ISSN 10637842, Technical Physics, 2010, Vol. 55, No. 11, pp. 1669–1679. © Pleiades Publishing, Ltd., 2010. Original Russian Text © P.V. Smirnov, A.A. Rusanov, 2010, published in Zhurnal Tekhnicheskoі Fiziki, 2010, Vol. 80, No. 11, pp. 122–132.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUE

Stability of the Crystal–Melt System in the Growing of Large Lithium Niobate Crystals by the Low Melt Level Czochralski Technique P. V. Smirnov* and A. A. Rusanov Ioffe PhysicalTechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia *email: [email protected] Received October 24, 2009; in final form, February 15, 2010

Abstract—The Lyapunov stability of the crystal–melt system and the solidification rate is studied for the growing of lithium niobate crystals by the low melt level Czochralski technique. The relaxation time is calculated as a func tion of growth conditions. The curves of transient processes are plotted at various variable perturbations. It is shown that, for optically homogeneous crystals to be produced, the stability of the solidification rate rather than the crys tal–melt system as a whole should be ensured. The stability of the solidification rate is found to be maximal at the maximum temperature gradient at the solidification interface and the minimum melt level in a crucible. DOI: 10.1134/S1063784210110198

1. INTRODUCTION The low melt level Czochralski (LMLCz) technique proposed in [1] is closest to the Czochralski technique with a double crucible and differs from it in the fact that the melt level is only several millimeters (Fig. 1). The setup for growing lithium niobate crystals by the low melt level technique differs from that of the Czochralski tech nique with a double crucible in the following: (i) a low crucible wall, which provides heat removal from the crystal surface adjacent to the solidification interface; (ii) a bottom heater; and (iii) feeding. The melt level is chosen so that conditions of the absence of free melt convection are met in the crystal– crucible gap (see [2]). Without crystal rotation, this condition is met at a melt level of 1–3 mm [3], all other lithium niobate crystal growth conditions being identi cal to those of the Czochralski technique. In the pres ence of rotation, this is likely to be somewhat higher. The LMLCz technique is interesting for the grow ing of lithium niobate crystals of a congruent compo sition, since it can increase the setup capacity by about three times and decrease the platinum consumption per unit capacity by about ~30 times. The crucible bottom located near the solidification interface ensures control of its shape. In the absence of free melt convection, the level of temperature fluctuations is lower. Thus, the LMLCz technique can become a more economical and promising technique for the growing of opticalgrade lithium niobate crystals than the Czochralski technique widely used for this pur pose.

The LMLCz technique is now too poorly under stood to be applied in practice: in particular, it is nec essary to estimate the optimum parameters of the growth unit and the growth conditions required for the growing of opticalgrade lithium niobate crystals. Lithium niobate crystals are the basic material for acoustic wave filters (cellular phones, television sets, etc.) and nonlinear (light transformation into har

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Fig. 1. Low melt level Czochralski technique: (1) grown crystal, (2) crucible, (3) ring insert in the crucible, (4) bot tom heater, (5) lateral crucible heater, (6) tube heater, (7) insulation, (8) feeding tube, (9) probe for measuring the melt level, (10) melt, and (11) thermocouples.

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monics, parametric generation) and integrated (inte grated optics chips for ultrafast communication lines) optics. The last two fields of application require high quality opticalgrade crystals. The growing of large opticalgrade crystals is a difficult technical problem and always requires optimal conditions for their growth, especially for a new technique. Growth striations and an optical heterogeneity (birefringence gradient) in lithium niobate crystals are mainly caused by a change in the distribution coeffi cients of melt components in a crystal due to devia tions of the solidification rate, the melt temperature gradient, and the diffusion layer thickness from their stationary values. To create growing conditions that ensure the max imum stability of the growth rate, one has to know the effect of these conditions on the stability of the crys tal–melt system and the solidification rate. The authors of [4, 5] studied the stability of the crystal–melt system as the resistance of steady motion of the crystal–melt system to small random indepen dent perturbations. In the simplest model of the crys tal–melt system for the Czochralski technique, steady motion means a constant crystal radius and a constant meniscus height, with only these variables being sub jected to perturbations. According to Lyapunov, the stability of the crystal– melt system is characterized by the time it takes for the parameter deviations having appeared before the ini tial time to disappear. The form of the evolution of variable perturbations in the crystal–melt system before the initial time is insignificant: it is only impor tant that all boundary conditions correspond to steady motion after the initial time. If several relaxation times exist, stability is estimated from the maximum relax ation time [4, 5]. As a variable for the crystal–melt sys tem in the Czochralski technique, Tatarchenko [4] chose the crystal radius and the melt meniscus height. Léibovich [5] added another variable, the melt level, to take into account finite crucible dimensions. The approach in [4, 5] can be used to optimize growing conditions to achieve the maximum resis tance (selfstabilization) of the crystal–melt system to random perturbations. To increase the stability of the system, Tatarchenko [4] recommends increasing the coefficient of heat removal from the crystal surface near the solidification interface, which results in an increase in average melt temperature gradient GL at the solidification interface. The authors of [6] considered methods to implement this recommendation for the growing of lithium niobate crystals by the Czochralski technique and showed that GL (subscript L stands for liquid; S, for solid)) increases when a reflecting dia phragm above the crucible edge, a gap between heat shields, or their combination is used. Since the melt level in the LMLCz technique is constant, it is even easier to apply the methods [6] for increasing the heat removal from the crystal surface than for the Czo chralski technique. Therefore, in this work we do not

take into account the methods of controlling temper ature gradient GL. For the Czochralski technique, the authors of [7] studied the dependence of the stability of the crystal– melt system during the growth of lithium niobate crys tals on the ratio of the crucible to the crystal diameter. We found that, when the distance between the crystal wall and the crucible wall is smaller than or equal to the shrinkage gap that forms in the absence of pulling, the crystal–melt system is stable under standard lith ium niobate crystal growth conditions. If the gap is smaller than the shrinkage gap (pulling is directed down), instability appears. The authors of [7] did not take into account the capillary effect of the crucible wall and estimated the gap at two capillary constants between the crystal and crucible at which this effect may be neglected. The authors of [8] studied the stability of the crys tal–melt system when lithium niobate crystals are grown at the crystal–crucible gap much smaller than the capillary constant with allowance for the capillary effect of the crucible wall and also showed that this sys tem is resistant to symmetric crystal radius perturba tions. Satunkin [9] developed a crystal–melt model to control crystal growth by making allowance for finite crucible dimensions. To this end, he considered per turbed motion of the crystal–melt system and took into account nonstationary boundary conditions. This approach is necessary to optimize control algorithms. For our purposes of optimizing growing conditions, boundary conditions after the initial time should be stationary. In [9], the solidification interface surface was taken to be a segment of a sphere, and the sphere radius was assumed to be much larger than the crystal radius. This assumption generates transient process fea tures that are insignificant for us; therefore, the solidifi cation interface in our work is taken to be planar. The purpose of this work is to find optimum lith ium niobate crystal growth conditions for the LML Cz technique provided that the crystal–melt system and the solidification rate are stable. To this end, we have to construct a model for the crystal–melt system in the LMLCz technique and to analyze the stability of the crystal–melt system and the form of transient processes as functions of growth conditions. 2. APPLICABILITY OF A ONEDIMENSIONAL HEAT CONDUCTION EQUATION FOR A CRYSTAL DURING THE GROWING OF LARGE LITHIUM NIOBATE CRYSTALS To obtain analytical estimates for the degree of sta bility, researchers use a onedimensional stationary heat conduction equation for a crystal. The applica tion of this equation requires quasistationary condi tions, a small Biot number, and a high absorption coefficient of infrared radiation near the melting tem perature. TECHNICAL PHYSICS

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The quasistationary condition means that solidifi cation interface relaxation rime τin is much longer than temperature relaxation time τ (subscript “in” stands for interface). The estimation of these times according to [4] yields τ in L H k L  ≈   2 ≈ 100, τT λL GL H where L is the latent heat of melting of unit volume, λL(λS) is the thermal conductivity of the melt (crystal), kL(kS) is the thermal diffusivity of the melt (crystal), H is the solidification interface height with respect to the crucible bottom, and GL(GS) is the average melt (crystal) temperature gradient at the solidification interface. Thus, the quasistationary condition is met. The condition of smallness of the Biot number Bi = μ*R/λS (where μ* is the coefficient of heat removal from the crystal or melt surface and R is the crystal radius) is met only approximately, Bi ≈ 1. Neverthe less, the onedimensional heat conduction equation can be applied for an approximate calculation of the temperature averaged over the cross section of the crystal. The radiant heat transfer in a lithium niobate crys tal near the melting temperature can easily be taken into account by introducing an effective thermal con ductivity, since the absorption length is much smaller than the crystal sizes at a temperature near the melting point [10]. 3. THERMAL FIELD IN THE MELT The thermal field in the melt is substantially deter mined by the melt motion induced by crystal rotation. The melt convection between a rotating solidification interface and a fixed crucible bottom can be consid ered by analogy with the problem of liquid motion in the case of disk rotation in a casing [11]. It was shown that the shear Couette flow, which is characterized by a linear distribution of the peripheral velocity over the height, forms in the gap when the sum of hydrody namic boundary layers on the rotating disk (δ ≈ 4 ν/ω ) and in the fixed casing (δ ≈ 8 ν/ω ) is much greater than the gap. Here, ν is the kinematic viscosity and ω is the disk rotation frequency. When we grew lithium niobate crystals 80–100 mm in diameter by the LMLCz technique, a nearplane shape of the solidification interface was obtained at a disk rotation frequency of about 1 rpm. Therefore, with allowance for the data in Table 1 for lithium nio bate, the sum of hydrodynamic layers is more than 40 times greater than the solidification interface height with respect to the crucible bottom. In other words, the conditions of the formation of the shear Couette flow are met. Schlichting [11] also presents an approximate rela tion between the hydrodynamic (δ) and thermal (δT) TECHNICAL PHYSICS

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Table 1. Physical properties of a lithium niobate crystal and its melt Parameter, dimension a, m L, J/m3 v, 10–5 m2/s ψ0, grad ρL, kg/m3 ρS, kg/m3 Pr λS, W/(K m) λL, W/(K m)

Lithium niobate

Source

0.004 2.11 × 109 1.123 0 3668 4630 13.6 1.1 3.09

[13] [14] [14] [13] [15] [14] [14] [16] [14]

boundary layers, δT ≈ δ/ Pr , where Pr is the Prandtl number. The thermal boundary layer is the region where a temperature distribution is linear along the normal to the surface on which this layer formed [12]. The sum of thermal boundary layers for the growing of lithium niobate crystals by the LMLCz technique is ~12 times greater than the solidification interface height with respect to the crucible bottom. Therefore, the temperature distribution between the solidifica tion interface and crucible may be considered as a lin ear distribution over the height, which allows us to remain in terms of analytical calculations. We neglect the Marangoni convection, which appears due to a gradient in the surface tension on the free melt surface, because of a small melt depth (1–3 mm). A Marangoni convection vortex can hardly move under the crystal, since it is rejected by forced convection. 4. EQUATIONS DESCRIBING THE CRYSTAL–MELT SYSTEM 4.1. Choice of Variables As the model variables for the crystal–melt system, we chose crystal radius R, meniscus height h, and solidification interface height H with respect to the crucible bottom (Table 2). For this choice of variables, a deviation of the solidification rate is only directly related to front height H. At a given front height H0, the difference H0 – h specifies the melt level in the cru cible. 4.2. Steady Motion In a stationary state, the process variables are con stant (R = R0, h = h0, H = H0); that is, their deviations from stationary values (δR = R – R0, δh = h – h0, δH = H – H0) are zero (Fig. 2).

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(without melt level feedback, δMf = Mf – Mf0 = 0), and obtain (3) δm· + δM· + δμ· = 0.

Table 2. Experimental conditions Parameter, dimension

Value

(0)

In the plane solidification interface approxima tion, the meniscus mass is

–3

V c , mm/h V, mm/h

3

2

M0, kg

0.067

R0, m

0.050

Rcruc, m

0.075

h0, m

0.0039

H0, m

t

∫

1

GL, K/m

–3000

0

2

0

For a deviation of the radius from a stationary value δR, we obtain δR· = – V tan ( δα ). (1) c

M = πρ L R cruc ( H – h ). Here, ρL(ρS) is the melt (crystal) density, a is the cap illary constant of the melt, Rcruc is the crucible radius, and the other designations are given in Fig. 2. 4.6. Stefan Conditions at the Interface With the temperature gradients averaged over the cross sections of the melt and crystal, we write the plane solidification interface approximation in the form –1

V c = L ( λ S G S – λ L G L ).

0

4.4. Continuity Equation The continuity equation in variables R, h, and H is written as – H· + V + V = 0. (2) c

4.5. Mass Balance Feeding mass Mf during growth changes into crys tal mass m, melt mass M, and melt meniscus mass μ, m· + M· + μ· = M· . In a stationary state, we have m· 0 + M· 0 = M· f0 . We introduce designations δm = m – m0, δM = M – M0, and δμ = μ – μ0, suppose that feeding is stationary δα0

αe

ψ0 R0

H0

H

R

δh < 0

ψ

h

δR < 0 δH > 0

α0

(4)

5. LINEARIZATION OF THE SET OF EQUATIONS IN THE VICINITY OF UNPERTURBED MOTION AND THE ESTIMATION OF THE COEFFICIENTS After the linearization of the set of equations (1)– (4) in δR, δh, and δH in the vicinity δR = 0, δh = 0, and δH = 0, we obtain the set of equations ˆ x· = Aˆ x, D where

f

h0

and

the melt mass in the crucible between the zero pressure level and the bottom is

4.3. Equation for Measuring the Crystal Radius The rate of change of the crystal radius is propor tional to the tangent of the expansion angle δα0 ≡ α0 – αe = ψ0 – ψ and solidification rate Vc, R· = – V tan ( δα ). c

2

m = – πρ S R V d dt,

0.0049

ω, ppm

2

μ = πρ L ( R h + a R cos ψ ), the crystal mass is

⎛ ⎞ ⎜ δR ⎟ x = ⎜ δh ⎟ ⎜ ⎟ ⎝ δH ⎠ is the state vector and ⎛ 1 0 0 ⎜ ˆ D = ⎜ 0 0 1 ⎜ ⎝ d 31 d 32 d 33

⎞ ⎛ A A ⎞ ⎟ ⎜ RR Rh 0 ⎟ ˆ ⎟ , and A = ⎜ A hR A hh A HH ⎟ ⎟ ⎜ ⎟ ⎠ ⎝ A HR 0 0 ⎠ 1

are composed of elements dnm and Aik, respectively. Subscripts n and m take values of 1, 2, and 3. Subscript

1 The linearization means the expansion into a Taylor series in small

Fig. 2. (left) Undisturbed and (right) disturbed state of the crystal–melt system.

deviations δR, δh, and δH and the truncation of all terms in which the powers of δR, δh, and δH are higher than the first power. TECHNICAL PHYSICS

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n means belonging to an equation (n = 3 means that d3m enters into Eq. (3)), and subscript m indicates belonging of a coefficient to a quantity (δ R· (m = 1), δ h· (m = 2), δ H· (m = 3)). Subscripts i and k take the

Table 3. Calculated values of the parameters for the grow ing of lithium niobate crystals under the conditions from Table 2

values of R, h, and H. Subscript i means belonging to an equation: i = R, belonging to the first equation in set (5) (see below); i = h, belonging to the second equation in set (5); and i = H, belonging to the third equation in set (5). Subscript k indicates belonging of a coefficient to a quantity: δR (k = R), δh (k = h), and δH (k = H). After reducing the set of equations to canonical ˆ x, we obtain form x· = Ω ⎛ ⎜  ⎜ ⎛ δR· ⎞ A   RR ⎜ 1 ⎜ ⎟ ⎜ δh· ⎟ = ⎜  ( A HR – d 33 A hR – d 31 A RR )  ⎜ d 32 ⎜ ⎟  A HR ⎜ ⎝ δH· ⎠  ⎜ ⎝ (5) ⎞ ⎟ A Rh 0 ⎟⎛ ⎞ ⎟ ⎜ δR ⎟ d 33 1 –  ( d 33 A Hh + d 31 A Rh ) – A HH ⎟ ⎜ δh ⎟ . d 32 d 32 ⎟ ⎟⎜ A HH ⎟ ⎝ δH ⎠ A Hh ⎟ ⎠

kS, 10–7 m2/s kL, m2/s μ*, W/(m2 K) ARR, 10–4 s–1 ARh, 10–4 s–1 AhR, 10–4 s–1 Ahh, 10–4 s–1 AHH, 10–4 s–1 AHR, 10–4 s–1

In a physical meaning, some Aik coefficients of the LMLCz technique model (namely, AhR, ARR, and ARh) coincide with the corresponding coefficients of the Czochralski technique from [4, 7]. Therefore, we take expressions for them from [4, 7], where these coeffi cients were calculated, ΔT e0 ∂G ∂G μ* A hR = 1 λ S S – λ L L ≈ 2  , ∂R ∂R L LR V 2 2μ* 2 +  4k S λ S R ∂δα 0 A RR = – V c   ∂R

R  a, R cruc – R > 2a

≈ 0,

∂δα 0 A Rh = – V c   ∂h

2 ≈ – V c . α Ra a cos e 2 Coefficient AHH controls the reaction of the heat flow through the solidification interface to a change in solidification interface height H with respect to the crucible bottom. It was found using the following assumptions: the thermal field in the melt is linear in the height from the crucible bottom to the solidifica tion interface, the crucible bottom temperature is fixed after the initial time, and the heat flow abstracted by the crystal reacts to a change in the solidification TECHNICAL PHYSICS

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Parameter

Value

Parameter

2.3 8.26 100 0 –4.2 0.31 –0.006 –9.0 –12

d31, kg/m d32, kg/m d33, kg/m ΔTe, K δT , m δ, m GS, K/m τ 31, min τ 32, min

Value 4.9 –37 31 100 0.029 0.104 –3447 18.9 497

interface height much weaker than the heat flow from the melt to the crystal does. We derived the simple expression ∂G ∂G λ G 1 A HH =  λ S S – λ L L ≈ L L . ∂h ∂h L H L Coefficient Ahh, which determines the reaction of the heat flow through the solidification interface from the melt side to a change in the melt meniscus height without a change in the position of the solidification interface with respect to the crucible bottom, was esti mated numerically using a onedimensional heat con duction equation for the melt, λL A Hh ≈ ΔT e kh , L where ΔTe is the difference in the radiation tempera tures of the melt meniscus and the growth unit walls (subscript e[roman] means external) with which it exchanges radiation and kh is a numerical coefficient. The value of Ahh estimated with a onedimensional heat conduction equation used for the melt meniscus is so small (see Table 3) that Ahh may be taken to be zero within the framework of the problem. For this purpose, the accuracy of its estimation by one order of magnitude was found to be sufficient; therefore, the use of the heat conduction equation for the melt is jus tified. Coefficients AHR, d31, d32, and d33 are obtained from mass balance. Similar calculations were per formed in [5, 9] to within the choice of variables and with/without allowance for the feeding error, 0

A HR = 2πρ S RV c , 2 2 ∂ψ , d 31 = πρ L 2Rh + a cos ( ψ ) – a R sin ( ψ )  ∂R 2 2 2 ∂ψ d 32 = – πρ L R cruc + πρ L R – πρ L a R sin ( ψ )  , ∂h 2

2

d 33 = πρ L R cruc – πρ S R .

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τ2

⎛ A A ⎞⎛ ⎞ ⎛ δR· ⎞ ⎜ ⎟ = ⎜ RR Rh ⎟ ⎜ δR ⎟ . ⎝ A hR A HH ⎠ ⎝ δh ⎠ ⎝ δh· ⎠

(a) 103

Hereafter, model (6) is designated as {R, h}. In con trast to the Czochralski technique model from [4], coefficient AHH depends on stationary solidification interface level H. Here, meniscus height h is measured from a fixed melt level. Dynamic system (6) for the LMLCz technique coincides with the corresponding system for the Czochralski technique in all features except for the expression for coefficient AHH.

(3)

τ2

102 (3)

(2)

τ1 , τ1 101 0 103

−2

−4 GL, 103 K/m

(b)

τ, min

(3) τ2

102 (3)

τ1 , τ1 101

0.06

0.08 Rcruc, m

0.10

(2)

τ2

103

(c)

(3)

τ2

102 (2)

6. RELAXATION TIME AS A FUNCTION OF THE GROWTH CONDITIONS

−6

(2) τ2

(2)

(3)

Crucible radius Rcruc, stationary solidification interface height H with respect to the crucible bottom, and average melt temperature gradient GL at the solid ification interface are important parameters for the stability of the crystal–melt system during the growth of a crystal with a given radius. A stationary value of the solidification rate is determined by the charge quality and cannot be chosen from stability consider ations. The relaxation times are the roots of characteristic polynomials for the set of equations (5) for model {R, h, H} and set (6) for model {R, h}. For the calculations, we used the physical properties of a lithium niobate crystal and its melt from Table 1 and the experimental conditions from Table 2. The numerical values of the matrix coefficients and other calculated quantities are given in Table 3. Figure 3 shows the calculated dependences of the relaxation time on (a) the axial component of melt temperature gradient GL, (b) crucible radius Rcruc, and (c) stationary solidification interface height H0 with respect to the crucible bottom. We found two roots with different real parts for both approximations ({R, h, H}, {R, h}). The longer relaxation time is desig (2) nated as τ 2 for twoparameter model {R, h} and as (3)

τ1 , τ1

τ 2 for threeparameter model {R, h, H}. The shorter (2)

101 0.004

(6)

0.008

0.012 H0, m (3)

(2)

(3)

Fig. 3. Dependences of relaxation times τ 1 , τ 1 , τ 2 , (2) and τ 2 on (a) temperature gradient GL, (b) radius Rcruc, and (c) stationary front height H0.

(3)

relaxation times are τ 1 and τ 1 , respectively. Under conditions close to those given in Table 2, we have (2) (3) τ1 ≈ τ1 .

crucible

Hereafter, model (5) is designated as {R, h, H}. In the case where Rcruc  R, the linearized dynamic sys tem can be approximately written as

We now consider the relaxation time as a function of average melt temperature gradient GL at the solidifi cation interface (Fig. 3a). At a small temperature gra dient (about –300 K/m) in model {R, h}, the two roots degenerate into one; as the temperature gradient increases, they split into two roots. The shorter relax ation time decreases as temperature gradient GL increases in both models, and the longer relaxation time increases with temperature gradient GL. TECHNICAL PHYSICS

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(3)

The values of τ 1 and τ 1 are independent of the crucible radius in the radius range under study (Rcruc ∈ (3)

[R + 2a, 2R], Fig. 3b). τ 2 decreases when the cruci ble radius approaches the crystal radius. A stability analysis at a crucible radius smaller than R + 2a requires the capillary effect of the crucible walls to be taken into account, which is beyond the scope of this (2) work. τ 2 is independent of the crucible radius, since model {R, h} does not take into account finite crucible dimensions. Figure 3c shows the dependence of the relaxation time on solidification interface height H with respect to the crucible bottom in the range from the capillary constant of the melt to 15 mm. If height H is smaller than the capillary constant, the entire melt concen trates into a meniscus and models (5) and (6) are inap plicable. When stationary solidification interface height H0 with respect to the crucible bottom (2)

(3)

(2)

(3)

decreases, τ 1 and τ 1 decrease and τ 2 and τ 2 increase (Fig. 3c). In [4, 5], the degree of stability was specified by the longer relaxation time. According to this criterion, the stability of the crystal–melt system is maximal at low temperature gradient GL (at the point of bifurcation of the roots, GL is about –300 K/m), a high melt level (several tens of millimeters), and a crucible radius approaching the crystal radius (Fig. 3). The first con dition is in obvious conflict with the conclusions for the Czochralski technique in [4, 7], where the stability of the system was found to increase with temperature gradient GL. 7. TRANSIENT PROCESSES We will consider the transient processes that occur at various initial parameter deviations to understand (3) (3) the role of relaxation times τ 1 and τ 2 and to find optimum growth conditions from the standpoint of a stable solidification rate. The transient process curves shown in Fig. 4 for the variables of the crystal–melt system (a–c) and the solidification rate (d) follow from the solutions to sets of equations (5) and (6) with the following initial con ditions: (a) an initial negative meniscus height perturbation δR = 0, δh = –0.2h0, and δH = 0 for model {R, h, H} (for brevity, –h(3)) and δR = 0, δh = –0.2h0 for model {R, h} (for brevity, –h(2)); (b) an initial front height perturbation δR = 0, δh = 0, and δH = 0.2h0 for model {R, h, H} (for brevity, H) and meniscus height perturbation δR = 0, δh = 0.2h0 for model {R, h} (for brevity, h(2)); (c) an initial crystal radius perturbation δR = 0.2h0, δh = 0, and δH = 0 for model {R, h, H} (for brevity, R(3)) and δR = 0.2h0, δh = 0 for model {R, h} (for brev ity, R(2)); and TECHNICAL PHYSICS

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(d) initial perturbations of meniscus height h(2) and h , crystal radius R(2) and R(3), and front height H. (3)

The curves in Fig. 4 are given on a logarithmic time scale. The variables of model {R, h} are designated as superscript (2) and those of model {R, h, H} are desig nated as superscript (3). The solidification rate decay time under conditions close to those given in Table 2 is the same for all tran (3)

(2)

–1

sient processes and is approximately τ 1 ≈ τ 1 ≈ A HH (see Fig. 4d), which corresponds to the meniscus height h relaxation time for model {R, h} and the solid ification interface H relaxation time for model {R, h, H}. The transient processes for model {R, h, H} have an oscillation character with exponential decay. The tran sient processes for model {R, h} only have decay expo nents. The longer time of the two values of relaxation time (3) τ2

specifies the decay of the deviations of the crystal radius and the melt meniscus height. When only the crystal radius is initially perturbed (Fig. 4c), the tran sient processes of models {R, h, H} and {R, h} are sim ilar in the deviation amplitude and the deviation decay time. The solidification rate deviation is negligibly small: it is no more than 6% of the initial value. At an initial negative meniscus height deviation (Fig. 4a), the crystal radius deviation amplitude is about 2 mm for model {R, h, H} and about 0.2 mm for model {R, h}. Thus, the δR(3)(t) amplitude is an order of magnitude higher than the δR(2)(t) amplitude. Under these condi tions, the solidification rate deviation is about 1% for model {R, h, H} and about 80% for model {R, h}; that is, the δVc(t)/Vc amplitude in model {R, h, H} is almost two orders of magnitude lower than the δVc(t)/Vc amplitude in model {R, h}. At the initial solidification interface height H per turbation (Fig. 4b), the relative solidification rate deviation δVc(t)/Vc in model {R, h, H} is maximal (80%). A positive front height H deviation transforms into a negative melt meniscus height deviation in time (3)

τ 1 . Then, the transient processes of the crystal radius (δR(3)(t)) and the meniscus height (δh(3)(t)) develop identically to those for a negative meniscus height per turbation (curves δR(3)(t) and δh(3)(t)) with almost the same shape and amplitude (Fig. 4a). The curve of the decay of the front height deviation δH(t) in model {R, h, H} resembles the decay shape of the meniscus height δh(2)(t) in model {R, h}. Front height deviation δH specifies the solidification rate deviation, and the (3)

decay times of their deviations coincide and are τ 1 .

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(3)

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2

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0

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−1 0 10

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0 h(2)(t)

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0 102 t, min

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100

102 t, min

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Fig. 4. Transient processes of (a–c) the deviations of process variables δR(2)(t), δh(2)(t), δR(3)(t), δh(3)(t), and δH(t) and (d) the relative deviation of solidification rate δVc(t)/Vc. The initial conditions are as follows: (a) negative meniscus height perturbation –h(3) and –h(2); (b) deviations of meniscus height h(2) and front height H; (c) deviation of crystal radius R(2) and R(3); and (d) deviations of crystal radius R(2) and R(3), meniscus height h(3) and h(2), and front height H.

8. ON THE CONTROL OF THE GROWTH OF OPTICALGRADE LITHIUM NIOBATE CRYSTALS IN THE LOW MELT LEVEL CZOCHRALSKI TECHNIQUE In the crystal radius control of crystal growth, a deviation of the crystal radius from a given value ini tiates an automatic control system. However, a crystal radius deviation from a given value weakly affects the deviation of the solidification rate from a stationary value (Figs. 4c, 4d). Heat channel control is per formed through solidification interface height H, whose change represents a solidification rate devia tion. Thus, the crystal radius falls in a given range due to a solidification rate deviation. For the production of opticalgrade crystals, we think that a constant solidi

fication rate is more important than the fact that the crystal radius should fall in a given range. Therefore, the rate of change of height H during control should be lower than the level of H fluctuations for crystal radius and meniscus height deviations; then, the solidifica tion rate deviations are also at the level of fluctuations. As follows from the curves in Fig. 4b, this addition does not exceed 10% of the solidification rate. To maintain a constant crystal radius during the growth of lithium niobate crystals, it is most often required to monotonically decrease the temperature (heating power). However, a usual PID crystal radius controller does not change the temperature (heating power) if the crystal radius corresponds to a stationary value and does not change. It becomes operative only TECHNICAL PHYSICS

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when a crystal is constricted. If the heating power is decreased against the background of the operative controller according to data on the earlier processes, the action of the PID controller in an analogous situ ation can be significantly weaker. In other words, it can have lower coefficients and a lower maximum allow able value for the power. The last circumstance decreases the controller action amplitude that is harmful to the crystal homogeneity. In the LMLCz technique, a melt level is constant and growth conditions are maintained at a constant level more strictly than in the Czochralski technique. To grow crystals as homogeneous as possible, we pro pose to refuse crystal diameter control in the cylindri cal part of the crystal at a control frequency compara (3) ble with 1/ τ 1 . For the current crystal diameter to correspond to a given diameter, we should smoothly and intentionally change the temperature, with the time interval between specifying the rate of change of (3) the temperature being much longer than τ 1 . 9. DISCUSSION OF RESULTS 9.1. Comparison of the Relaxation Times in the LMLCz and Czochralski Techniques When stability was analyzed for the Czochralski technique, the melt temperature after the initial time was assumed to be fixed at the level of zero pressure [4, 5]. This assumption is based on the fact that the thermal inertia of the major portion of the melt is greater than that of the meniscus. However, when large lithium niobate crystals are grown, free melt convection is pre dominant and it is difficult to find the site of tempera ture fixation and to estimate the temperature because of the strong nonlinearity of the Navier–Stokes equa tions that describe the melt temperature distribution. The relaxation times calculated from the crystal– melt system model depend substantially on the site of temperature fixation, which results in a low (up to an order of magnitude) accuracy of an analytical estima tion of the relaxation times for the Czochralski tech nique for the case of growing large lithium niobate crystals. For the LMLCz technique, the site of tem perature fixation is the crucible bottom, the convection is much simpler, and the estimates are much more accurate. The low accuracy of the analytical estimation of the relaxation times in the Czochralski technique models does not allow us to compare the relaxation times for the Czochralski and LMLCz techniques. 9.2. Closeness of the LMLCz Technique to the Techniques of Axial Heat Flux Close to the Phase Interface When lithium niobate crystals are grown by the LMLCz technique, the calculation conditions for the linearity of the thermal field of the melt in the gap TECHNICAL PHYSICS

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between the crucible bottom and the solidification interface are met. The linearity of the heat front near the solidification interface is the main property of the te chniques of axial heat flux close to phase interface (AHP techniques) [12]. In these techniques, a crystal is placed below the melt. A heater is located before the solidification interface to decrease the radial tempera ture gradient in the melt to a level at which free melt convection is absent. The thermal field in the melt is homogeneous in the vertical direction. At present, the LMLCz technique is the only AHP version where a crystal is situated above the melt and forced melt con vection takes place. 9.3. Inequality of the Parameters The solidification rate deviation in LMLCz tech nique is equal to the rate of change of solidification interface height H with respect to the crucible bottom. Perturbation of H directly affects the solidification rate (Fig. 4), and perturbations of the crystal radius and the meniscus height affect it indirectly, via their effect on an H deviation. The effects of the crystal radius and meniscus height deviations on the solidification rate are much weaker than the effect of the front height H devia tion having the same magnitude. The decay of the solid ification rate deviation in time is controlled by the decay of the solidification interface height H deviation. 9.4. Simple Estimation of the Stability of the Solidification Rate for the LMLCz Technique Under conditions that are close to those in Table 2, the relaxation time of the solidification rate is approx imately –1 L  H , A HH ≈  λL GL where H is the solidification interface height with respect to the crucible bottom and GL is the average melt temperature gradient at the solidification interface. 9.5. Sensitivity of the Solidification Rate to Feeding Nonuniformity The sensitivity of the solidification rate to a deviation in the melt meniscus height in the LMLCz technique is several times lower (Fig. 4d, curve h(3)) than that of the Czochralski technique (Fig. 4d, curve h(2)), which means that the solidification rate has a lower sensitivity to feeding nonuniformity. This also leads to lower selfstabilization of the crystal radius in the LMLCz technique compared to the Czochralski technique. However, this effect plays an insignificant role due to its long duration. 9.6. Stability Criteria According to [4, 5], the stability of the crystal–melt system is determined by the maximum relaxation

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time. According to this criterion, the stability is maxi (3) (3) mal at the minimum value of time t 2 or t 1 depend ing on the chosen model. It follows from Fig. 3 that, irrespective of the model, the longer relaxation time is minimal at (1) an absolute temperature gradient GL of about 300 K/m, (2) the maximum melt level, and (3) the minimum crucible radius for model {R, h, H}. As is seen from the consideration of the transient processes of the solidification rate (Fig. 4d), the stabi lization of the solidification rate in both models is con trolled by the minimum time between the two relax ation times. The shorter relaxation time is minimal at the maximum value of temperature gradient GL and the minimum melt level. The crucible radius is insignificant for the stability of the solidification rate (Fig. 3b). Thus, the traditional estimation of the stability of the crys tal–melt system as the maximum relaxation time [4, 5] did not allow us to choose growth conditions for the best stabilization of the solidification rate for both the LMLCz and Czochralski techniques. A correct criterion for this purpose is the minimization of the minimum time between the two relaxation times. 9.7. Term Diameter Stabilization It is seen from the curves of transient processes in (3) Fig. 4 that shorter relaxation time τ 1 specifies the decay time of the deviation of solidification interface height H with respect to the crucible bottom and the decay time of the solidification rate deviation. Time (3) τ 2 specifies the decay times of the deviations of the crystal radius and the meniscus height. As temperature gradient GL increases or stationary solidification inter (3)

face height H0 decreases, time τ 1 decreases whereas (3)

time τ 2 increases. In other words, the growth condi tions most favorable from the standpoint of a stable crystal radius are far from optimal conditions for the stabilization of the solidification rate. This circum stance comes into conflict with the generally accepted terminology “diameter selfstabilization,” “shape sta bility,” “diameter stability,” etc., since the conditions that promote a larger amplitude of crystal radius oscil lations in our case favor a constant solidification rate. 9.8. Relation between the Shape and Homogeneity of a Crystal The transient processes in the crystal radius δR(3)(t) and the meniscus height δh(3)(t) at an initial meniscus height perturbation (Fig. 4a) almost coincide with δR(3)(t) and δh(3)(t) at an initial front height perturba tion (Fig. 4b). However, the corresponding solidifica tion rate deviation amplitudes differ by more than an

order of magnitude (Fig. 4d; curves h(3), H). This means that the presence of folds on the crystal surface in the LMLCz technique (model {R, h, H}) is not always an indication of its inhomogeneity, since they can be caused by a deviation of the melt level because of nonuniform feeding. A smooth crystal surface even at a smoothly changed radius indicates a homogeneous crystal, since a solidification rate perturbation is accompanied by a change, though small, in the crystal radius. In con trast, for the Czochralski technique (model {R, h}), it is difficult to imagine a crystal radius perturbation without a preceding meniscus height perturbation. Therefore, folds are most likely to reflect a nonuni form solidification rate. For both methods, a constant crystal diameter (crystal diameter falling in a given range) does not guarantee an optical quality of the crystal and the absence of growth striations related to solidification rate oscillations. 10. CONCLUSIONS We constructed a threeparameter model for the crystal–melt system to grow large lithium niobate crystals by the low melt level Czochralski technique, calculated the relaxation time as a function of growth conditions, and plotted the curves of transient pro cesses at various variable deviations. The decay times of crystal radius and melt meniscus height deviations were shown to be significantly higher than the decay time of a solidification rate deviation, which is only related to the decay of the deviation of the solidifica tion interface height with respect to the crucible bot tom for the LMLCz technique. It was shown that a crystal radius deviation is weakly related to a solidifi cation rate deviation and that the growth conditions favorable for a constant crystal diameter do not pro vide a stable solidification rate and, hence, the forma tion of optically homogeneous crystals. To produce optically homogeneous crystals, the maximum stability of the solidification rate is more important than optimum control of the crystal diame ter. This stability is reached under the following growth conditions: (a) the minimum stationary melt level, (b) the maximum average melt temperature gradi ent at the solidification interface, and (c) a limited rate and amplitude of a temperature action in controlling the crystal diameter. A stability analysis also demonstrates that the sen sitivity of the solidification rate to melt level oscilla tions in the LMLCz technique is lower than in the Czochralski technique with a double crucible. Thus, in this sign, the LMLCz technique is better for pro ducing opticalgrade crystals than the Czochralski technique with a double crucible. TECHNICAL PHYSICS

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ACKNOWLEDGMENTS This work was supported by the Science and Edu cation Committee at the St. Petersburg government. REFERENCES 1. P. V. Smirnov, RF Patent No. 2320791 (8 June2010). 2. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon, New York, 1987). 3. P. V. Smirnov and P. I. Antonov, in Proceedings of the National Conference on Crystal Growth, Moscow, 2006, p. 482. 4. V. A. Tatarchenko, Steady Growth of Crystal (Nauka, Moscow, 1988) [in Russian]. 5. V. S. Leibovich, in Growth of Crystals, Ed. by E. I. Givargizov and S. A. Grinberg (Nauka, Moscow, 1986; Consultants Bureau, New York, 1988), Vol. 15, p. 143. 6. V. T. Gabrieyan, V. B. Smirnov and P. V. Smirnov, in Proceedings of the 4th International Conference on Single Crystal Growth and Heat and Mass Transfer, Obninsk, 2001, pp. 258267.

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7. V. T. Gabrieyan, V. B. Smirnov, P. V. Smirnov, and Kov alevskii, Laser Investigations in St. Peterburg State Uni versity (SpbGU, St. Peterburg, 2004), No. 3, p. 255. 8. D. V. Kovalevskii and P. V. Smirnov, Laser Investigations in St. Peterburg State University (SpbGU, St. Peterburg, 2007), No. 5, p. 68. 9. G. A. Satunkin, J. Cryst. Growth 154, 172 (1995). 10. C. Marin, A. G. Ostrogorsky, D. Foulon, D. Jundt, and S. Motakef, Appl. Phys. Lett. 78, 1379 (2001). 11. H. Schlichting, GrenzschichtTheorie (Verlag G. Braun, Karlsruhe, 1970; Nauka, Moscow, 1974). 12. V. D. Golyshev and M. A. Gonik, Cryst. Prop. Prep. 36–38, 623 (1991). 13. B. S. Red’kin, G. A. Satunkin, V. N. Kurlov, V. A. Tatarchenko, in Crystal Growth (Nauka, Moscow, 1986), Vol. 15, pp. 210216 [in Russian]. 14. N. J. Imaishi, J. Chem. Eng. Jpn. 27/1, 25 (1994). 15. Y. Anzai, S. Kimura, T. Sawada, T. Rudolph, and K. Shigematsu, J. Cryst. Growth 134, 227 (1993). 16. Z. Galazka, J. Cryst. Growth 178, 345 (1997).

Translated by K. Shakhlevich