant decrease in grad T s : Larger

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the value of g , the more significant the decrease in grad T s : After the crystal starts to grow at a constant radius for L46.5/5.5 cm (g ¼ 70 =g ¼ 100 ) the temperature gradient increases, tending finally towards a constant value corresponding to that of a semi-infinite cylinder.  This figure suggests that the behavior of grad T s  should also be taken into account when considering the pulling rate at which a crystal grows during the initial stage, because the growth interface may be prone to morphological destabidecrease in lization. In particular, a very significant grad T s  takes place for g ¼ 100 —the minimal value E70 K/cm is about three times smaller than that of the steady-state mode E220 K/cm when the crystal length has become sufficiently large. This means that if the threshold value for the onset of constitutional supercooling starts at   grad T s ¼ 150 K=cm (for a particular value of the pulling rate V) a very significant part of the crystal will be grown under constitutional super cooling mode (grad T s o150 K/cm) giving grainlike, columnar, dendtritic growth structures or inclusions of other phases for materials with a more complex phase diagram. Finally let us note, that the larger the angle of crystal extension the more the initial stage seems to be prone to this kind of growth destabilization and related defect structures.

4. Conclusions The considered model reveals that during the initial stage of crystal growth from the initial seed a very significant compressive stress may be induced. This stress is due to the specific temperature distribution along the conical part of the growing crystal, characterized by a negative value of the second derivative of the temperature which induces high compressive stress components, overcoming tensile stress. Thermal analysis explicitly shows effects responsible for stress formation. In particular, this analysis shows that higher the shoulder angle, higher the compressive component which under some specific operational conditions may compensate tensile stress in the vicinity of the growth interface.

327

Additionally, the initial stage of crystal extension into a maximum diameter crystal is shown to be characterized by a low-temperature gradient at the solidification interface. Larger the crystal extension angle, smaller the temperature gradient at the growth interface, and smaller the pulling rate which should be applied in order to inhibit the onset of the morphologically destabilizing constitutional supercooling in front of the growth interface and consequent formation of defect structures in the growing crystal.

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