Experiment and simulation on interface shapes of an yttrium ...

research papers Journal of

Applied Crystallography ISSN 0021-8898

Experiment and simulation on interface shapes of an yttrium aluminium garnet miniature molten zone formed using the laser-heated pedestal growth method for single-crystal fibers

Received 6 April 2009 Accepted 8 May 2009

P. Y. Chen,a C. L. Chang,b K. Y. Huang,b C. W. Lan,c W. H. Chenga and S. L. Huangb,d* a Department of Photonics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan, bInstitute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan, cDepartment of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, and dDepartment of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan. Correspondence e-mail: [email protected]

# 2009 International Union of Crystallography Printed in Singapore – all rights reserved

A two-dimensional simulation was employed to study the melt/air and melt/solid interface shapes of the miniature molten zone formed in a laser-heated pedestal growth (LHPG) system. Using a non-orthogonal body-fitting grid system with the control-volume finite-difference method, the interface shape can be determined both efficiently and accurately. During stable growth, the dependence of the molten-zone length and shape on the heating CO2 laser is examined in detail under both the maximum and the minimum allowed powers with various growth speeds. The effect of gravity on the miniature molten zone is also simulated and the possibility of horizontally oriented LHPG is revealed. Such a horizontal system is good for the growth of long crystal fibers.

1. Introduction Crystal growth technology for solidifying bulk crystals from a melt has been developed over the course of the past century (Scheel, 2000). In contrast, single-crystal fibers have become the subject of intense study only recently (Feigelson, 1988). They have been recognized to possess remarkable characteristics, such as a low defect density leading to a high optical damage threshold, longer mode confinement leading to high nonlinear wavelength conversion efficiency and better laser efficiency for transition-metal-doped fibers. These advantages make possible applications as light guides (Love et al., 1991; Tsai et al., 2008; Lai et al., 2008) in the field of passive devices; as fiber lasers (Digonnet et al., 1986; Lo et al., 2002), amplifiers (Davis et al., 1991; Lo et al., 2005) and amplified spontaneous emission light sources (Lo et al., 2004; Chen et al., 2007; Huang, Hsu & Huang, 2008) in the field of active devices; and as harmonic generators (Lee et al., 2007) in the field of nonlinear fiber devices (Yoon, 2004). Among the floating-zone methods (Rudolph & Fukuda, 1999), pedestal growth (Poplawsky, 1961) has been proved to be the most versatile method so far, in particular when coupled with laser heating (Gasson & Cockayne, 1970) in a process that is also known as the laser-heated pedestal growth (LHPG) technique (Fejer et al., 1984; Feigelson et al., 1985; Feigelson, 1986). The main advantages of this crucible-free technique include high pulling rates, low production cost, and the feasibility of growing materials with very high melting J. Appl. Cryst. (2009). 42, 553–563

points, high purity and low stress. The syntheses of a range of materials and properties of crystal growth have been explored using the LHPG method (Yen, 1999). For the purpose of growing high-quality crystal fibers, the stability of the molten zone (Coriell et al., 1977; Coriell & Cordes, 1977) can be revealed by its specific interface shape during the growth process. In static conditions (Li et al., 1993), the molten zone is only affected by gravity and surface tension. However, the stability of the molten-zone shape is the result of the dynamic equilibrium between hydrostatic pressure, surface tension and gravity. The interface shape depends strongly on the physical and chemical properties of the liquid phase and may be a control parameter to prevent problems associated with composition, stress and the smoothness of the perimeter of grown crystal fibers. How to describe it accurately by theoretical estimation (Green, 1964) or by experimental imaging processes (Kim et al., 1979; Ardila et al., 2004) has been the subject of studies for many years. For the miniature molten zone fabricated using the LHPG method, convections can be induced by the buoyancy force as natural convection, by mass transfer as mass-transfer convection and by surface-tension gradients on the melt/air interface as thermocapillary convection. The natural convection is determined by the density distribution. The masstransfer convection is a function of the reduction ratio according to the continuity equation. The thermocapillary convection is significantly and locally enhanced by high power input optically at the melt/air interface. There is high doi:10.1107/S0021889809017361

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research papers convection speed inside the molten zone as a result of mass and thermocapillary convections. The variation in pressure induced by the stable convections can be expressed by the shape, volume and stability of the melt at various laser powers. Therefore, a range of laser powers needs to be specified to ensure the stability and symmetry of the miniature molten zone. Bulk crystal growth using the floating-zone method has been modeled in a rod (Duranceau & Brown, 1986) and a thin sheet (Young & Chait, 1989) to determine the interface shapes. Heat transfer in the melt was assumed to be a result of conduction and radiation. The melt/air interface was calculated with the help of the Young–Laplace equation for a static fluid (Kozhoukharova & Slavchev, 1986; Lie et al., 1988; Lie & Walker, 1989; Riahi & Walker, 1989). A non-orthogonal bodyfitting grid system (Lan & Kou, 1990, 1991a) has been successfully employed to predict more accurately the interface shapes via a two-dimensional model describing heat transfer and fluid flow in the steady state. In order to describe the growth of a single-crystal fiber using the LHPG method, several numerical models have been used (Fejer, 1986; Feigelson, 1986). The thermal profiles, the mean positions of the flat melt/solid interface, the meniscoid melt/air interface, the melt volume and the fiber diameter have been predicted by time-dependent one-dimensional equations of motion and heat transfer (Young & Heminger, 1997). In addition, the melt/air interface has been analyzed using the Young–Laplace equation for a static fluid (Gu et al., 2001), empirical relations (Li & Liu, 1995) and molecular dynamics simulations (Nijmeijer & Landau, 1997). However, there is still no accurate simulation model for describing these deformed interface shapes of the miniature molten zone using the LHPG method. In this paper, we report two-dimensional simulation results, which are compared with the miniature molten zone formed using the LHPG method. The numerical model is modified from that for growing bulk crystals using the floating-zone method (Lan & Kou, 1990, 1991a, 1993; Lan, 1994, 1996; Lan & Liang, 1997). Using a non-orthogonal body-fitting grid system with the control-volume finite-difference method, it is possible to obtain more accurate estimations near the interface in order to reduce the computation time required to make

the comparison. Moreover, the laser intensity profile and boundary conditions are revised because the heat source is replaced by a carbon dioxide (CO2) laser. The parameters needed to produce a useful crystal fiber are determined. In addition, the variations of the molten-zone length as a function of laser power are described at various reduction ratios for different feed speeds. The influences on the melt/air interface shape are characterized using the curvature radius and the inflection point. Finally, the effect caused by gravity along the growth direction is examined.

2. Experimental approach Fig. 1 illustrates the experimental layout of the growth system (Huang, Hsu, Jheng et al., 2008) using the LHPG method. The source rod was cut from an yttrium aluminium garnet (Y3Al5O12, YAG) crystal and a seed rod in the h111i direction was utilized to determine the crystallographic orientation. The YAG single crystal used here contained a mixture of 0.25 wt% chromium(III) oxide (Cr2O3) and less than 0.1 wt% calcium(II) oxide (CaO) (Ishibashi et al., 1998). With such a low doping level, the impact of the dopant on the interface shapes is neglected. A tightly focused CO2 laser was the heat source employed to melt the source rod as the pedestal and the seed rod was then dipped into it to form a miniature molten zone. In the growth process, a crystal fiber can be grown by pulling the seed rod and feeding the source rod upward simultaneously with the melt bridge located between them. The laser focal spot, and consequently the miniature molten zone, remain stationary during the growth. The reduction ratio is defined as the diameter of the crystal fiber divided by that of the source rod. Furthermore, the speed ratio is defined as the growth speed divided by the feed speed. According to mass conservation, the reduction ratio can be determined as the inverse square root of the speed ratio. The ground source rods were about 400  400 m in square cross section. A new source rod of smaller diameter can be obtained by a pre-growth process. Inside the growth chamber, the optics were designed to focus the laser beam onto the miniature molten zone with

Figure 1 The experimental layout of the LHPG system for growing single-crystal fibers. Mh and Mc are the mirrors for the He–Ne laser and CO2 laser, respectively, BC is the dichroic mirror, BS is the 90/10 beam splitter, PAZ is the polarizer–analyzer–attenuator, Lz1 and Lz2 are ZnSe lenses with focal lengths of 10 and 60 cm in the 6 beam expander, OB is the zoom stereo microscope system, NDF is the neutral density filter, CCD is the charge couple device, RI is the inner-cone reflaxicon, RO is the coaxial outer-cone reflaxicon, W is the ZnSe window, and PM is a parabolic mirror.

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J. Appl. Cryst. (2009). 42, 553–563

research papers Table 1 The physical properties of YAG material and other related input parameters. Symbols Values

Units

Descriptions

Yttrium aluminium garnet; YAG s 3.685 g cm3 m 4.3 g cm3 2243.2 K Tm
H 455.5 J g1 ks 0.1 W cm1 K1 km 0.1 W cm1 K1 Cps 1.0 J g1 K1 Cpm 0.39 J g1 K1  780 dyn cm1 @/@T 0.035 dyn cm1 K1 m 0.4 g cm1 s1 m 6.5  105 K1 "s 0.7 – "m 0.5 –

Density of solid Density of melt Melting point Melt/solid latent heat Thermal conductivity of solid† Thermal conductivity of melt† Specific heat of solid† Specific heat of melt† Surface tension† Surface-tension-temperature coefficient† Viscosity of melt Thermal expansion coefficient of melt Radiation emissivity of solid Radiation emissivity of melt

Other input parameters See Table 4 mm Ds,c Us,c See Table 4 mm s1 L 5 cm h 1.1  103 W cm2 K1 fm 1.00–1.88 – am 0.5 –

Diameter of rod and fiber Speeds of feed and growth Rod length Heat transfer coefficient Factor of equivalent melt absorption Graybody factor

† At melting point.

uniform distribution azimuthally. The incident laser beam in Gaussian profile is transformed into a ring-shaped beam in near semi-Gaussian profile by the reflaxicon, which consists of an inner cone surrounded by a coaxial outer cone. The ringshaped beam is reflected by a planar mirror at an incident angle of 45 to a parabolic mirror at normal incidence. A donut-shaped focal spot of diameter 25 mm FWHM with neardiffraction-limit quality on the melt/air interface can be obtained by means of the parabolic mirror. These three diamond-turned optical components are made from oxygenfree copper. A gold coating on the copper surface enhances the reflectance and protects the copper substrate. Holes of diameter 25.4 mm in the centers of the parabolic and planar mirrors allow both the crystal fiber and the source rod to pass through. The inner cone of the reflaxicon, of diameter 31.75 mm, is mounted on the center of a zinc selenide (ZnSe) window of diameter 101.6 mm. The parabolic mirror, of diameter 76.2 mm, has a focus length of 25 mm and an acceptance angle of 39 . For the entire growth system, a 100 W, 10.6 mm, linearly polarized, water-cooled CO2 laser system (Spectral Laser, Series 100) driven by high direct-current voltage in continuous-wave operation served as the heat source. A built-in visible helium–neon (He–Ne) laser beam was combined with the CO2 laser beam using a dichroic mirror to assist beam alignment. The beam quality was M2 ’ 1.1. A polarizer– analyzer–attenuator (II-VI Inc., Model PAZ Plates 6) was utilized to adjust the laser power. This device consists of a stacked series of ZnSe plates placed at Brewster’s angle to the incoming beam. At each plate, all the p-polarized components are transmitted virtually, while most of the s-polarized components are reflected. After the beam traverses the six J. Appl. Cryst. (2009). 42, 553–563

plates, an extinction ratio of 500:1 and a transmission of 98% is obtained. Therefore, the laser power can be adjusted without power fluctuation by rotating the motor-driven attenuator at the fixed system output. After passing through a ZnSe telescope functioning as a 6 beam expander, the laser beam with a diameter of 27 mm enters the chamber with a beam divergence of 2.5 mrad. Outside the chamber, the imaging system consists of an 8 bit digital camera alongside a zoom stereo microscope system, which comprises a 1 objective lens with a working distance of 90 mm and a 20 eyepiece (Olympus SZX7, AXH1X and WHSZ20X), affording a total magnification of 16–112 with a field-of-view of 2.9–20 mm. The imaging system to monitor the crystal growth is located at the same level as the molten zone. A neutral density filter is used in front of the imaging system to attenuate the strong thermal emission and so avoid detector saturation. In order to acquire a uniform crystal fiber of smaller diameter or longer length, stable growth conditions, such as smoother movement, better collimation along the growth direction, higher power stability and lower beam-pointing fluctuation, must be achieved. The crystal fiber and the source rod fastened by a copper adapter were loaded on a programmable linear stage driven by a stepping motor (Autolab Inc.) with a pulse-type motion control card (Advantech PCI-1240) to allow the system to be moved up and down smoothly. The whole translation stage loaded on an aluminium pillar was locked onto an optical table with the aim of damping vibration. The maximum length of the grown crystal fiber was 50 cm, which is limited by the translation distance. The allowed feed speeds range from 0.375 to 37.5 mm min1, which results in various reduction ratios from 10 to 100%. Automation of the entire system involves using the feedback control of the laser power for power stability and geometrical uniformity, using the remote control of the translation stage for smooth pedestal movement, and using the imaging of the molten zone and the crystal-fiber diameter for in-situ diagnostics. Consequently, the power fluctuation can be optimized from 5% to less than 1%, and the core diameter of the crystal fiber has been demonstrated to be 10 mm (Huang, Hsu, Jheng et al., 2008) with a double-cladding structure (Lo et al., 2005).

3. Mathematical formulation The major differences compared with previous floating-zone simulations (Lan & Kou, 1990, 1991a; Lan, 1996) are the heat source and the thermal boundary conditions at the melt/air interface. The thermal boundary conditions on the source rod, melt and crystal fiber should be considered at the same time, when the ring-shaped laser beam incident on the melt/air interface is absorbed and then dissipated. The dissimilarity of the streamline is enhanced by the thermocapillary effect near the melt/air interface and mass-transfer convection at a specific reduction ratio. The physical properties of the singlecrystal YAG and other related input parameters are shown in Table 1 (Brandon & Derby, 1992; Fratello & Brandle, 1993; Lan & Tu, 2001). P. Y. Chen et al.



Interface shapes of miniature molten zones

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research papers Fig. 2 is a schematic sketch of the miniature molten zone formed using the LHPG method. The laser power is absorbed on the constant azimuthal area of the melt/air interface at normal incidence without fluctuations in power and beam pointing. The azimuthal area, determined by the source-rod diameter and the reduction ratio, is fixed to be 2RLi sec L, where Li is the axial length of the azimuthal area whose value is 15–30 mm and R is the radius of the molten zone. Owing to the small azimuthal area, the heat re-absorption of the multiple-reflection lights inside the chamber can be neglected after the laser beam penetrates the melt. Therefore, the ambient temperature can be considered constant without a Gaussian distribution (Duranceau & Brown, 1986). As in the floating-zone method for growing bulk crystals (Lan & Kou, 1990, 1991a; Lan, 1996), thermal convection is assumed to be symmetric axially, laminar and in a pseudo-steady state. The oscillatory thermocapillary convection is not considered (Lan & Kou, 1990, 1991a). Gravity is considered, and cylindrical coordinates are used in the two-dimensional simulation. The three governing equations in the two-dimensional model are listed below: Equation of motion:       @ !@ @ !@ @ 1@ ðm r!Þ  þ @r r @z @z r @r @r r @r   @ 1@ @T ¼ 0: ð1Þ þ ðm r!Þ  m m g @z r @r @r Stream equation:     @ 1 @ @ 1 @ þ þ ! ¼ 0: @z m r @z @r m r @r Energy equation:     @ @ @ @ C T Cpm T  @r pm @z @z @r     @ @T @ @T rks;m rks;m þ ¼ 0: þ @z @z @r @r

1 @ ; m r @z

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(2) At the melt/solid interface near the source rod (denoted the feed front below) or crystal fiber (denoted the growth front below): T ¼ Tm ;

ð8Þ

    ks nf;g  rT jS  km nf;g  rT jL   þ s
HUs;c nf;g  ez ¼ 0;

ð9Þ

where Tm is the melting temperature of the YAG, nf;g is the unit normal vector at the feed or growth front toward the melt, respectively, ez is the unit vector along the z axis, s is the density of the solid,
H is the latent heat, whose values are positive at the feed front and negative at the growth fronts, Us,c is the feed speed or growth speed at the melt/solid interfaces, respectively, and S and L represent the solid and liquid phase, respectively. (3) On the surfaces of the source rod, the melt and the crystal fiber (Lan, 1996; Rivas et al., 1992):

ð3Þ

ð4Þ

¼

1 @ ; m r @r

ð5Þ

!¼

@u @  : @z @r

ð6Þ

Four thermal boundary conditions are involved as follows: (1) Along the z axis:

556

ð7Þ

ð2Þ

is the stream function, ! is the vorticity, m is the viscosity of the melt, m is the density of the melt, m is the thermal expansion coefficient of the melt, Cpm is the specific heat of the melt, T is the temperature and ks,m is the thermal conductivity of the solid or melt. The radial velocity u, the axial velocity  and the vorticity ! are defined as follows: u¼

@T ¼ 0: @r

Interface shapes of miniature molten zones

Figure 2 A schematic sketch of the miniature molten zone formed using the LHPG method. z is the axial coordinate along the growth direction, r is the radial coordinate, z0 and z3 refer to the axial positions at infinite distance, z1 and z2 are lower and upper boundaries of the simulation region far away from the melt, z = 0 is the axial position onto which the laser beam projects, 0 is the contact angle at the air/melt/solid tri-junction, L is the incident angle of the laser beam related to the radial axis, g is the gravitational acceleration, Us and Uc are the feed speed and growth speed at the melt/ solid interfaces, respectively, Rs and Rc are the radii of the source rod and the crystal fiber, respectively, nf;g is the unit normal vector at either the feed or the growth front toward the melt, nm is the unit normal vector described by a 1  2 matrix at the melt/air interface, sm is the unit tangential vector described by a 1  2 matrix at the melt/air interface, and b is the unit vector along the laser propagation direction. J. Appl. Cryst. (2009). 42, 553–563

research papers   ks;m ns;m  rT ¼ hðT  Ta Þ þ "s;m ðT 4  Ta4 Þ  Ia ðnm  bÞfm am ;  "  2 #1=2 dr dr nm ¼ er  ez ; 1þ dz dz b ¼ er cos L  ez sin L ;

Table 2

ð10Þ

The coefficients a, b, c and d in equation (18). ’

a

b

c

d

0

1/(m r)

1

!

!

1/r

1/r

m r

  m gm @z @T @z @T   @ @ @ @ J

T

Cpm

km r

1

0

ð11Þ

ð12Þ

where ns;m is the unit normal vector at the solid/air or melt/air interface, respectively, for the deformed shapes, fm is the factor of equivalent melt absorption, am is the graybody factor, Ia is the laser intensity profile on the melt/air interface, h is the heat transfer coefficient, Ta is the ambient temperature, "s;m is the radiation emissivity of the solid or melt, respectively, is the Stefan–Boltzmann constant, and er and ez are the unit vectors in the radial and axial directions, respectively. The term Ia ðnm  bÞ means the equivalent laser intensity on the melt/air interface at normal incidence as illustrated in Fig. 3(a). Moreover, the term Ia ðnm  bÞfm am denotes the exact absorption of laser energy in the melt by taking into account the absorption conditions and material properties. For some phenomena, such as refraction, reflection and absorption that occur at the deformed melt/air interface under heating by the laser beam (Fejer et al., 1984; Goodman, 1968), the temperature in the melt is typically higher than that on the melt/air interface as a result of the better absorption near the central annular section (Liu et al., 2002). Furthermore, absorption saturation occurs, caused by the nonlinear effect under high laser intensity (Longtin & Tien, 1997). Therefore, the factor of equivalent melt absorption fm is added to enhance the heat absorption term in particular situations. Its value is set to 1 in case A as defined in x4 but 1.00–1.88 in the other cases. The graybody factor am, whose value is set to 0.5, is employed to reflect laser heating characteristics of translucent objects. It is assumed that heat absorption occurs only through the melt/air interface. (4) Far away from the melt:   T ¼ T1=2 s;c ; ð13Þ

where ðT1=2 Þs;c is the temperature at z = z1 in the source rod or at z = z2 in the crystal fiber. These two points are sufficiently far away from the melt that heat transfer becomes onedimensional beyond them. The boundary condition has been implemented previously (Lan & Kou, 1990). Three boundary conditions of melt flow are considered as follows: (1) Along the z axis: ¼0

and ! ¼ 0:

The stream function is set to zero as a reference, and zero vorticity is obtained since @u=@z ¼ @=@r ¼ 0. (2) At the melt/solid interface:  2 ð15Þ ¼ ð1=2Þs Rs Rs;c Us r2 ; where Rs and Rc are the radius at the feed front and at the growth front, respectively. The above equation for the stream function is obtained by integrating equations (4) and (5) from the z axis. (3) At the melt/air interface: ¼ ð1=2Þs Us R2s ;

A schematic sketch of the boundaries c and s in the curvilinear coordinates at the growth and feed front, respectively, for (a) the physical domain and (b) the computational domain. and are the nonorthogonal axes transformed from z and r, respectively. J. Appl. Cryst. (2009). 42, 553–563

ð16Þ

with the vorticity satisfying nm  ts  sm ¼

@ ðs  rT Þ; @T m

ð17Þ

where ts is the second-order stress tensor represented by a 2  2 matrix and is the surface tension–temperature coefficient. The term nm  ts  sm can then be obtained as a scalar quantity representing the shearing stress (Block, 1962). Equation (17) is the shear-stress balance at the melt/air interface. Fig. 3 shows the two-dimensional curvilinear coordinates applied instead of the cylindrical coordinates used above; the former are introduced in order to allow more precise and efficient calculations, because the interfaces are not flat but curved. The governing equations (1)–(3) can be transformed into the following form (Thompson et al., 1974):       @ @ @ @ b @2 ðc’Þ @2 ðc’Þ a’ a’ þ g11  þ g22 @

@ @ @

J @ 2 @2 þ dPQ þ dnor þ dor ¼ 0;

Figure 3

ð14Þ

ð18Þ

where a, b, c and d are the coefficients given in Table 2 for ’ representing , ! or T. Other coefficients in equation (18) are defined as follows (Lan & Kou, 1991a, 1993): P. Y. Chen et al.



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research papers 

dPQ

dnor

 @ðc’Þ @ðc’Þ þ Qð ; Þ  bJ Pð ; Þ ; @

@

   2bg12 @2 ðc’Þ 1 @b @b @ðc’Þ þ g22  g12  J @

@ @

J @ @   @b @b @ðc’Þ ; þ g11  g12 @ @ @

ð19Þ

represents summation over all grid points. max denotes the maximum of all grid-point values. i, j and t are specific locations and the iteration step. Parameter

ð20Þ

 2  2 @r @z g11  þ ; @

@

ð22Þ

ð23Þ

      @r @r @z @z g12  þ ; @ @ @ @

ð24Þ

      @r @z @z @r  ; J @ @ @ @

ð25Þ

ð26Þ

ð27Þ

ð28Þ

The transformed boundary conditions of fluid flow are also described in the same way, besides the vorticity. The vorticities !1 at the melt/solid interface and !2 at the melt/air interface are as follows: ( "  #) 2 g11 @2 @r @2 r !1 ¼   s Us;c þr 2 ; ð29Þ @ @ m rJ 2 @2 ð30Þ

The kinematic condition nm  ðuer þ ez Þ ¼ 0 applies because of the constant stream function and because @ =@ ¼ 0 along the melt/air interface; this condition is determined by the following equation of normal-stress balance:   1 1 n m  ts  n m ¼  þ ð31Þ  Pa ; R1 R2

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jRtþ1  Rti jmax =jRti j < 107 i

Temperature

jTi;jtþ1  Ti;jt jmax < 104 K P tþ1 jhc  htc j < 105 cm P tþ1 jhf  htf j < 105 cm

Growth-front position

where R1 and R2 are the radii of curvature horizontally and vertically, respectively,  is the surface tension, and Pa is the ambient pressure. The term nm  ts  nm can be obtained as a scalar and represents the outward pressure in the melt. Equation (31) is written after coordinate transformation as follows: "  3 #  "  1 # d2 R @2 z @z dR g22 S @z   T g3=2  ¼ 0; 22 2 2 d @ @ d @ R  ð32Þ

The transformed thermal boundary conditions are expressed in the same way, except for equation (7), which becomes

    1 @ @T 2 @z @u @r @  þ : !2 ¼  @T @ g22 @ @ @ @ m g1=2 22

Molten-zone radius

Feed-front position

 2  2 @r @z þ ; g22  @ @

@T ¼ 0: @

Vorticity

Convergence criterion P tþ1 P j i;j  ti;j j= j ti;j j < 105 P tþ1 P j!i;j  !ti;j j= j!ti;j j < 105

Stream

ð21Þ

     2 @ r @z @2 z @r @z @2 r @r @2 z    2g Q  g22 12 @ 2 @ @ 2 @

@ @ @ @ @ @  2  2 @ r @z @r @ z  þ g11 J 3 : @2 @ @ @2

The parameter convergence criteria. P

dor  Jd;

     2 @ z @r @2 r @z @r @2 z @z @2 r    2g P  g22 12 @ 2 @ @ 2 @ @ @ @ @ @ @  2  2 @ z @r @z @ r  þ g11 J 3 ; @2 @ @ @2

Table 3

Interface shapes of miniature molten zones

where ST is as follows (Lan & Kou, 1991a):    @z @u @r @  ST ¼  ðP  Pa Þ þ 2m g22 @ @ @ @

  @r @ @z @u  þ g12 ðJg22 Þ1 : @ @ @ @

ð33Þ

P is determined by integrating a special form of the equation of motion in terms of the stream function and the vorticity from the air/melt/solid position as tri-point along the melt/air interface. Equation (33) is the total normal stress ST including the relative pressure difference (P  Pa), the static pressure due to gravity and the dynamic pressure due to melt flow. As shown in Fig. 3, c and s are the boundaries of the curvilinear coordinates at the growth and feed fronts, respectively. The melt/air interface can be determined by integrating equation (31) with the boundary conditions R = Rc at  = c and R = Rs at  = s. The unknown reference pressure Pref at the tri-point is determined using the following contact-angle constraint:  1 dR @z ¼ tan 0 : ð34Þ d @ Equation (32) with the contact-angle constraint in equation (34) is solved by the Newton–Raphson method (Lan, 1994). The numerical method for solving equation (18) is a controlvolume finite-difference method (Lan & Kou, 1991b) for solving the governing equations (1)–(3) (Gosman et al., 1969). Each time the melt/solid interfaces are located and updated, the melt/air interface is then also updated by solving equation (32) again. The calculation loop is repeated until the convergence criteria are satisfied as shown in Table 3. Owing to the non-uniform grid spacing in the physical domain, finer fitting J. Appl. Cryst. (2009). 42, 553–563

research papers Table 4 The samples used in the experiments. Sample index

A

B

C

D

Source rod size (mm)† Crystal fiber diameter (mm) Feed speed (mm min1) Growth (mm min1) Reduction ratio (%)

400  400 290 4 9.6 64

Ø300 300 1.2 1.2 100

Ø300 105 1.2 9.6 35

Ø300 75 1.2 19.2 25

† Ø denotes diameter.

near the interface allows the higher melt flow and temperature at these locations to be measured more accurately. Conversely, the grid spacing in the transformed domain is uniform with


=
 = constant, as shown in Fig. 3. In this way, second-order accuracy is retained in the finite difference approximations (Lan & Kou, 1990, 1991a,c). In the experiments, the total lengths of the source rod and the crystal fiber are usually much longer than their radii. In order to reduce the computational load without compromising the accuracy at the same time, the system is divided into three regions, namely an outer region in the crystal fiber (z2 < z z3), an outer region in the source rod (z0 z < z1), and an inner region comprising the melt and solid material near the melt/solid interface (z1 z z2). The two-dimensional temperature distribution is essentially uniform in the inner region, and heat transfer becomes one-dimensional in the two outer regions. For the miniature molten zone, the length of the inner region is long enough to be 32 times the diameter of the source rod; further length increases produce no significant changes. The two outer regions have lengths 65 times longer than that of the source rod (Lan & Kou, 1990).

the examples shows a semi- and an asymmetrical Gaussian distribution as shown in the inset of Fig. 4. Fig. 4 shows a comparison of molten-zone length between experiments and simulations at various laser powers in case A with semi- and asymmetric Gaussian distributions. For all cases, it is assumed that vaporization is negligible. Stable growth is defined as occurring under conditions where usable crystal fibers can be produced successfully. The molten-zone length required to achieve stable growth varies depending on the laser power. The slope of the linear dependence is defined as . Greater means that there is more energy stored in the melt under dynamic equilibrium to extend the molten-zone length and raise the average temperature. Therefore, the magnitude of can correspond to the grown material indexed by the Prandtl number Pr = / , where  = m /m is the kinematic viscosity and is the thermal diffusivity. The slope of 210 mm W1 acquired for both distributions in the simulations is consistent with that obtained in the experiments, especially at the higher allowed powers. The Prandtl number of the slightly doped YAG material is typically 7–8 (Lan & Tu, 2001; Schwabe et al., 2004). For similar distributions with the same total energy, the laser intensity profile only slightly affects . At the lower allowed powers, the molten zones in the experiments were a little longer than those in the simulations for all cases owing to the higher laser energy stored in the melt. It is noted that a little more laser power absorbed on the concave melt/air interface resulted in a smaller radius of curvature in the experiments. Moreover, the heat dissipation of the molten zone is mainly through the conduction of the melt/air interface; therefore, the asymmetric Gaussian distribution in the simulations is chosen for convenience. The higher allowed powers are preferred for production of crystal fibers in practice.

4. Results and discussion 4.1. Comparison between experiments and simulations

Table 4 summarizes the samples used in the experiments as cases A, B, C and D. Two types of source rod are used; they are square with a cross section of 400  400 mm in case A and a rod with a diameter of 300 mm in the other cases. The ambient temperature is 575 K in case A and 650 K in the other cases. Case A is employed to verify the source rod whose dimension approximates that of the bulk crystal. In equation (10), the laser intensity profile with a Gaussian distribution is defined as follows:

 Ia ¼ Aq exp  a ðz=a Þ2 ;

ð35Þ

where Aq and  a are the amplitude and the half-width (1/e2) of the Gaussian distribution at z = 0, and a is a beam-shape factor to define the symmetry of the laser intensity profile above (z > 0) and below (z < 0) the z = 0 plane. An asymmetrical Gaussian distribution can be expressed by the combination of two Gaussian distributions with a = 1 and  a = 12 mm for z > 0, and a = 1 and  a = 6 mm for z < 0. Similarly, a semi-Gaussian distribution can be defined with a = 1 and  a = 24 mm for z > 0, and a = 106 and  a = 12 mm for z < 0. One of J. Appl. Cryst. (2009). 42, 553–563

Figure 4 Miniature molten-zone lengths at various laser powers in case A with semi- and asymmetric Gaussian distributions in both experiments and simulations. The solid and dashed lines indicate the profile in an asymmetric and a semi-Gaussian distribution. The square dots represent the experimental data. The inset shows the two laser intensity profiles. The black lines denote the laser intensity profile and the gray lines denote the intensity profile on the melt/air interface. P. Y. Chen et al.



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research papers allowed power, look similar. The reasons are that the contact angle at the tri-point and the surface tension depend on the ability of the material to restrict variations in the melt/air interface shape. For fixed contact angle, the shapes of the melt/air interface and the length of the molten zone can be varied by changing the laser power or reduction ratios. From examination of simulations like the inset in Fig. 7, the growth front appears to change from convex to flat and then concave toward the melt as the reduction ratio increases. It is noted that the heat transfer and the release of latent heat at the melt/solid interface affect the shape of the growth front according to equation (9). For the same feed speed, higher reduction ratios result in a lower growth speed due to continuity, and then less release of latent heat at the growth front, and vice versa. Therefore, a growth front that is Figure 5 more convex toward the melt can be obtained Miniature molten-zone lengths at various laser powers for (a) case A, (b) case B, (c) case C at lower growth speed. Although the growth and (d ) case D in both experiments and simulations. The square dots represent the experimental data and each dot is assigned an index number of (1)–(18). The solid lines in front can also be influenced by varying the laser gray denote the simulation results. The gray arrows labeled Hi and Li are the higher and power, it is determined by both the temperalower allowed laser powers, where the subscript i represents case A, B, C or D. ture gradient normal to the growth front and the release of the latent heat. Fig. 7 shows a comparison between experiFig. 5 shows a comparison between experiments and simuments and simulations on the normalized radial and axial lations on the molten-zone lengths at various laser powers for positions where the melt/air interfaces are located at HC in all cases. The allowed powers and the value of are 5.3–6.7 W case C. For clear comparison, the positions are normalized by and 210 mm W1 in case A, 2.0–2.2 W and 330 mm W1 in the length of the molten zone axially and the maximum radius case B, 1.5–1.7 W and 390 mm W1 in case C, and 1.2–1.5 W of the molten zone radially. The azimuthal area and shape of and 420 mm W1 in case D. When normalized to the sourcethe melt/air interface are in good agreement between the rod diameter, the allowed molten-zone lengths are 1.56–2.88 experiments and simulations. The inset shows the image in case A, 1.07–1.53 in case B, 1.07–1.72 in case C and 0.67–1.46 observed experimentally in Fig. 6 (13) overlapped by in case D. With a smaller source rod, the allowed power range reduces but increases oppositely. Furthermore, the length and volume of the molten zone are also decreased. By decreasing the reduction ratios for the same source rod, the allowed laser power is lower, while is roughly constant at the feed speed of 1.2 mm min1. We are also aware that the maximum length of the molten zone divided by the minimum length is 2.0 (5) for stable growth. Similar empirical criteria for the miniature molten zone in a trapezoid shape (Gu et al., 2001) and the floating zone for growing bulk crystals (Young & Heminger, 1997) have been introduced previously. Fig. 6 shows the captured images of the thermal radiation from the melt designated in Fig. 5 experimentally. The contact angle 0 at the tri-point is about 11 (1) . According to the images on the same row, the length of the molten zone extends by increasing the laser power as a result of there being more energy stored in the melt. Furthermore, the melt/air interface shapes evolve from concave to convex because the outward Figure 6 pressure in the melt is increased. Images of the miniature molten zones in Fig. 5 captured experimentally. For images (9) and (14), at the lower allowed power, the The index numbers in each image correspond to those in Fig. 5. The upper melt/air interface becomes more concave with a decrease in and lower arrows in white are the directions of motion of the crystal fiber the reduction ratio, while images (13) and (18), at the higher and the source rod, respectively.

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research papers isotherms on the right side and streamlines with double flow loops on the left side, as is typically observed in the melts obtained in the simulations. The melt/air interface shape is consistent, and the deformed shape of the melt/solid interfaces can be obtained both experimentally and numerically for comparison.

the greater contribution to heat dissipation. Therefore, when increasing the feed speed, becomes higher owing to the larger cross section at the growth front.

4.2. Molten-zone shape

4.3. Curvature of melt/air interface

Fig. 8 shows at various reduction ratios for different feed speeds in both experiments and simulations. If the feed and growth speeds are zero, the heat energy is stored in the melt, and this situation results in the largest value of for a reduction ratio of less than 70% (Fig. 8). This occurs because only thermocapillary convection is available to form the double flow loops for heat dissipation shown on the left side of the inset in Fig. 7. With the same source rod and feed speed, the increase in

becomes saturated by decreasing the reduction ratio, because the cross section at the growth front becomes smaller. This can be elucidated by the fact that the lower flow loop of the thermocapillary convection obstructs the mass convection when the feed speed is not zero. Therefore, the heat transfer out of the melt is mainly through the growth front rather than the feed front. The parabolic trend of indicates that is inversely proportional to the cross section of the growth front, i.e. an inverse square proportion of the reduction ratio. By increasing the feed speed, decreases at a reduction ratio of less than 70% but increases at a higher reduction ratio using the same source rod. The reason is that heat removal via mass-transfer convection is enhanced by increasing the feed speed at the lower reduction ratios. However, there is a trade-off between the mass transfer and the cross section at the growth front due to continuity. At the higher reduction ratio, not mass flow but cross section makes

Fig. 9 shows the radial positions of the melt/air interface and the curvature radius at various axial positions for all cases in the simulations. An inflection point can be identified by the peak value of the curvature radius at various axial positions. The melt/air interface is wine-bottle shaped, except for case B at a reduction ratio of 100%. In case B, the inflection points cannot be recognized without diameter reduction. The melt/ air interface shape is slightly convex toward the melt, in agreement with that shown in the second row of Fig. 6. The higher peak value of the curvature radius means a smoother profile at the inflection point. Moreover, the axial position of the inflection point reveals the degree to which the molten zone extends radially at the bottom. In case A, all the inflection points are located at z > 0. As the molten-zone length extends with rising laser power, the inflection point moves upward and the peak value of the curvature radius increases. In cases C and D, the inflection point moves from ‘z < 0’ to ‘z > 0’ and the peak value of the curvature radius decreases as the laser power rises. For the LHPG method, it is noted that the laser heating efficiency for the same melt volume can be improved, i.e. fm increases, if the laser beam is projected onto the position near the location of the peak value of the curvature radius.

Figure 8 Figure 7 The normalized radial and axial positions of the melt/solid interface in case C at HC in both experiments and simulations. The square dots in black represent the experimental results and the solid line in gray denotes the simulation result. The inset shows the image observed in the experiments overlapped by isothermal lines on the right side and streamlines on the left side in the simulations. J. Appl. Cryst. (2009). 42, 553–563

The slopes at various speed ratios or reduction ratios for different feed speeds in both experiments and simulations. The solid, dashed and center lines denote the simulation data of Us = Uc = 0 mm min1, Us = 1 mm min1 and Us = 1.2 mm min1, respectively. The solid triangles represent the experimental data of Us = 1.2 mm min1 in cases B, C and D. The error bars of the solid triangles indicate the standard errors of the linear fitting. P. Y. Chen et al.



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research papers 4.4. Effect of gravity on the miniature molten zone

Although the melt/air interface can be determined by equations (10)–(12), (16) and (17), the static Bond number Bos is a good indicator for evaluating the influence of gravity on the surface tension. The static Bond number is defined as Bos = mgLm2/, where Lm is the molten-zone length. Typically, the static Bond numbers are 1.5  102 in case A and 2.2  103 in the other cases. With the same material and gravity, the static Bond numbers can be determined by the lengths of the molten zones. Fig. 10 simulates the effect of gravity on the melt/air interface in case C. The length of the molten zone is increased by Figure 9

0.53Rs from the lower (LC) to the The radial positions and the curvature radius of the melt/air interface at various axial positions in higher (HC) allowed laser powers in the simulations for (a) case A, (b) case B, (c) case C and (d ) case D. The black and the gray lines represent the radial positions of the melt/air interface and its curvature radius at various axial Fig. 5(c). The difference in the static positions, respectively. The solid lines and the dashed lines indicate Hi and Li in Fig. 5, respectively. Bond number between LC and HC is 3

3  10 . At LC, a shorter moltenzone length is obtained. There is almost power, remote control of the translation stage, and image no difference between normal and near-zero gravity. At HC, processing of the molten-zone and crystal-fiber diameter, is surface tension shows an observable reduction in molten-zone reported. The stable conditions utilized for producing a useful length by 0.072Rs with near-zero gravity. The difference in the crystal fiber are defined. The effects caused by thermocapillary static Bond number between normal gravity and near-zero and mass-transfer convection in the melt and the trade-off gravity is 4.9  104, which is an order of magnitude less between mass transfer and cross section at the growth front for than that between LC and HC. heat dissipation are discussed. The melt/air interface is estiThe variation in the length of the molten zone due to gravity mated and optimized for efficient laser absorption according is much smaller than that effected by varying the laser power. to curvature radius and the inflection point. Finally, there is no The same results can also be inferred from the corresponding significant change if the gravity is varied along the growth difference in the static Bond number, which is an order of direction, especially when the laser power is low. The influence magnitude smaller than that from LC to HC. Therefore, a of gravity is discussed with reference to the feasibility of variation in gravity along the growth direction will not growing a crystal fiber horizontally. obviously affect the stable conditions for growing crystal fibers. Furthermore, the strength of gravitational force acting on the melt or the surface tension associated with the melt/air interface can be measured experimentally using a fluid accelerometer or a tensiometer. The feasibility of growing crystal fibers using the LHPG method in the horizontal plane could also be further investigated.

5. Conclusions A two-dimensional simulation on the miniature molten zone fabricated using the LHPG method for growing single-crystal fibers was successfully modified from that for growing bulk crystals using the floating-zone method. Using a non-orthogonal body-fitting grid system and the control-volume finite difference method, a more accurate determination of the molten-zone interfaces can be achieved with reasonable computer consumption. The automatic crystal-fiber growth system, which involves the feedback control of the CO2 laser

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Figure 10 The radial positions of the melt/air interface at various axial positions in case C in the simulations. The black and the gray lines represent the conditions of normal and zero gravity, respectively. The solid and dashed lines indicate HC and LC in Fig. 5, respectively. J. Appl. Cryst. (2009). 42, 553–563

research papers This work was sponsored in part by the National Science Council, People’s Republic of China, under contract No. NSC 96-2628-E-002-042-MY3.

References Ardila, D. R., Cofre´, L. V., Barbosa, L. B. & Andreeta, J. P. (2004). Cryst. Res. Technol. 39, 855–858. Block, H. D. (1962). Introduction to Tensor Analysis. New York: C. E. Merrill Books Press. Brandon, S. & Derby, J. J. (1992). J. Cryst. Growth, 121, 473–494. Chen, J. C., Lin, Y. S., Tsai, C. N., Huang, K. Y., Lai, C. C., Su, W. Z., Shr, R. C., Kao, F. J., Chang, T. Y. & Huang, S. L. (2007). IEEE Photonic Technol. Lett. 19, 595–597. Coriell, S. R. & Cordes, M. R. (1977). J. Cryst. Growth, 42, 466–472. Coriell, S. R., Hardy, S. C. & Cordes, M. R. (1977). J. Colloid Interface Sci. 60, 126–136. Davis, G. M., Yokohama, I., Sudo, S. & Kubodera, K. (1991). IEEE Photonic Technol. Lett. 3, 459–461. Digonnet, M. J. F., Gaeta, C. J. & Shaw, H. J. (1986). J. Lightwave Technol. LT-4, 454. Duranceau, J. L. & Brown, R. A. (1986). J. Cryst. Growth, 75, 367– 389. Feigelson, R. S. (1986). J. Cryst. Growth, 79, 669–680. Feigelson, R. S. (1988). Mater. Sci. Eng. B, 1, 67–75. Feigelson, R. S., Kway, W. L. & Route, R. K. (1985). Opt. Eng. 24, 1102–1107. Fejer, M. M. (1986). PhD dissertation, Stanford University, USA. Fejer, M. M., Nightingale, J. L., Magel, G. A. & Byer, R. L. (1984). Rev. Sci. Instrum. 55, 1791–1796. Fratello, V. J. & Brandle, C. D. (1993). J. Cryst. Growth, 128, 1006– 1010. Gasson, D. B. & Cockayne, B. (1970). J. Mater. Sci. 5, 100–104. Goodman, J. W. (1968). Introduction to Fourier Optics. San Francisco: McGraw Hill Press. Gosman, A. D., Pan, W. M., Runchal, A. K., Spalding, D. B. & Wolfshtein, M. (1969). Heat and Mass Transfer in Recirculating Flows. London: Academic Press. Green, R. E. Jr (1964). J. Appl. Phys. 35, 1297–1301. Gu, J., Shen, Y., Chen, S. & Zhao, W. (2001). J. Mater. Sci. Eng. 19, 20– 23. Huang, K. Y., Hsu, K. Y. & Huang, S. L. (2008). IEEE J. Lightwave Technol. 26, 1632–1638. Huang, K. Y., Hsu, K. Y., Jheng, D. Y., Zhuo, W. J., Chen, P. Y., Yeh, P. S. & Huang, S. L. (2008). Opt. Express, 16, 12264–12271. Ishibashi, S., Naganuma, K. & Yokohama, I. (1998). J. Cryst. Growth, 183, 614–621. Kim, K. M., Dreeben, A. B. & Schujko, A. (1979). J. Appl. Phys. 50, 4472–4474. Kozhoukharova, Z. & Slavchev, S. (1986). J. Cryst. Growth, 74, 236– 246.

J. Appl. Cryst. (2009). 42, 553–563

Lai, C. C., Tsai, H. J., Huang, K. Y., Hsu, K. Y., Lin, Z. W., Ji, K. D., Zhuo, W. J. & Huang, S. L. (2008). Opt. Lett. 33, 2919–2921. Lan, C. W. (1994). Int. J. Numer. Methods Fluids, 19, 41–65. Lan, C. W. (1996). J. Cryst. Growth, 169, 269–278. Lan, C. W. & Kou, S. (1990). J. Cryst. Growth, 102, 1043–1058. Lan, C. W. & Kou, S. (1991a). J. Cryst. Growth, 108, 351–366. Lan, C. W. & Kou, S. (1991b). J. Numer. Methods Fluids, 12, 59–80. Lan, C. W. & Kou, S. (1991c). J. Cryst. Growth, 114, 517–535. Lan, C. W. & Kou, S. (1993). J. Cryst. Growth, 132, 578–591. Lan, C. W. & Liang, M. C. (1997). J. Cryst. Growth, 180, 381–387. Lan, C. W. & Tu, C. Y. (2001). J. Cryst. Growth, 223, 523–536. Lee, L. M., Pei, S. C., Lin, D. F., Tsai, M. C., Tai, T. M., Chiu, P. C., Sun, D. H., Kung, A. H. & Huang, S. L. (2007). J. Opt. Soc. Am. B, 24, 1909–1915. Li, G. & Liu, L. (1995). J. Synth. Cryst. 24, 208–211. Li, G., Liu, Z. & Lin, Y. (1993). J. Synth. Cryst. 22, 32–35. Lie, K. H. & Walker, J. S. (1989). Int. J. Heat Mass Transfer, 32, 2409– 2420. Lie, K. H., Walker, J. S. & Riahi, D. N. (1988). Physicochem. Hydrodyn. 10, 441–460. Liu, L. H., Tan, H. P. & Yu, Q. Z. (2002). Int. J. Heat Mass Transfer, 45, 417–424. Lo, C. Y., Huang, K. Y., Chen, J. C., Chuang, C. Y., Lai, C. C., Huang, S. L., Lin, Y. S. & Yeh, P. S. (2005). Opt. Lett. 30, 129–131. Lo, C. Y., Huang, K. Y., Chen, J. C., Tu, S. Y. & Huang, S. L. (2004). Opt. Lett. 29, 439–441. Lo, C. Y., Huang, P. L., Chou, T. S., Lee, L. M., Chang, T. Y., Huang, S. L., Lin, L., Lin, H. Y. & Ho, F. C. (2002). Jpn. J. Appl. Phys. 41, L1228–L1231. Longtin, J. P. & Tien, C. L. (1997). Int. J. Heat Mass Trans. 40, 951– 959. Love, J. D., Henry, W. M., Stewart, W. J., Black, R. J., Lacroix, S. & Gonthier, F. (1991). IEE Proc. J. Optoelecton. 138, 343–354. Nijmeijer, M. J. P. & Landau, D. P. (1997). Comput. Mater. Sci. 7, 325– 335. Poplawsky, R. P. (1961). J. Appl. Phys. 33, 1616–1617. Riahi, D. N. & Walker, J. S. (1989). J. Cryst. Growth, 94, 635–642. Rivas, D., Sanz, J. & Va´zquez, C. (1992). J. Cryst. Growth, 116, 127– 138. Rudolph, P. & Fukuda, T. (1999). Cryst. Res. Technol. 34, 3–40. Scheel, H. J. (2000). J. Cryst. Growth, 211, 1–12. Schwabe, D., Sumathi, R. R. & Wilke, H. (2004). J. Cryst. Growth, 265, 440–452. Thompson, J. F., Thames, F. C. & Mastin, C. W. (1974). J. Comput. Phys. 15, 299–319. Tsai, C. N., Lin, Y. S., Huang, K. Y., Lin, Y. S., Lai, C. C. & Huang, S. L. (2008). Jpn. J. Appl. Phys. 47, 6369–6373. Yen, W. M. (1999). Phys. Solid State, 41, 770–773. Yoon, D. H. (2004). Opto-electronics Rev. 12, 199–212. Young, G. W. & Chait, A. (1989). J. Cryst. Growth, 96, 65–95. Young, G. W. & Heminger, J. A. (1997). J. Cryst. Growth, 178, 410– 421.

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