Heat transfer in growing Bi4Ge3O12 crystals under weak convection: II—radiative–conductive heat transfer

ARTICLE IN PRESS

Journal of Crystal Growth 262 (2004) 212–224

Heat transfer in growing Bi4Ge3O12 crystals under weak convection: II—radiative–conductive heat transfer V.D. Golyshev, M.A. Gonik* Centre for Thermophysical Researches ‘‘Thermo’’, Institutskay St. 1, 601650 Alexandrov, Russia Received 13 August 2003; accepted 3 October 2003 Communicated by T. Hibiya

Abstract Experimental and calculation investigations of the features of radiative–conductive heat transfer during the growth of bismuth germanate (Bi4Ge3O12) single crystals are carried out. It is demonstrated that due to the radiative–conductive character of heat transfer, the temperature regime of the crystallization of perfect crystals has non-conventional character. In order to keep the growth rate constant during crystal growth under condition of weak thermal convection, it is necessary to decrease the seed temperature at the beginning of growth; then temperature should be kept constant; and finally, starting from a certain length of the crystal (Lcr E30 mm), seed temperature should be slightly increased. It is shown that the presence of defects in crystal decreases Lcr value. The strong effect of supercooling of the interface on the features of heat transfer during the growth of Bi4Ge3O12 crystals is demonstrated. r 2003 Elsevier B.V. All rights reserved. PACS: 81.10.Fq; 44.10.+i; 44.40.+a; 68.45.
v; 44.30.+v Keywords: A1. Heat transfer; A2. Growth from melt; B1. Bismuth germanate

1. Introduction Bi4Ge3O12 (BGO) single crystals are grown by Bridgman method [1], Czochralski one [2], and others. Large crystals of rather high quality were obtained by the Low Thermal Gradient Czochralski Technique (LTG Cz method) [3]. However, the problem of production of large perfect single crystals still remains urgent for BGO. It would hold true for growing crystals at low temperature gradient too. Under conditions of *Corresponding author. Fax: +7-095-584-5816. E-mail address: [email protected] (M.A. Gonik).

low temperature gradient, BGO crystals are characterized much more pronounced tendency to faceting [4], therefore one should take into consideration the phenomena of the facet interfacial kinetics in heat transfer calculations during the BGO crystal growth. An efficient approach for the calculation of radiative heat transport in axisymmetric domains of arbitrary shape with both diffuse and specular (Fresnel) reflective boundaries was developed [5] and applied to the simulation of global heat transfer during the growth of BGO crystals by the LTG Cz method [6]. By now, there are good achievements in the area of numerical methods for

0022-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2003.10.008

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

facet crystal growth [7,8]. However, the analysis of the melt–crystal interface supercooling, as well as of the effect of optical properties of the melt and crystal on heat transfer character for BGO has not been carried out yet. At the same time, the absorption coefficient of BGO and BSO complex oxides was found to be strongly dependent on the quality and grade of the crystal [9]. To solve this problem, coupled radiative–conductive heat transfer (RCT) and natural convection in the melt along with the phenomena of the facet interfacial kinetics have to be considered. This problem cannot be solved at present in general formulation for crystal growth of dielectric materials because of its complexity. However, the problem to find a behaviour of heat transfer in BGO crystal growth can be successfully solved in certain situations. First of all, due to high viscosity of the BGO melt [2,10], natural convection is weak and can be neglected, when a small height of the melt and/or a small temperature gradient are realized during crystal growth. Thus the problem can be taken as the problem of RCT only. It should be noted that the LTG Cz method and a technique for crystal growth under conditions of the axial heat flux close to the crystallization front (AHP—Axial Heat Processing method) [11] meet the abovementioned situation. Secondly, the basic mechanism of heat transfer in the BGO crystal is thermal radiation (see Part I [12]); therefore, one can expect that because of large refractive index of the crystal (n ¼ 2:15 [10]) the thermal radiation from hot parts of crystal goes through a crystal like as in a light guide. Hence, there is no large loss of thermal radiation through the side surface of the growing crystal, and for this portion of radiation, the problem seems to be close to one-dimensional one. Third, RCT parameters for high temperature have been determined for BGO [12] as well as the functional dependence of growth rate on supercooling [13]. In the present work, the problem of interface faceting and influence of crystal optical properties on heat transfer during BGO crystal growth has been studied with the use of AHP method. Suppression of natural convection and one-dimensional heat flux [11,14,15] are specific features of

213

this method. These factors along with providing collection of precise data on the temperature boundary conditions enable the AHP method to be a good tool for testing the validity of RCT description during crystal growth. The goal of the work was: (i) to calculate and experimentally investigate the RCT during growing of BGO single crystals by means of AHP technique, (ii) to determine temperature regime and thermal conditions providing crystallization with constant growth rate along with taking into account supercooling on the growing facet; (iii) to verify the found regularities in growing crystals with low-gradient Czochralski technique.

2. Instrumentation and procedure of experimental researches 2.1. Investigation of the character of heat transfer during BGO single crystal growth by means of AHP 2.1.1. A brief description of the AHP procedure A schematic of the set-up for growing crystals by the crucibleless AHP2a method [14] is shown in Fig. 1. A seed with a diameter being equal to that of the AHP-heater is placed between the bottom of AHP-heater casing and the pedestal (on the Pt disk). Then the seed is heated to temperature somewhat lower than the BGO melting point Tm ; after this, the AHP-heater is switched on, the top part of the seed melts forming a melt layer of the height h: The necessary temperature gradient in the melt is established. Then the charge is fed to the AHPheater where it melts, and the pulling of the seed downward is started at a constant rate v: The melt is suspended at the AHP-heater by surface tension forces. Since temperatures in the centre of bottom of the AHP-heater casing, T1 and in its periphery, T2 (see Fig. 1) are kept equal each other and constant during the growth by means of control computer system, the hot boundary of the melt is isothermal and its temperature Thot ¼ const: While pulling, the temperature in points T3 and T4 (accordingly, the temperature Tcool of the cool boundary of the

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

214

Melted charge

Feeder Guide tube

AHPheater

Charge Background heater

T5 T2 T1

Optical devise (cathetometer)

h

Annular gap T4

Thermal insulation

T3

Pt casing of heater Melt Crystal Thermocouples

Pt disk Pedestal

Fig. 1. Diagram of the set-up for growing crystals by AHP crucibleless method.

crystal being in contact with a thin Pt disk) changes in such manner to maintain h¼ const: As v¼ const and h¼ const; the growth rate V is constant too (V ¼ v). A set of time dependencies of temperature in points shown in Fig. 1 T1ðtÞ; T2ðtÞyT5ðtÞ; where t is time, that provides h¼ const; V ¼ const; grad T m ¼ const; the given shape of interface, as well as constant overheating of the melt DT ¼ Thot
Tm ¼ const; is called as a temperature regime of the AHP-crystallization (TRC). Features of crystal growing according to the AHP2a procedure are described in more detail in Ref. [14].

dimensional temperature field in the region close to crystallization front. The geometry of calculation domain for the AHP2a method is shown in Fig. 2. A system of the melt and the crystal is located between two horizontal parallel-sided planes. Since the hot surface I is isothermal (radial gradient, grad Tr along the bottom of AHP-heater does not exceed 0.1 K/cm), the melt layer is heated from above and its thickness is small (ho1 cm), natural convection is suppressed. Estimations of the Rayleigh number (Rar ¼ gbDTr h3 =an; where g is the gravitational acceleration, b is the coefficient of thermal expansion, DTr ET2
T1 is the destabilizing actual temperature difference, a is the thermal diffusivity, and n is the kinematic viscosity), give the value of no more than 2–3. For this Rar ; natural convection does not bring any contribution into heat transfer [16] and can be neglected in the heat transfer model. Due to this fact, only RCT model was used in crystal growth of BGO by the AHP method. At the beginning of growth (Fig. 2a), the height of melt–crystal system d is small: d=Do0:25; where D is the diameter of crystal. For such d=D values with the surface I being isothermal, heat transfer at the axis of the system is described by one-dimensional model [17]. At the increase of crystal height H (Fig. 2b), it is necessary to take into account heat exchange on the side surface. Due to strong effect of complete reflection in BGO crystal, the axial heat flux is noticeably

I

I h

Melt

d

Interface H

2.1.2. Methods of heat transfer calculations TRC depends on the character of heat transfer in melt and in crystal; it also depends on the boundary conditions. The AHP methods were developed specially for simplifying the description of heat transfer [11,14]. This simplification is achieved by: (i) suppressing natural convection, (ii) simple geometry, (iii) including only melt and crystal into the calculation domain, (iv) measuring temperature at the boundaries of the calculation region, (v) creating conditions for obtaining one-

Thot

Crystal

II Tcool

(a)

II (b)

Fig. 2. Geometry of the calculation domain in the beginning of growth (a) and at large crystal length (b), when radiation from the interface partially leaves the crystal.

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

215

H is its height) and on the radiant emittance of boundaries, e:

predominant. Only 8–10% of thermal radiation from the interface does not reach the cold boundary of the crystal (surface II in Fig. 2). The amount of heat leaving the crystal through side walls does not change with increasing of the crystal length, starting from its certain value LD40 mm (L ¼ Dtgy; where y is an angle of complete reflection). Because of this, one may assume that the application of one-dimensional model gives good quantitative results at the axis of the system for crystals of a large height, too. An advantage of an idea to study temperature behaviour at the axis of the system is that the general regularities of changing the interface position with changing the growth conditions can be investigated without complicated numerical calculations. To solve the RCT problem, the approach described in Ref. [18] was used. In the model involved, heat transfer by radiation in the melt was neglected [12]; radiation in the crystal was calculated by approximating methods [12] depending on the optical thickness of the crystal t ¼ kav H (kav is absorption coefficient of the crystal averaged over the spectrum in transmission band,

lm eff

ðThot
Tm Þ þ VJrm h ðTm
Tcool Þ : ¼ qcrad ðe; tÞ þ lccond H

ð1Þ

Here lm eff is the effective thermal conductivity of the melt [12] responsible for heat transfer by both conduction and radiation, lccond is the thermal conductivity of the crystal, V is the rate of crystal growth, J is the heat of crystallization, rm is the density of the melt, Tm ; Thot ; Tcool and h were described earlier. It is necessary to verify the proposed model for several reasons: first, because we use onedimensional approximation; second, we do not take into account possible deviation of temperature profile from the linear one; and third, there is incompleteness in parameters of a model. The properties necessary for calculations are listed in Table 1.

Table 1 Physical properties of bismuth germanate and platinum (material of the container) Properties

Crystal

Melting temperature, K Density, kg/m3 Thermal capacity, J/kg K Heat of transaction, kJ/kg Coefficient of volume thermal expansion, 1/K

1323 7090

Melt 6650 410.0

140 7.6  10
5
6

Thermal conductivity, W/mK Thermal diffusivity, m2/s

8  10 1.2 50.6  10
8

2

Kinematic viscosity, m /s Pr number Rarad number (h ¼ 1 mm, D ¼ 83 mm, DTrad ¼ 2 K) Absorptivity, cm
1 Refractive index close to the melting point Emissivity of platinum in vacuum, at 900 K Effective emissivity factor of two boundaries (in BGO medium): Both platinum at 900 K Platinum at 900 K; melt-crystal interface, at 1323 K

0.03 2.1

0.14 3.67  10
8 8.13  10
6 130 0.035 >150–200 1.9 0.116

0.410 0.407

Data source [10] [10] [20] [20] [21] [20] [12] [20] Calculations based on [21] [21]

[12] Our estimations [22] Calculation on data [23]

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2.1.3. Experimental procedures involved in AHP method In the present work, the AHP method is used as an instrument to investigate the features of heat transfer during crystal growth. To verify the model, the position of the interface determined experimentally was compared with that calculated using the RCT heat transfer model. Experimental procedure was performed by two methods. The first one is a determination of the position of marks in the grown crystal. The marks were made by switching the AHP-heater off for a short period of time during crystal growth; these marks were observed later, at the longitudinal section of the grown crystal. Another way is a direct measurement of the position of interface during growth. A cathetometer was used for this purpose (Fig. 1). With its help, one can clearly see a contrasting boundary between non-transparent melt and transparent crystal. Since the AHP growth leads to the formation of the plain interface [14], measurements on the side surface give information on the position of the interface also at the centre of the crystal. The casing of AHP-heater as well as the disk placed between the seed and the pedestal were both made of platinum. Temperature was measured with platinum thermocouples with the help of the computer control system, which included high-precision digital multimeter and CAMAC modules [19]. The BGO crystals were grown at a rate of 1–6 mm/h at temperature gradient in the melt 50–200 K/cm. The height of grown crystals did not exceed 50 mm (d=D was varied within the range 0.2–0.7).

2.2. Experimental investigation of the temperature regime of BGO crystallization by means of low-gradient Czochralski technique A scheme of set-up for growing the crystals by LTG Cz technique is shown in Fig. 3. A specific feature of the method is that the crystal is placed in a closed volume bordered by platinum walls; small axial gradient of temperature is established along the side surface of it. So, a crystal grows at some small temperature drop over the whole crystal.

Fig. 3. Diagram of set-up for growing crystals by LTG Cz procedure (a) and a typical shape of as grown crystal (b).

During this process, similarly to the conventional Czochralski procedure, the constancy of growth rate is carried out with the help of weight sensor by changing the heater power during the

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

growth of a crystal to keep the diameter of the crystal constant. This variation of power results in the change of the heater temperature to provide necessary changes of heat transfer in the growth system at the increase of the crystal height. So, the character of relative change in the temperature of heaters describes the character of change in heat transfer during crystal growth.

1200 Conduction (W/m2)

3 1000 2 800 4

1

600 5

400 0

3. Results and discussion

50

(a)

3.1. Heat transfer in crystal and in melt

100

150

Crystal height (mm) 50000 2

Radiation (W/m2)

3.1.1. Crystal 3 A dimensionless parameter N ¼ 8n2 sThot = klcond characterizes the portion of radiation qrad in total heat transfer. For BGO single crystal, NE1500: This means that much more heat is transferred by radiation in comparison with conduction. The heat conduction qcond is determined by thermal conductivity lcrys and temperature gradient grad T in the crystal. The qrad value is determined by t; temperature and optical properties of boundaries. We assume that one of the boundaries of the crystal (cool) is in contact with the platinum disc characterized by the emissivity of ePt : Another boundary (hot) is the interface between the liquid and solid phases, with its radiative emissivity being ei : Since BGO melt is opaque [12], the ei of this boundary can be determined due to the known Fresnel equation that gives ei E1: The temperature of the hot boundary of the crystal was assumed to be constant and equal to Tm . The t value depends on H and kav : Hence, the character of heat transfer in crystal depends on the kind of crystal and on the presence of defects in it. Fig. 4 shows the dependencies of qrad and qcond on H for different kav ; while Table 2 shows their dependence on temperature gradient for a set of H values. It was assumed in calculations that the temperature gradient in crystal, ðgrad TÞcryst ¼ ðTm
Tcool Þ=H does not change during the crystal growth and corresponds to that established in the furnace ðgrad TÞfurnace : One can see that qrad bqcond ; the ratio between the fluxes being changed with the changes in H:

217

40000 5

30000 1 20000 10000 0

50

100

150

Crystal height (mm)

(b)

Fig. 4. Changes of the conductive (a) and radiative (b) heat fluxes in the BGO crystal (1—k ¼ 0:04 cm
1, 2—k ¼ 0:03 cm
1, 3—k ¼ 0:005 cm
1, 4—k ¼ 0:002 cm
1, 5—k ¼ 0:001 cm
1) during its growth under the constant temperature gradient averaged over the height in the melt–crystal system: Thot ¼ 1543 K, Tcool is a linear function of time.

Table 2 Conductive qcon and radiative qrad heat fluxes in single crystal with a height of H depending on temperature gradient
1 ðgrad TÞcryst ; e ¼ 0:116; lcryst cond ¼ 1:15 W/m K, k ¼ 0:03 cm grad T; K/cm 1 5 50

qcon ; W/m2 120 600 6000

Qrad ; W/m2 H ¼ 10 mm

H ¼ 50 mm

H ¼ 100 mm

1173 2837 55,457

5946 29,062 224,410

12,095 57,784 344,019

The total heat flux, noticeably following crystal height and growth by an order

qcryst ¼ qcond þ qrad increases for qrad with increase of the can change during crystal of magnitude. In particular,

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Temperature (K)

1325 1320

2

3

1315 1310 1 1305 1300 0

0.2

0.4

0.6

0.8

1

Dimensionless crystal length Fig. 5. The distribution of temperature over BGO single crystal with a length of: 1—50 mm, 2—100 mm, and 3—300 mm.

for an ordinary gradient in the furnace ðgrad TÞfurnace ¼ 10 K/cm qcryst ¼ 1:7  104 W/m2 for Ho3 cm and 1.3  105 W/m2 for HE30 cm. Depending on kav ; the requirement to0:1 (the model of weakly absorbing medium) can be valid within the whole H range up to 30 cm. It is true, for instance, for top quality BGO material with absorption coefficient of 0.002 cm
1. As regards the BGO sample under examination with kav ¼ 0:03 cm
1, the internal radiation in crystal begins affecting heat transfer starting from the crystal height of 3 cm. Already at the crystal height of 10 cm, the temperature profile essentially differs from the linear one (Fig. 5). One can clearly see in Fig. 5 that, neglect of the S-shaped profile of the temperature curve in the case of long crystal results in almost a double error (underestimation) of the temperature gradient at the crystallization front. It is necessary to note that for crystals with defects the substantial role of RCT appeared to be dominating at a length, which is less by 20% than that for defect-free crystals. This is due to the fact that defects results in increase of absorption coefficient kav [12], therefore, the same optical length (t ¼ kav H) for the crystal of poor quality corresponds to smaller value of a crystal length H in comparison with the perfect one. 3.1.2. Melt Large absorptivity of the melt [12] allows describing the radiative heat transfer in melt using

conception of radiative thermal conductivity. For the melt, N ¼ 1:5; that indicates that heat fluxes transferred by conduction and by thermal radiation are of the same order. Because of this, all calculations of conductive heat transfer in the BGO melt, as well as calculations of various criteria using the thermal conductivity value should be based on the effective thermal conductivity lm eff found in Ref. [12] to be equal to 0.2 W/m K. Such a small value of lm eff leads to a very small total heat flux in the melt qm ¼ leff ðgrad TÞm at an ordinary temperature gradient in the melt. For the discussed above value of the temperature gradient in the furnace being equaled to 10 K/cm, qm ¼ 200 W/m2. Thus, one may conclude that for one and the same temperature gradient in the melt and in the crystal, the heat transferred through the melt is two–three orders of magnitude less than that in the crystal. 3.2. Features of heat transfer in growing crystals by AHP method 3.2.1. AHP growth at TRC: Thot ¼ const; Tcool ðtÞ ¼ T0 —at (a¼ const; t-time) Experiments on AHP growth of BGO show that, for the initial position (at d0 ¼ 10 mm), it was impossible to establish a relatively large melt layer h0 even at a large temperature difference DT ¼ Thot
Tcool ¼ 30240 K. In all the experiments, the h0 value was within 1–2 mm. For the TRC under consideration, h decreased with increasing crystal length. Such a behaviour of the melt–crystal system is explained by the distinction in the mechanisms of heat transfer in melt and in crystal. The heat fluxes in both melt and crystal should be equal to each other at the interface: m cryst lm ; eff ðgrad TÞ þ ql ¼ q

ð2Þ

where ql is the latent heat of crystallization. Since qcryst is large because of intensive radiation from the interface, and lm eff is small, then, in order to meet requirement (2), a large temperature gradient should be established in the melt. But melt overheating ðDTÞm ¼ Thot
Tm is small; hence,

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melt thickness h ¼ DT m =ðgrad TÞm should be small too. For example, with temperature drop of DT ¼ 10 K at the crystal and overheating ðDTÞm ¼ 20 K, it follows from Eq. (2) that h ¼ 0:5 mm. Thus, large heat flux by radiation forces the melt–crystal interface to move closer to the hot boundary with the temperature Thot : If Tcool decreases during the crystal growth, the increase in H leads at the same time to the increase of temperature drop at the crystal DT cryst ¼ Tm
Tcool : In turn, an increase of DT cryst causes an increase of heat flux leaving the interface by radiation; the thickness of melt layer h decreases. The calculated dependence of the change of h with time for different conditions is shown in Fig. 6a. In the experiment on BGO growth [14], due to variations of thermal conditions at hot and cool boundaries, the growth rate have sharply changed that made it possible to obtain a set of marks 6

4 3

1

2 2

1 0 10

15

(a)

20 25 Crystal length (mm)

30

35

ð3Þ

where R ¼ 1
e; s is Stephan–Boltzmann constant. Good coincidence of the one-dimensional model with experiment confirms the assumptions concerning the role of complete reflection in crystal and allows one to analyse the features of thermal conditions on the basis of Eq. (2). For the selected type of TRC (constant temperature gradient in the furnace), the growth rate and temperature gradient in the melt were found to strongly change during the growth process (Fig. 6b). It is obvious that these changes are most substantial in the initial period of growth. As early as H reaches 30 mm when temperature at the cold boundary decreases by only 4.5 K, the temperature gradient near the crystallization front changes by several times, with the thickness of the melt layer decreasing from 5 to 1 mm. The changes are the larger the larger is the temperature gradient in the furnace and correspondingly the higher is the

250

3.5

200

3

150

1

2.5

100

2

50 0 10

15

20 25 Crystal length (mm)

30

35

Fig. 6. Change of the thickness of melt layer (a), growth rate and temperature gradient in the melt (b) during BGO crystallization at constant overage temperature gradient: 1—cooling rate: 0.5 K/h, 2—1 K/h (pulling rate is 2 mm/h; initial crystal thickness equals to 10 mm).

4

7

3.5

6 5

3

2

4

2.5

3

2

2

1.5

1

1

Optical density (r.u.)

300

2

4

1.5 (b)

4  ðTm4
Tcool Þ;

350

4.5

grad T (K/cm)

Growth rate (mm/hr)

5

connected with impurity capture at the interface. The comparison of the revealed positions of interface with those calculated in the present work with account of RCT shows that the position of the interface can be determined at accuracy in the mean of 5% (Fig. 7) with the help of Eq. (2), in which the radiative term is described in the approximation of optically thin layer [12]:   1
R 2 1
4R þ R2 qcrad ðe; tÞ ¼ sn2
t 1þR 3 ð1 þ RÞ2

Crystal growth rate (mm/h)

Melt layer (mm)

5

219

0

1 0

5

10 15 20 Crystal height (mm)

25

30

Fig. 7. Comparison of the calculated data on the interface position (corresponds to maxima of growth rate—curve 1) with the experimental marks (over crystal zones with increased concentration of the captured impurity—curve 2).

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220

cooling rate of the pedestal, i.e. the higher is the rate of decrease of temperature Tcool : 3.2.2. The character of TRC providing constant thermal conditions for the AHP technique In the AHP technique, to make the growth rate V constant it is necessary to provide DT m ¼ Thot
Tm ¼ const; h¼ const and n ¼ const that means keeping constant the temperature gradient in the melt ðgrad TÞm too. It means that heat flux m in melt qm ¼ lm eff ðgrad TÞ must be constant, and, in accordance with Eq. (2), one should realize the conditions of crystal growth providing constant heat flux in the crystal (qc ¼ const). Since qcryst is non-linear function of optical density t ¼ tðHÞ; then, for linear change of H; it is necessary to change temperature Tcool in a special non-linear manner in the course of crystal growth. Calculated according to one-dimensional RCT problem, the functional dependence of Tcool ðtÞ providing constant growth rate is shown in Fig. 8. One can see that it is necessary to change the temperature of the cold boundary (bottom of the seed) Tcool ðtÞ in a complicated manner in order to provide qcryst ðtÞ¼ const; as the crystal length growths. Such a character of the TRC is explained as follows. For small H (up to 30 mm), the contribution of heat conduction in total heat transfer is still crystc noticeable: qcrystc ¼ qcond þ qcrystc rad ðqcond ¼ l½Tm
Tcool ðtÞ=HðtÞ). Consequently, to provide qcrystc ¼ const; at first stage crystal growth it is necessary to decrease Tcool as usual. Starting from H ¼ 30 mm, radiative component rapidly increases in compar1319.5

Tcool (K)

2 1319

1 1318.5 5

25

45 65 85 Crystal length (mm)

105

Fig. 8. Character of temperature change at the cool boundary of the crystal providing constant growth rate for the AHP procedure: 1—k ¼ 0:03 cm
1, 2—k ¼ 0:06 cm
1.

ison with conduction, and the total heat flux becomes approximately equal to that transferred crystc by radiation: qcrystc Eqcrystc rad : Hence, qrad should be constant to provide constant heat removal from interface, and it is reached by keeping (at this stage of crystal growing) a constant temperature drop across the crystal: DTint
cool ¼ Tint
Tcool ; where Tint is the interface temperature. Thus, it should be noted the feature of heat transfer when, starting from some small H ¼ Lcr ; the requirement qcrystc ¼ const is provided not by constancy of the temperature gradient but due to constancy of temperature drop. Further on, at achieving H; for which t ¼ kH > 0:1; the heat transfer character becomes complicated as internal radiation of matter itself makes a contribution into RCT, especially for well reflecting boundaries. The radiation increases faster than DTint
cool does, so, in order to maintain the value of qcrystc constant, it rad is necessary even to increase Tcool a little (Fig. 8). Attention should be paid (Fig. 8, curve 2) to the fact that the presence of defects in crystal leads to more intensive heat transfer by radiation in crystal; that is why critical Lcr value in this case is somewhat less than in high-quality (perfect) crystal. Therefore, the decrease of Tcool for crystal with defects should be stopped earlier and the slope of the temperature curve should be slightly steeper. 3.3. Features of heat transfer in the low-gradient Czochralski procedure The character of TRC shown in Fig. 8 should become apparent in full measure while growing the crystals of large length. Therefore, obtained results were verified according to the procedure described in Section 2.2, for BGO crystal growth by LTG Cz method up to 400 mm long. The appearance of the crystal grown during investigations is shown in Fig. 3b. Experimentally recorded dependence of temperature variations of the heater DTHeater during crystal growth at a constant rate (as indicated by the weight sensor) is shown in Fig. 9 [24]. The oscillating character of the experimental curve was caused by the feed-back with respect to weight signal. A good qualitative coincidence is observed between smoothed

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

∆Theater (K)

2

1

1 0 -1 0 -2

100

200

300

2

-3 Crystal length (mm) Fig. 9. Change of temperature at a heater DT ¼ TðtÞ
Tðt0 Þ during BGO crystal growth by LTG Cz technique: 1—actual data; 2—faired curve. Experimental data.

experimental data (Fig. 9) and the character of the calculated one (Fig. 8). Thus, it is possible to conclude that the discussed approach describes properly the principal features of heat transfer during BGO crystal growth by LTG Cz method. There are two reasons which provided a good fit of the model offered to the experiment. The first one is a satisfactory description of heat transfer by the one-dimensional model. Second, there is a very small contribution of natural convection into heat transfer in both of these methods. In the AHP procedure, convection is small because of small thickness of the melt layer and small radial temperature gradients in it, while a small temperature drop over the whole height of the melt layer is realized in the LTG Cz method. However, one can see that there is a difference between experimental (Fig. 9) and calculated values (Fig. 8) of both the temperature changing value and the crystal height corresponding to the point of the temperature curve inflexion that may be explained not only by a difference in points of temperature measurement location (at the heater in experiment and at the crystal seed in calculations), but also by affect of interface supercooling phenomenon. Its influence may be important because, as a rule (and always when using LTG Cz method), BGO crystal grows by facet mechanism [4] and value of supercooling is large enough [13]. 3.4. The effect of facet on heat transfer To find the character of TRC providing constant thermal conditions at the melt–crystal

interface and constant growth rate in presence of interface supercooling, the model above discussed (Eq. (1)) was used in present work. It was supposed that the facet arises immediately at the moment of the beginning of crystallization, with facet area Sfacet increasing linearly with time Sfacet ðtÞ ¼ Scryst f ðtÞ; so that after some time tend the faceted interface covers the whole crystal cross section: Sfacet ðtend Þ ¼ Scryst : Within the time interval 0ototend ; the time dependence of the facet area can be written as a function of the height of the growing crystal: Sfacet ðHÞ ¼ Scryst ½1
ðH0 =HÞ2 ;

ð4Þ

where H0 is the height of the seed. In the case under consideration, H0 ¼ 5 mm; taking into account the results of Ref. [18], where the dynamics of facet size is considered, we suppose the facet to be formed over the whole cross section of the crystal, as early as at H=30–40 mm. During the growth of facet, supercooling DTint ; facet which determines the rate of facet growth Vint ¼ facet f ðDTint Þ; is formed; or in another form: Vint ¼ bfacet DTint : Experimentally measured data [13] characterizing the dependencies between supercooling averaged over the facet and the growth rate are presented in Fig. 10; these dependencies are well described by functions facet Vint ¼ fV1 ðDTint =DT1 Þ
0:46 gDTint facet Vint p3:5 mm=h;

for

ð5aÞ

facet Vint ¼ fV2 ðDTint =DT2 Þ0:4 gDTint facet Vint X3:5 mm=h

for

Vint (mm/hr)

3

221

ð5bÞ

Vint facet = {V2(∆Tint/∆T2)0.4}∆Tint

12 10 8 6 4 2 0

2

1

Vint facet = {V1(∆Tint/∆T1)-0.46}∆Tint 0

2 4 Supercooling (K)

6

Fig. 10. Dependence of the rate of BGO (1 1 0) facet growth on supercooling of the melt–crystal interface: 1—experimentally measured values [11], 2—approximation of experimental data by the analytical equation.

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

222

(the character of power dependence b on DTint ; and also the V1;2 and DT1;2 values are chosen empirically). The facet average temperature is Tfacet ¼ Tm
DTint ; average temperature at the crystallization front is Tint ¼ Tm
DTint Sfacet ðtÞ= Scryst or, in terms of H: Tint ¼ Tm
DTint ½1
ðH0 =H 2 Þ;

ð6aÞ

Tint ¼ Tm
DTint :

ð6bÞ

Eq. (6a) is used while facet is steel growing; after facet is formed completely, Eq. (6b) must be used. Thus, to describe heat transfer during the crystal growth by faceted mechanism, it is necessary to introduce a correction into Eq. (1) to account changing in temperature of the interface during crystallization in accordance with Eqs. (5) and (6). In addition, it is necessary to decrease the temperature of the hot boundary Thot by the same value as DTint in order to maintain constant overheating of the melt with respect to the interface, and correspondingly to keep the temperature gradient on it during the whole crystal growth process. Finally, the following system of two equations has to be solved: lm eff

fThot
DTint ½1
ðH0 =HÞ2 g
fTm
DTint ½1
ðH0 =HÞ2 g h

þ Vint Jrm ¼ qcrad ðe; tÞ þ lccond

fTm
DTint ½1
ðH0 =HÞ2 g
Tcool ; H ð7aÞ

Vint ¼ f ðDTint Þ:

ð7bÞ

It is impossible within the framework of the onedimensional model to consider the interface having a complicated shape with areas, which grow by different growth mechanisms. Therefore, the plane interface with mean temperature averaged over the interface was considered to perform calculations due to Eq. (6a). In that equation supercooling of rough rough surface DTint is supposed to be rather less than that of faceted one and can be neglected. However, defining the mean value of growth rate for such an interface, one should take into account that the growth rate of zones growing by rough mechanism is essentially higher in comparison with that growing by facet one for the same

supercooling. Therefore, at the beginning of the crystallization, when facet is still forming and growing, the crystal growth by rough mechanism rough ¼ should be considered in calculations too: Vint rough rough b DTint : This means that in the model describing growth rate dependence on supercooling, it is necessary to consider in Eq. (6a) a contribution of both growth mechanisms when accounting facet size changing due to Eq. (4) during crystallization: rough Vint ¼ bfacet DTint ½1
ðH0 =HÞ2  þ brough DTint ðH0 =HÞ2 ;

ð8Þ where ðH0 =HÞ2 indicates a part of the interface area growing by rough mechanism. As crystal grows, this area diminishes rapidly and no longer affects the steady-state growth rate. The model offered is not quite realistic but, being applied, allows to prevent an appearance of the sudden change in temperature curve of Tcool without solving the non-stationary problem of the facet origin and growth. To exclude ambiguity in description of the growth rate dependence on supercooling (5), the solution of the system of equations was tried for Vint ¼ 3:5 mm/h that corresponds to the range of growth rates, in which the functional dependence (5a) is still valid. In the calculations the following equation was finally used: rough m1 Vint ¼ fa1 DTint gDTint ½1
ðH0 =HÞ2  rough m3

þ fa3 DTint

gDTint ðH0 =HÞ2 ;

ð9Þ

in which the conventional approximation for kinetic coefficient b ¼ am DT m was introduced to Eq. (8). Values of a1;3 and m1;3 were selected in such a manner to satisfactory fit data on experimental dependence Vint ¼ Vint ðDTint Þ providing a rapid decrease of contribution of rough growth component of crystal growth over the interface with an increase of H: With account of Eq. (9) the system of Eq. (7) was solved depending on the length of the grown crystal. Fig. 11 shows distinction of temperature Tcool calculated in presence of supercooling in comparison with data obtained without taking into account interface supercooling (Fig. 8). Found solution shows (Fig. 11) that, as facet is

ARTICLE IN PRESS V.D. Golyshev, M.A. Gonik / Journal of Crystal Growth 262 (2004) 212–224

Temperature deviation (K)

0 0

100

200

300

-0.5 -1

1 -1.5 -2

2 Crystal length (mm)

Fig. 11. Distinction of temperature on hot (curve 1) and cool (curve 2) boundaries of the melt–crystal system, when facet growth is taken into account, from the values Thot and Tcool calculated in the absence of a facet. Requirement Vint ¼ const is met over the whole crystal length.

forming, one should decrease the temperature of the crystal cold boundary Tcool rather more than it might be done in absence of supercooling. When facet formed all over the cross section of the crystal, a maximum value of supercooling corresponding to the given value of growth rate Vint establishes; and distinction of temperature curve Tcool ðtÞ in its character from that found in Section 3.2.2 (Fig. 8) disappears. Obtained result (Fig. 11) is appreciably closer to experimental data shown in Fig. 9. It demonstrates that one can satisfactory describe heat transfer in BGO crystal growth by AHP technique and LTG Cz method using onedimensional model, if supercooling of the interface is taken into account. Besides, it is obvious from Fig. 11 that in case of facet growth it is necessary to reduce the temperature of hot boundary Thot ðtÞ too, until maximum supercooling establishes over all the interface.

4. Conclusion Investigations have shown that the AHP method of growing crystals is an efficient one for experimental studies and calculations of the features of RCT during the crystal growth of dielectrics. Having been carried out for growth of large size BGO single crystals, the analysis of RCT made it possible to find a temperature regime of crystallization providing the constant thermal conditions at the interface, which cannot be selected in a proper way by means of trial-and-

223

error method. Supercooling of the melt-crystal interface was found to be considered in calculations of heat transfer. The character of heat transfer during the growth of large-sized BGO single crystals is determined by peculiar thermal and optical properties of the melt and crystal. Under conventional thermal conditions, heat flux in the melt is small due to small thermal conductivity of the opaque melt. In high transparent crystal, because of thermal radiation, heat flux is considerable. The crystal functions as light guide, therefore heat elimination occurs directly from the interface. Under these conditions, the use of the conventional Czochralski technique leads to the appearance of cone shape of crystallization front. It is difficult to correct this situation due to more overheating of the melt because of its low thermal conductivity. The only way is to decrease heat flux in the crystal providing small temperature drops on it. Thus, LTG Cz Technique proved to be highly efficient method for growing large BGO crystals. The results obtained have clearly demonstrated the importance of knowing thermophysical properties and studying RCT problem. The BGO single crystal can be recommended as a model material to investigate into connection between heat transfer and interface kinetics because its thermophysical properties and features of heat transfer in crystal growth are well studied at present.

Acknowledgements The research described in this publication was made possible in part by Grant 02-02-17128a from the Russian Foundation for Basic Research, and in part by Award No. RE1-2233 of the US Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF) and in part by INTAS Project No. 2000-263. Authors would like to express great gratitude to Ya.V. Vasilyev and V.N. Shlegel from Institute of Inorganic Chemistry SB RAS (Novosibirsk, Russia) for presented BGO crystals for studies, information on LTG Cz method and apparatus, and fruitful discussion.

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