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The effect of applied magnetic field on flow structures in liquid phase electroepitaxy—a three-dimensional simulation model
Journal of Crystal Growth 244 (2002) 12–26
The effect of applied magnetic field on flow structures in liquid phase electroepitaxy—a three-dimensional simulation model Y.C. Liua, Y. Okanob, S. Dosta,* a
Crystal Growth Laboratory, Rel. Tech. Department of Mechanical Engineering, Center for Advanced Materials, University of Victoria, P.O. Box 3055, Victoria, BC, Canada V8W 3P6 b Department of Materials Science and Chemical Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, 432-8561, Japan Received 15 April 2002; accepted 26 June 2002 Communicated by K. Nakajima
Abstract A three-dimensional numerical simulation for the liquid phase electroepitaxial growth of GaAs under a vertical stationary magnetic field was carried out. The effect of magnetic field intensity and non-uniformity on the flow field in the liquid solution was investigated. Numerical results show that the flow patterns exhibit three distinct stability characteristics: a stable flow field up to a magnetic field level of Ha ¼ 150; a transitional flow between Ha ¼ 150 and 220; and an unstable flow above Ha ¼ 220: In the stable region, the applied magnetic field suppresses the flow field, and the flow intensity decreases with increasing magnetic field exhibiting a power law of Umax pHa5=4 relationship for the maximum velocity (Umax ). In the transitional region, the flow intensity increases dramatically with the increase in magnetic field strength. The flow patterns are significantly different from those in the stable region. The flow field is no longer axisymmetric but still stable. In the unstable region, the flow structure and intensity change with time. Under a strong magnetic field, the flow cells are confined to the vicinity of the vertical wall and exhibit significant non-uniformity near the growth interface. Such strong flow fluctuations and non-uniformities near the growth interface may have an adverse effect on the growth process and lead to an unsatisfactory growth. In this region, the maximum velocity (Umax ) obeys approximately a power law Umax pHa5=2 : Results show that for a successful growth the effect of applied magnetic field must be optimized. r 2002 Elsevier Science B.V. All rights reserved. Keywords: A1. Computer simulation; A1. Magnetic fields; A2. Growth from solutions; A3. Electroepitaxy; B2. Semiconducting materials
1. Introduction Liquid phase electroepitaxy (LPEE), being a solution growth, has a number of advantages over *Corresponding author. Tel.: +1-250-721-8900; fax: +1250-721-6051. E-mail address: [email protected] (S. Dost).
other bulk crystal growth techniques. For instance, the relatively low temperatures used in LPEE, and its ability to provide well-controlled growth of crystals with the desired properties, make this technique technologically very promising (see for instance Refs. [1–9]). LPEE has a great potential to become a commercial technique in growing high quality, bulk crystals; particularly
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 2 ) 0 1 6 0 3 - 2
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
ternary crystals such as GaInAs, GaInSb, and CdZnTe (see Refs. [10–13]). In LPEE, growth is achieved by passing an electric current through the growth cell at a constant furnace temperature. The applied electric current is the sole driving force for growth, and gives rise to two growth mechanisms known as ‘‘electromigration’ and ‘‘Peltier cooling/heating’’. The electromigration of species in the solution takes place due to electron-momentum exchange and electrostatic field forces, and sustains a controlled growth [1,2]. The Peltier heating/cooling, on the other hand, is a thermoelectric effect occurring when an electric current passes through an interface of two materials with different Peltier coefficients. The Peltier cooling at the growth interface supersaturates the solution in the immediate vicinity of the substrate and leads to epitaxial growth. The Peltier heating at the dissolution interface, on the other hand, causes the dissolution of the source material into the solution and provides constantly the needed feed material for growth. The growth rate is proportional to the applied electric current. Although the furnace temperature is constant, the combined effect of the Joule heating in the solid crystals, and the Peltier heating/cooling at the interfaces leads to small temperature gradients in the solution. Such temperature gradients together with concentration gradients result in a significant natural convection in the solution. This convective flow in the solution has an adverse effect on the LPEE growth process [14]. It leads to interface instability and consequently stops the growth. Recently, we were interested in the LPEE growth of single crystals under a static applied magnetic field [8,13–20]. The objective was to minimize the effect of convection in the solution. Lower convection in the solution will allow the use of higher electric current densities that will be translated into higher growth rates. The theoretical model, and results of twodimensional (2D) numerical simulations for LPEE under an applied magnetic field can be found in Refs. [14,15,19,20]. Simulation results have shown that the level of applied magnetic field has a significant effect on the intensity and structure of the convective flow in the solution. As expected,
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stronger applied magnetic field leads to weaker natural convection in the solution. The shape of the growth interface becomes more uniform at higher magnetic field levels. Our previous 2D numerical simulations have shed light on various aspects of the applied magnetic field in LPEE. However, these earlier models, being 2D, have naturally neglected the contributions of the circumferential velocity and magnetic body force components. Therefore, the results obtained earlier were qualitative and did not include information about three-dimensional (3D) effects such as deviations from axisymmetry, variations of the magnetic field components, and mixing in the solution. In order to shed light on these 3D effects, in this study we present a 3D model and numerical simulation results for the LPEE growth of a binary system. To the best of our knowledge, there is no 3D model developed previously for the LPEE. The flow field in the solution, which is often related to phenomena such as macro-segregation and striations, and hydrodynamic instabilities, plays an important role in a crystal growth process (see, e.g. Refs. [21,22]). The application of an external magnetic field to suppress convection has been found to be very attainable and efficient. The applied magnetic field could be static or rotating, and vertical or transverse. A detailed account of the subject and related literature can be found in Refs. [23,24]. Here, only a very brief review is given. Most of the 2D models developed thus far to study the effect of an applied magnetic field in crystal growth (see, e.g. Refs. [14,15,19,20,25–30]) have focused on the suppression of flow fluctuations and concentration uniformity. An important related work with emphasis on the flow structures and intensity, was given in Ref. [31]. The results of the 2D numerical study of a horizontal Bridgman configuration have shown that higher applied magnetic field intensity levels (higher Hartmann numbers (Ha), a measure of the applied magnetic field strength) do not only result in further suppression of the convective flow in the liquid, but also lead to a progressive change of the overall structure of the flow followed by the appearance of Hartmann layers in the vicinity of the rigid walls.
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They have also demonstrated that when Ha is large enough (Ha > 10) the decrease of the strength of the convective flow asymptotically approaches a power-law dependence in Ha; and the flow in the central region becomes more and more unidirectional. Some recent studies [32–34] have focused on the influence of thermal-electromagnetic convection (TEMC) [35] that is generated by an induced magnetic force measured by the product of the electromotive force and the local temperature gradient in the presence of an applied magnetic field. As mentioned in these studies, the TEMC may become significant under certain conditions, for instance, when the solution and solid have different thermoelectric power and the growth interface is not isothermal (e.g. GeSi and HgCdTe), or when the growth interface is faceted where the local temperature gradient is large due to interface undercooling, or when the thermoelectric power in solution is a function of a non-uniform liquid composition. Note that most of the TEMC studies are related to reduced gravity or microgravity conditions [34] under which the natural convection is weaker. To the best of our knowledge, the literature on the 3D modeling of crystal growth under an applied magnetic field is not so rich. We would like to cite here some of the related 3D studies. For instance, 3-D models have been developed for the Czochralski growth process to examine the oxygen transport and fluid flow in silicon melt under vertical [35–37], horizontal [36,38], and cusp [38,39] magnetic fields. Numerical simulations for the LiNbO3 melt flow under a uniform vertical magnetic field were carried out in Ref. [40]. Simulations showed that the spoke pattern flow that appeared at azimuthal planes was due to the Marangoni instability, and the rotating motion was driven by the magnetic body force that was induced by the interaction between the vertical magnetic field and the radial motion [23,24]. A study for temperature contours under a transverse magnetic field for a Czochralski system was given in Ref. [41], where a quite peculiar convection pattern under a certain magnetic field was obtained; velocity components were all downwards in a vertical cross section parallel to the magnetic field, while those in a plane perpendi-
cular to the magnetic field were natural convection dominant, which indicates the possible complexity of flow field and its dependence on certain growth conditions in the presence of a magnetic field. Other 3D simulation studies, under the presence of a magnetic field and often without a particular growth system, mainly focus on the flow field in a certain geometrical system. The natural convection in a differentially heated cubical box with three different orientations of the magnetic field along the coordinate axes was investigated numerically in Ref. [42]. It was found that the applied magnetic field suppressed the natural convection most efficiently when the magnetic field is imposed perpendicular to the heated vertical walls. Numerical results in a cylindrical geometry under a constant magnetic field were compared with experiments in Ref. [43], and used towards the improvement of the numerical model introduced for better predictions. The numerical simulation of the flow of liquid gallium melt in a cylindrical container in Ref. [44] showed varying temperature fluctuations of nearly sinusoidal form when the temperature difference is 0.65 K and the applied vertical magnetic field is 20 G. Results were verified by experiments. The suppression of such temperature fluctuations by increasing the magnetic intensity was also demonstrated. Furthermore, through numerical simulation, they found evidence that the induced magnetic field follows the oscillatory convection and temperature variation. The effect of a constant magnetic field on electrically conducting liquid-metal flows has been numerically (in three dimensions) investigated in Ref. [45]. Results have again, similar to their 2D analysis [31], showed that when Ha reaches a certain critical value, which is found to depend on the direction of the applied magnetic field, the decrease of the flow intensity takes on an asymptotic power-law form, and the flow patterns have significant changes. They have separated the flow structure into three regions: the core flow, Hartman layers which develop in the immediate vicinity of the rigid horizontal boundaries or at the end walls, and parallel layers appearing in the vicinity of the sidewalls. Flow transitions in isothermal metallic liquids confined in cylindrical cavities under a rotating magnetic field were also
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
simulated in Ref. [46]. A parametric study for the change of aspect ratio of the geometry, Hartman number, and rotating Reynolds number was provided. For the sake of brevity, a review of experimental investigations on the effect of an external magnetic field on flow patterns and crystal growth processes is not given here. The reader is referred to Refs. [23,24,47]. It is however important to point out that both experimental and numerical studies carried out to study the effect of a magnetic field in crystal growth are often related to unsteady convection, especially when free surfaces exist in the growth system, since one of the advantages of applying a magnetic field is to suppress such unsteady flows. However, if the magnetic body force becomes much larger than other flow driving body forces (for instance due to buoyancy and surface tension), it will likely result in a stronger convection than the one with no magnetic field effect. Some studies have indeed demonstrated this point [48,49]. Our experimental interest is focused on the LPEE growth of a ternary In0.04Ga0.96As system under a static vertical magnetic field. In the present article, however, since the emphasis is on the variations of flow patterns and intensities, the present 3D simulation study is carried out for the growth of a binary system (GaAs) for computational simplicity. Simulation results are compared with other studies. The effect of magnetic field non-uniformity is also investigated.
2. The simulation model The LPEE growth process under an applied magnetic field can be influenced by many factors, such as fluid flow, heat and mass transfer, electric and magnetic fields, various thermoelectric effects and their interactions in the solution, and heat and electric conduction with various thermoelectric effects in the solid. For a typical schematic LPEE growth configuration, see Fig. 1 and Refs. [14,20]. The development of the fundamental equations of the LPEE growth process for binary and ternary systems can be found in Refs. [8,14,19,50].
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2.1. The governing equations 3D time-dependent governing equations describing the fluid flow, heat and mass transport in the liquid solution (binary) are given as follows: Continuity: 1q 1 qv qw ðruÞ þ þ ¼ 0: ð1Þ r qr r qj qz Momentum: qu qu v qu qu v2 þu þ þw qt qr r qj qz r u 2 qv 1 qp FrM 2 þ ; ¼n r u 2 2 r r qj r qr r qv qv v qv qv uv þu þ þw þ qt qr r qj qz r v 2 qv 1 qp FjM þ ; ¼ n r2 v 2 þ 2 r r qj r qj r qw qw v qw qw þu þ þw qt qr r qj qz 1 qp FzM þ ¼ nr2 w r qz r gbt ðT T0 Þ þ gbc ðC C0 Þ: Mass transport: qC qC v qC qC þu þ þw qt qr r qj qz qC qC þ mEz ¼ Dr2 C: þ mEr qr qz Energy: qT qT v qT qT þu þ þw ¼ ar2 T qt qr r qj qz
ð2Þ
ð3Þ
ð4Þ
ð5Þ
ð6Þ
with the Laplacian operator 1q q 1 q2 q2 r r2 ¼ þ 2 2 þ 2; r qr qr r qj qz where u; v and w are, respectively, the velocity components in the radial ðrÞ; circumferential (j) and vertical (z) directions. n is the kinematic viscosity, p is the pressure, T is the temperature, C is the arsenic (As) concentration in the Ga–As solution, r is the solution density, bt and bc denote, respectively, the thermal and solutal expansion coefficients, T0 and C0 represent the
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
16
Fig. 1. Schematic view of the LPEE growth crucible, and the coordinate system.
reference temperature and concentration, respectively, g is the gravitational constant, a is the thermal diffusivity. m and D denote, respectively, the solute mobility and the diffusion coefficient. FrB ; FzB ; and FjB are the magnetic body force components, respectively, along the r; z and j directions. These magnetic force components are introduced in detail in the next section. Er and Ez are the electric field intensities in the r and z directions, which are obtained by solving an axisymmetric quasi-steady electric field (f) equation: q2 f 1 qf q2 f þ ¼0 þ qr2 r qr qz qf qf with Er ¼ ; Ez ¼ : qr qz
ð7Þ
In the derivation of the governing equations it was assumed that the solution is incompressible and Newtonian, and the well-known Boussinesq approximation—assuming a constant fluid density in all equations except in the body force term due to buoyancy—holds. The boundary conditions associated with the governing Eqs. (1)–(7) are as follows: Along the vertical wall: u ¼ 0; v ¼ 0; w ¼ 0; z z qf qC 0 T ¼ Tg ¼ 0; ¼ 0: DT; qr qr H
ð8Þ
Along the growth interface: u ¼ 0; v ¼ 0; w ¼ 0; qT qT kl ¼ pJ; ks qz qz C ¼ C1 :
s
qf ¼ J; qn ð9Þ
Along the dissolution interface: u ¼ 0; v ¼ 0; w ¼ 0; qT qT ks ¼ þpJ; f ¼ 0; C1 ¼ C2 : kl qz qz
ð10Þ
In the above equations p is the Peltier coefficient, J is the electric current density, s is the electrical conductivity of the liquid, H is the height of the solution zone along the vertical axis, ks and kl are the thermal conductivities of the solid and liquid, respectively. Also, C1 and C2 are, respectively, the solute concentrations at the growth and dissolution interfaces, which are determined by the interfacial equilibrium conditions (see Ref. [51]). DT is used to denote the temperature drop due to the heat loss when heat is transferred from the outside wall of the furnace to the vertical wall of the cell. Its value is estimated by considering the heat transfer through the whole LPEE system that was computed in our previous articles [51,52]. It was assumed that the contribution of latent heat is negligible since the growth rate is very low.
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
The initial conditions are C ¼ C0 ; u ¼ 0; v ¼ 0; w ¼ 0 and T ¼ Tg at t ¼ 0:
ð11Þ
2.2. Magnetic body force components The applied static magnetic field is considered to be perfectly aligned with the vertical axis of the growth system B ¼ Bez ;
ð12Þ
where B is the magnitude of the applied magnetic field. The magnetic body force acting on the points of the metallic liquid solution is then expressed by FM ¼ J B
ð13Þ
and the current density vector J is assumed to be given by Ohm’s law as follows: J ¼ sðE þ v BÞ;
ð14Þ
where E and v represent, respectively, the electrical field and velocity vectors acting on fluid points. The induced electric field due to the applied magnetic field and the fluid motion was neglected (see Ref. [24] for justification). The solution of Eq. (7) together with the corresponding boundary conditions lead to the following expression for E: qf J E ¼ E z ez ¼ ð15Þ ez ¼ ez : qz s Eq. (15) implies that the electrical field intensity is zero along the r and j directions, and is constant along the z-direction. The use of Eq. (15) in Eq. (14) leads to J J ¼ sðBver Buej þ ez Þ ð16Þ s and then the magnetic body force in Eq. (13) becomes M M FM ¼ FM r e r þ F j ej þ F z ez
¼ sB2 ðuer þ vej Þ;
ð17Þ
where the components of the magnetic body force, used in Eqs. (2)–(4), are then given by FrM ¼ sB2 u; FjM ¼ sB2 v; FzM ¼ 0:
ð18Þ
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Deviation from axisymmetry: Our measurements show that the magnetic field distribution in the magnet opening where the growth crucible is to be located is almost uniform. However, when the growth system is lowered into the magnet opening, it is not possible to measure the field distribution, and it is likely that the field uniformity would be altered. Furthermore, the center of the growth system may not be aligned perfectly with the vertical axis due to possible unintentional experimental errors. This will result in further deviations from axisymmetry. This in turn may affect the growth adversely and lead to unsatisfactory growth. It was therefore desirable to investigate the influence of such asymmetries. In order to accommodate them in the model, it was assumed that the center of the magnetic field is offset with respect to the symmetry axis, and is located at P0 ðR0 ; y0 ; zÞ in the horizontal plane (see Fig. 1). The magnitude of the magnetic field is assumed to be in the form of B ¼ B0 ð1 þ AR0 Þ with R0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðR sin y R0 sin y0 Þ2 þ ðR cos y R0 cos y0 Þ2 ; ð19Þ where A is the coefficient measuring the nonuniformity of the magnetic field, B0 is the magnitude of the applied magnetic field at the field center. R0 is the distance of a point PðR; y; zÞ in the r j plane to the center of the magnetic field at P0 ; R is the distance of the point P to the center of the circular plane Oð0; 0; zÞ; and R0 is the distance of the point P to point O.
3. Numerical technique The commercial CFX software is used to solve the governing equations of the model. The computation mesh in the liquid is 120 40 80 in the r; j; and z directions, respectively, which is demonstrated to be sufficient for an accurate and stable solution. Since the focus was on the flow field only, the evolution of the growth and dissolution interfaces was not included in our computations. The mass transport equation in the solution was solved simultaneously
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Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
in order to take into account the influence of concentration field on the flow field (through solutal Grashof number); however, the concentration field is not presented in this article. Transient terms are considered in the heat, mass and momentum transfer equations in order to account for possible unsteady flows and their influence. The simulation results are presented at t ¼ 1:0 h since certainly the flow field has fully developed by that time.
4. Results and discussion The physical properties and operating parameters of the binary GaAs system are the same as those used in [51,52]. The physical and geometrical data used are summarized in Table 1 for the sake of completeness. Values of the dimensionless numbers, namely the thermal Grashof number GrT ; the solutal Grashof number GrC ; the Prandtl number Pr; and the Hartmann number Ha; i.e., bt gDTH 3 b gDCH 3 ; GrC ¼ c 2 ; 2 n n rffiffiffiffiffi n s ; Pr ¼ ; Ha ¼ BH a rn GrT ¼
Table 1 Physical and growth parameters of the LPEE GaAs system Parameter
Symbol
Value
Growth temperature Electrical current density Peltier coefficient Electrical conductivity Thermal conductivity of the solution Thermal diffusivity Solutal diffusivity Solutal viscosity Solution density Solutal mobility Thermal expansion coefficient Solutal expansion coefficient Crystal radius Solution height Magnitude of magnetic field Non-uniformity coefficient Thermal Grashof number Solutal Grashof number Hartmann number Prandtl number
Tg J
1073 K 3 104 A/m2
p s kl
0.3 V 2.5 106 O–1 m1 52.6 W/mK
a D n r m bt
0.3 104 m2/s 4 109 m2/s 1.21 107 m2/s 5.63 103 kg/m3 0.027 104 m2/V s 9.85 105 K1
bc
84.0 kg/m3
Rc H B
0.0120 m 0.0103 m 0–12 kG
A
0–4
GrT
6.84 104
GrC Ha Pr
7.10 103 0–871.5 4 103
ð20Þ 4.1. Effect of applied magnetic field strength
are also given in the table. It must be mentioned that all our simulations are carried out in half of the cylindrical cell for computational efficiency. However, to make sure that the half domain solution represents fully the flow structure of the full domain, a full domain solution was carried out for a case that asymmetry was assured (under a large magnetic field, B ¼ 4 kG). Results demonstrated that the halfplane treatment is reliable. Computed temperature distributions in the horizontal plane near the growth interface and also in the vertical plane at j ¼ 0 are given in Fig. 2. Temperature distributions agree with our earlier 2D solutions. Since electric current is passing through the source material the structure of isotherms indicate that the shape of the growth interface will be single-humped (concave towards the crystal) as expected [51,52].
To have a better sense for the flow field in the growth cell, the computed simulation results are presented in three distinct planes, the vertical (r z) plane at j ¼ 0 that represents typical flow structures along the growth direction, the horizontal (r j) plane at the middle of the growth cell (z ¼ 5:15 mm) where the changes in the flow field will be more prominent for radial and circumferential velocity components representing mixing, and finally the horizontal (r j) plane near the growth interface (z ¼ 0:4 mm) where the flow field is often closely related to the quality of the grown crystals. Fig. 3 summarizes the computed results for the flow field that presented in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) for four levels of magnetic field strengths (B ¼ 0:0; 0.5, 1.0,
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
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Fig. 2. Temperature distribution in K: (a) near the growth interface in the horizontal plane, (b) in the vertical plane at j ¼ 0:
Fig. 3. Flow field in the horizontal plane in the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) at: (a) B ¼ 0:0 kG, (b) B ¼ 0:5 kG, (c) B ¼ 1:0 kG, and (d) B ¼ 2:0 kG.
and 2.0 kG), while Fig. 4 show the results for the flow field in the horizontal plane near the growth interface (left column) and in the vertical plane at j ¼ 0 (right column) for two levels of magnetic field intensities (B ¼ 2:5 and 3.0 kG), and in the
horizontal plane at the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) for B ¼ 4:0 and 8:0 kG. Flow intensities are computed in m/s, and the scales of flow strengths are shown in each figure.
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Fig. 4. Flow field in the horizontal plane near the growth interface at z ¼ 0:4 mm (left column) and in the vertical plane at j ¼ 0 (right column) for (a) B ¼ 2:5 kG, and (b) B ¼ 3:0 kG, and in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane (right column) for (c) B ¼ 4:0 kG, and (d) B ¼ 8:0 kG.
Results shown for the vertical plane along the growth direction appear more complex than those of the 2D results of [51,52] given for the same growth system (Fig. 3a). This is assumed to be mainly due to the inclusion of the contribution of the circumferential velocity component. In addition, the size of the growth crucible used in this study is twice the one used in [47,48]; this might have also contributed towards these differences. Such differences in the flow field of 2D and 3D models have also been observed in [20,31,46]. The strongest flow is seen in the lower part of the crucible cell, near the centre of the half domain along the r-direction. This is similar to what was observed in Ref. [20]. A weak flow cell forms just
above the strongest flow cell. The flow in the horizontal planes is nearly homocentric. The flow in the horizontal plane near the growth interface is stronger in the middle region along the radial direction and becomes weaker and weaker near to the growth cell wall or to the axial center (Fig. 3a—left column). On the other hand, in the horizontal plane at the middle of the growth cell (Fig. 3a—left column) there are two maximum points for the flow intensity along the radial direction, with a quite strong flow in the central region. Notice that the flow field in Fig. 3a is not strictly axisymmetric, especially in the middle of the growth cell, although the flow is stable. This result suggests that our system is near the
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
axisymmetry breaking of the flow [53], although the Grashof number is a little bit lower than the critical Grashof number (2.5 105) under the same conditions (Prandtl number and geometry aspect ratio) given in Ref. [53] in the analysis of axisymmetry breaking of natural convection in a vertical Bridgman growth configuration. Figs. 3b–d and Figs. 4a and b show the influence of the applied magnetic field on the flow field. Results are also summarized in Figs. 5 and 6 for various aspects of the flow field. Simulation results show three distinct characteristics depending on the level of applied magnetic field: (a) the ‘‘weak’’ magnetic field, with intensities from 0.0 to 2.0 kG,
21
(b) the ‘‘intermediate’’ magnetic field, with levels from 2.0 to 3.0 kG, and (c) the ‘‘high’’ magnetic field, with intensities above 3.0 kG. Flow characteristics are quite different at each of these field levels. Let us first focus on Figs. 3b–d which represent results for magnetic field levels from 0.5 to 2.0 kG. In this category, the weak magnetic field level, and an increase in the applied magnetic field intensity, results in significant reduction in the flow intensity, which is, in general, desirable for a stable and controlled crystal growth. The flow cells are strongest near the vertical wall and forms the socalled Hartmann layer [25,45–47]. At higher
Fig. 5. Variation of the flow strength U ¼ ðu2 þ v2 þ w2 Þ1=2 along the radial direction at j ¼ p=2 under various magnetic field strengths: (a) B ¼ 0:0 kG, (b) B ¼ 1:0 kG, (c) B ¼ 2:0 kG, and (d) B ¼ 3:0 kG. Solid lines at z ¼ 5:05 mm (in the middle of the growth cell), and dashed lines at z ¼ 0:4 mm (near the growth interface).
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Fig. 6. Variation of the maximum velocity with magnetic field intensity: three distinct regions of stability of the flow field are obvious. The flow field is stable up to Ha ¼ 150; the region between Ha ¼ 150 and 220 the flow is transitional, and above Ha ¼ 220; the flow is unstable.
magnetic field strengths, the relative intensity of these flow cells becomes further stronger compared with the flow in the rest of the growth cell domain, and the Hartmann layer becomes thinner which is in accordance with the scaling analysis in Ref. [31]. Furthermore, the strongest flow cells appearing in the lower part of the growth cell (Fig. 3a) move further down towards the growth interface with increasing intensity, and new flow cells form near the dissolution interface. These strong flow cells form very visible boundary layers near the growth and dissolution interfaces (the so-called ‘‘end layers’’ [31,47]), and hence give rise to strong vertical velocity gradients in the vicinity of the growth interface, which may have an adverse effect on the crystal growth process. One may conclude that the application of a magnetic field may not always be beneficial for the growth process [25,30]. It was shown in Ref. [25] that the radial nonuniformity in vertical Bridgman is the most significant at the intermediate levels of magnetic field strength. Flow fields become perfectly homocentric (and hence axisymmetric) with the increase of magnetic
field strength, and hence the flow intensity decreases. The flow velocities in the central region of the cell (both in the r and z directions) become more uniform at higher magnetic field levels, and form a core region of uniform flow field [25,45], leading to a domain in which the flow is suppressed with the application of an applied magnetic field. Indeed the applied magnetic field at the weak intensity levels can suppress fluid flows in a growth system, as shown in Fig. 3. It must be mentioned here that a magnetic field level higher than 8.0 kG is very strong for an LPEE application. It is not possible for us in this study to present any correlation (or a comparison) between the present numerical results and our ongoing experiments. Only one thing can be said, though, that thus far it was not possible to exceed the level of 4.5 kG in experiments. It led to unsatisfactory growth due to strong convection in the liquid zones. However, the growth rate was increased about four times under a magnetic field of 4.5 kG (experimental results will be reported soon). Fig. 4 represents the simulation results for the intermediate magnetic intensity levels selected as B ¼ 2:5; 3.0, 4.0, and 8.0 kG. In this case, flow intensity increases with magnetic field strength. This result has not been widely reported in the relevant literature; however, it is reasonable due to the following reasoning. The magnetic body force and the body force due to buoyancy (maybe others such as surface tension) acting on the points of the liquid, although act in different directions, balance each other in a close container (growth cell for instance). Up to a certain level of magnetic body force, the magnetic body force suppresses the fluid flow by counterbalancing the buoyancy force. However, when the level of magnetic field exceeds a certain value, it surpasses the buoyancy force and becomes an excessive force, feeding the convection in the liquid. Thus, the flow strength increases further. Some experimental and numerical studies indeed demonstrated the enhancement of heat transfer (and hence flow strength) in a melt under a stationary magnetic field [48,49,54]. As can be seen from Fig. 4, the flow patterns show dramatic changes, two flow cells were formed in each half of the vertical plane, with
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
the upper cells getting larger and lower cells getting smaller with the increasing magnetic field intensity. In the vertical plane, some strong unidirectional flows appear in the middle region along the r-direction, and some with very weak intensity in the middle forming a small cylindrical region where the flow is nearly stationary. The flow fields are no longer axisymmetric and homocentric. Such an axisymmetry breaking may be caused by the unsteadiness of the flow field according to Ref. [53]. Although it is not presented here, the flow stability was also examined (at B ¼ 2:5 and 3.0 kG levels) comparing the results at different times, and found that the flow field is essentially stable in spite of some small changes with time. It is then speculated that the reason of the axisymmetry breaking of the flow field at these magnetic field levels may be due to a very delicate balance between the buoyancy and magnetic forces even though the flow still remains stable. The variation of the flow strength along the radial direction for B ¼ 0:0; 1.0, 2.0, and 3.0 kG levels were shown in Fig. 5. Solid and dashed lines represent, respectively, the values at the middle and the lower regions of the growth cell. As can be seen, the flow strength fluctuates and shows differences in these regions. However, at higher magnetic field levels, the difference becomes less obvious. The variation of the flow strength in the radial direction near the growth interface is almost symmetric in the absence of a magnetic field. However, this symmetry is broken with the increasing magnetic field intensity. Although higher magnetic field intensity levels, higher than 4.5 kG, appears to be not practical for the LPEE growth process, to the best of our knowledge at the moment of course, for the sake of completeness of the present numerical study and also for the purpose of comparison with other studies, higher magnetic field intensity levels were also examined. The levels of B ¼ 4:0 kG and B ¼ 8:0 kG were selected. The flow patterns become dramatically different and show large fluctuations with time, and temperature distributions show asymmetric behavior. These results are not presented here for the sake of space. However, the maximum velocity values
23
are presented in Fig. 6 for all magnetic field levels. In Fig. 6, the three regions of flow stability are clearly presented. The flow velocity decreases with increasing magnetic field in the stable region (up to Ha ¼ 150), but in the intermediate and unstable regions (between Ha ¼ 150 and 220, and above Ha ¼ 220; respectively) the flow velocity increases. If one examines the logarithmic plot of the maximum velocity (Umax ) of the flow field as a function of the Hartmann number, given in Fig. 6, can see that within the stable region, the relationship between the flow intensity and the Hartmann number obeys a power law of Umax pHa5=4 ;
ð21Þ
which has been demonstrated by many authors [25,31,45–47], although the index of the power law is slightly different due to different system parameters and conditions of the present study. In the unstable region, this relationship becomes Umax pHa5=2 :
ð22Þ
In the transitional (intermediate) region, on the other hand, as one expects, the change of the maximum velocity (Umax ) of the flow field with the Hartmann number is so dramatic that the power law is not suitable. It is important to point out that the results given here for the intermediate and unstable regions are new and, to the best of our knowledge, were not encountered in the literature. 4.2. Influence of magnetic field non-uniformity As mentioned in Section 1, in order to see the effect of field non-uniformities, certain non-uniformity levels were incorporated into the simulation model. Selecting y0 ¼ 0 and R0 ¼ 0:008 m, which, together with the crystal radius of R ¼ 0:012 m, gives the maximum difference of the magnetic field as 2A% (see Eq. (19), for instance A ¼ 1:0 and 3.0, correspond to the nonuniformity levels of 2.0% and 6.0%, respectively. These non-uniformity levels were examined for the magnetic field level of B ¼ 1:0 kG. The results of numerical simulations were presented in Fig. 7.
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Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
Fig. 7. Flow field under B ¼ 1:0 kG and three different non-uniformity values: (a) A ¼ 0; (b) A ¼ 1:0; and (c) A ¼ 3:0: The left column describes the flow patterns in the horizontal plane near the growth interface at z ¼ 0:4 mm, and the right column shows the flow patterns in the vertical plane at j ¼ 0:
Results show that even for a 6% the nonuniformity changes in the flow field are small, indicating that the natural convection becomes weaker under the application of an external magnetic field, and the influence of such nonuniformities at this level of external magnetic field is not significant. However, the flow field becomes asymmetric with increasing non-uniformity (see Fig. 7c). As one expects, at higher levels of applied magnetic field such non-uniformities may have significant effects on the flow field; however, due to the practical insignificance, the simulation results conducted for B ¼ 8:0 kG are not presented here.
5. Conclusion A 3D numerical simulation model for the liquid phase electroepitaxial (LPEE) growth of GaAs
under a vertical stationary magnetic field was presented. Computations were carried out using the commercial CFX software package. The effect of magnetic field intensity and non-uniformity on the flow field in the liquid solution was investigated. Numerical results show that the flow patterns exhibit three distinct stability characteristics: a stable flow field up to a magnetic field level of Ha ¼ 150; a transitional flow between Ha ¼ 150 and 220; and an unstable flow above Ha ¼ 220: In the stable region, the applied magnetic field suppresses the flow field, and the flow intensity decreases with increasing magnetic field exhibiting a power law Umax pHa5=4 relationship for the maximum velocity (Umax ). The flow in the horizontal plane is perfectly homocentric, and the flow in the central region of the growth cell becomes more uniform and weaker with increasing magnetic field levels. Boundary layers appear in
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
the vicinity of the growth and dissolution interfaces, and a Hartmann layer forms along the cell wall and becomes thinner with the increasing magnetic field strength. In the transitional region, the flow intensity increases dramatically with the increase in magnetic field strength. The flow patterns are significantly different than those in the stable region. The flow field is no longer axisymmetric but still stable. In the unstable region, the flow structure and intensity change with time. Under a strong magnetic field, the flows cells are confined to the vicinity of the vertical wall and exhibit significant non-uniformity near the growth interface. Such strong flow fluctuations and non-uniformities near the growth interface may have an adverse effect on the growth process and lead to an unsatisfactory growth. In this region, the maximum velocity (Umax ) obeys approximately a power law of Umax pHa5=2 : Results show that for a successful growth the effect of applied magnetic field must be optimized. The numerical study of the effect of nonuniformity in the applied magnetic field shows that at low magnetic field intensities (less then 150 Ha), the influence of non-uniformities on the flow field is insignificant. Under a strong magnetic field, however, such non-uniformities may become significant and affect the growth process adversely. A comparison with 2D simulation results for the same growth system shows that the flow patterns of the 3D model seem much complex. Observations in the flow structures at high magnetic field strengths could not be observed through the 2D models, partially due to the omission of the velocity and magnetic force components along the circumferential direction. Acknowledgements The financial support provided by the Canadian Space Agency Microgravity Science Program is gratefully acknowledged. References [1] L. Jastrzebski, H.C. Gatos, A.F. Witt, J. Electrochem. Soc. 123 (1976) 1121.
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[2] L. Jastrzebski, Y. Imamura, H.C. Gatos, J. Electrochem. Soc. 125 (1978) 1140. [3] A. Okamoto, L. Lakowski, H.C. Gatos, J. Appl. Phys. 53 (1982) 1706. [4] T. Bryskiewicz, C.F. Boucher Jr., J. Lagowski, H.C. Gatos, J. Crystal Growth 82 (1987) 279. [5] T. Bryskiewicz, P. Edelman, Z. Wasilewski, D. Coulas, J. Noad, J. Appl. Phys. 68 (1990) 3018. [6] T. Bryskiewicz, A. Laferriere, J. Crystal Growth 129 (1993) 429. [7] K. Nakajima, J. Appl. Phys. 61 (9) (1987) 4626. [8] S. Dost, Z. Qin, Int. J. Electromagn. Mech. 7 (2) (1996) 109. [9] Z.R. Zytkiewicz, J. Crystal Growth 172 (1996) 259. [10] K. Nakajima, J. Crystal Growth 98 (1989) 329. [11] K. Nakajima, T. Kusunoki, C. Takenaka, J. Crystal Growth 113 (1991) 485. [12] Z.R. Zytkiewicz, J. Crystal Growth 131 (1993) 426. [13] S. Dost, H. Sheibani, in: J.S. Yang, G.A. Maugin (Eds.), Mechanics of Electromagnetic Materials and Structures in Studies, IOS Press, Amsterdam, Appl. Electr. Mech. 19 (2000) 17. [14] S. Dost, Z. Qin, J. Crystal Growth 153 (1995) 123. [15] Z. Qin, S. Dost, N. Djilali, B. Tabarrok, J. Crystal Growth 153 (1995) 131. [16] S. Dost, Appl. Mech. Rev. 49 (12) (1996) 477. [17] Z. Qin, S. Dost, Int. J. Electromagn. Mech. 7 (2) (1996) 129. [18] S. Dost, Z. Qin, J. Crystal Growth 187 (1998) 51. [19] S. Dost, ARI—Bull. ITU 51 (1999) 235. [20] S. Dost, Y. Liu, B. Lent, J. Crystal Growth 240 (2002) 39. [21] L. Davoust, M.D. Cowley, R. Moreau, R. Bolcato, J. Fluid Mech. 400 (1999) 59. [22] Y.C. Liu, W.C. Chen, C.R. Huo, P.W. Ge, Acta Phys. Sin. 8 (1999) 881. [23] R.W. Series, D.T.J. Hurle, J. Crystal Growth 113 (1991) 305. [24] D.T.J. Hurle (Ed.), Handbook of Crystal Growth 2: Bulk Crystal Growth, Part B: Growth Mechanisms and Dynamics, North-Holland, Amsterdam, 1994. [25] D.H. Kim, P.M. Adornato, R.A. Brown, J. Crystal Growth 89 (1988) 339. [26] S. Motakef, J. Crystal Growth 104 (1990) 833. [27] C.K. Ghaddar, C.K. Lee, S. Motakef, D.C. Gillies, J. Crystal Growth 205 (1999) 97. [28] H. Ozoe, M. Iwamoto, J. Crystal Growth 142 (1994) 236. [29] V. Galindo, G. Gerbeth, W. von Ammon, E. Tomzig, J. Virbulis, Energy Convers. Manage. 43 (2002) 309. [30] S. Dost, Y.C. Liu, B. Lent, R.F. Redden, A numerical simulation study for the effect of applied magnetic field in growth of CdTe single crystals by the traveling heater method. Int. J. Electromagn. Mech., submitted for publication. [31] H. Ben Hadid, D. Henry, J. Fluid Mech. 333 (1996) 23. [32] S. Yesilyurt, L. Vjusic, S. Motakef, F.R. Szofran, A. Croell, J. Crystal Growth 211 (2000) 360. [33] Y.Y. Khine, J.S. Walker, F.R. Szofran, J. Crystal Growth 212 (2000) 584.
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[34] S. Yesilyurt, L. Vjusic, S. Motakef, F.R. Szofran, M.P. Volz, J. Crystal Growth 207 (1999) 278. [35] J.A. Shercliff, J. Fluid Mech. 91 (1979) 231. [36] M. Oshima, N. Taniguchi, T. Kobayashi, J. Crystal Growth 137 (1994) 48. [37] K. Kakimoto, K.W. Yi, M. Eguchi, J. Crystal Growth 163 (3) (1996) 238. [38] D. Vizman, J. Friedrich, G. Muller, J. Crystal Growth 230 (2001) 73. [39] H. Hirata, K. Hoshikawa, J. Crystal Growth 125 (1992) 181. [40] C.J. Jing, N. Imaishi, S. Yasuhiro, T. Sato, Y. Miyazawa, Int. J. Heat Mass Transfer 43 (2000) 4347. [41] M. Akamatsu, M. Higano, H. Ozoe, Int. J. Heat Mass Transfer 44 (2001) 3253. [42] H. Ozoe, K. Okada, Int. J. Heat Mass Transfer 32 (1989) 1939. [43] J. Baumgartl, G. Muller, Calculation of the effects of magnetic field damping on fluid flow—comparison of magnetohydrodynamic models of different complexity, Proceedings of the VIIIth European Symposium on
[44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
Materials and Fluid Sciences in Microgravity, Noordwijk, The Netherlands, 1992, pp. 161–164. J. Baumgartl, A. Hubert, G. Muller, Phys. Fluids A 5 (1993) 3280. H. Ben Hadid, D. Henry, J. Fluid Mech. 333 (1996) 57. H. Ben Hadid, S. Vaux, S. Kaddeche, J. Crystal Growth 230 (2001) 57. L. Davoust, M.D. Cowley, R. Moreau, R. Bolcato, J. Fluid Mech. 400 (1999) 59. T. Tagawa, H. Ozoe, J. Heat Transfer 119 (1997) 265. T. Tagawa, H. Ozoe, J. Heat Transfer 120 (1998) 1027. S. Dost, H.A. Erbay, Int. J. Eng. Sci. 33 (1995) 1385. N. Djilali, Z. Qin, S. Dost, J. Crystal Growth 149 (1995) 153. Z. Qin, S. Dost, N. Djilali, B. Tabarrok, Int. J. Numer. Methods Eng. 38 (1995) 3949. A.Yu. Gelfgat, P.Z. Bar-Yoseph, S. Solan, J. Crystal Growth 220 (2000) 316. K. Terashima, J. Nishio, S. Washizuka, M. Watanabe, J. Crystal Growth 84 (1987) 247.
The effect of applied magnetic field on flow structures in liquid phase electroepitaxy—a three-dimensional simulation model Y.C. Liua, Y. Okanob, S. Dosta,* a
Crystal Growth Laboratory, Rel. Tech. Department of Mechanical Engineering, Center for Advanced Materials, University of Victoria, P.O. Box 3055, Victoria, BC, Canada V8W 3P6 b Department of Materials Science and Chemical Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, 432-8561, Japan Received 15 April 2002; accepted 26 June 2002 Communicated by K. Nakajima
Abstract A three-dimensional numerical simulation for the liquid phase electroepitaxial growth of GaAs under a vertical stationary magnetic field was carried out. The effect of magnetic field intensity and non-uniformity on the flow field in the liquid solution was investigated. Numerical results show that the flow patterns exhibit three distinct stability characteristics: a stable flow field up to a magnetic field level of Ha ¼ 150; a transitional flow between Ha ¼ 150 and 220; and an unstable flow above Ha ¼ 220: In the stable region, the applied magnetic field suppresses the flow field, and the flow intensity decreases with increasing magnetic field exhibiting a power law of Umax pHa5=4 relationship for the maximum velocity (Umax ). In the transitional region, the flow intensity increases dramatically with the increase in magnetic field strength. The flow patterns are significantly different from those in the stable region. The flow field is no longer axisymmetric but still stable. In the unstable region, the flow structure and intensity change with time. Under a strong magnetic field, the flow cells are confined to the vicinity of the vertical wall and exhibit significant non-uniformity near the growth interface. Such strong flow fluctuations and non-uniformities near the growth interface may have an adverse effect on the growth process and lead to an unsatisfactory growth. In this region, the maximum velocity (Umax ) obeys approximately a power law Umax pHa5=2 : Results show that for a successful growth the effect of applied magnetic field must be optimized. r 2002 Elsevier Science B.V. All rights reserved. Keywords: A1. Computer simulation; A1. Magnetic fields; A2. Growth from solutions; A3. Electroepitaxy; B2. Semiconducting materials
1. Introduction Liquid phase electroepitaxy (LPEE), being a solution growth, has a number of advantages over *Corresponding author. Tel.: +1-250-721-8900; fax: +1250-721-6051. E-mail address: [email protected] (S. Dost).
other bulk crystal growth techniques. For instance, the relatively low temperatures used in LPEE, and its ability to provide well-controlled growth of crystals with the desired properties, make this technique technologically very promising (see for instance Refs. [1–9]). LPEE has a great potential to become a commercial technique in growing high quality, bulk crystals; particularly
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 2 ) 0 1 6 0 3 - 2
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
ternary crystals such as GaInAs, GaInSb, and CdZnTe (see Refs. [10–13]). In LPEE, growth is achieved by passing an electric current through the growth cell at a constant furnace temperature. The applied electric current is the sole driving force for growth, and gives rise to two growth mechanisms known as ‘‘electromigration’ and ‘‘Peltier cooling/heating’’. The electromigration of species in the solution takes place due to electron-momentum exchange and electrostatic field forces, and sustains a controlled growth [1,2]. The Peltier heating/cooling, on the other hand, is a thermoelectric effect occurring when an electric current passes through an interface of two materials with different Peltier coefficients. The Peltier cooling at the growth interface supersaturates the solution in the immediate vicinity of the substrate and leads to epitaxial growth. The Peltier heating at the dissolution interface, on the other hand, causes the dissolution of the source material into the solution and provides constantly the needed feed material for growth. The growth rate is proportional to the applied electric current. Although the furnace temperature is constant, the combined effect of the Joule heating in the solid crystals, and the Peltier heating/cooling at the interfaces leads to small temperature gradients in the solution. Such temperature gradients together with concentration gradients result in a significant natural convection in the solution. This convective flow in the solution has an adverse effect on the LPEE growth process [14]. It leads to interface instability and consequently stops the growth. Recently, we were interested in the LPEE growth of single crystals under a static applied magnetic field [8,13–20]. The objective was to minimize the effect of convection in the solution. Lower convection in the solution will allow the use of higher electric current densities that will be translated into higher growth rates. The theoretical model, and results of twodimensional (2D) numerical simulations for LPEE under an applied magnetic field can be found in Refs. [14,15,19,20]. Simulation results have shown that the level of applied magnetic field has a significant effect on the intensity and structure of the convective flow in the solution. As expected,
13
stronger applied magnetic field leads to weaker natural convection in the solution. The shape of the growth interface becomes more uniform at higher magnetic field levels. Our previous 2D numerical simulations have shed light on various aspects of the applied magnetic field in LPEE. However, these earlier models, being 2D, have naturally neglected the contributions of the circumferential velocity and magnetic body force components. Therefore, the results obtained earlier were qualitative and did not include information about three-dimensional (3D) effects such as deviations from axisymmetry, variations of the magnetic field components, and mixing in the solution. In order to shed light on these 3D effects, in this study we present a 3D model and numerical simulation results for the LPEE growth of a binary system. To the best of our knowledge, there is no 3D model developed previously for the LPEE. The flow field in the solution, which is often related to phenomena such as macro-segregation and striations, and hydrodynamic instabilities, plays an important role in a crystal growth process (see, e.g. Refs. [21,22]). The application of an external magnetic field to suppress convection has been found to be very attainable and efficient. The applied magnetic field could be static or rotating, and vertical or transverse. A detailed account of the subject and related literature can be found in Refs. [23,24]. Here, only a very brief review is given. Most of the 2D models developed thus far to study the effect of an applied magnetic field in crystal growth (see, e.g. Refs. [14,15,19,20,25–30]) have focused on the suppression of flow fluctuations and concentration uniformity. An important related work with emphasis on the flow structures and intensity, was given in Ref. [31]. The results of the 2D numerical study of a horizontal Bridgman configuration have shown that higher applied magnetic field intensity levels (higher Hartmann numbers (Ha), a measure of the applied magnetic field strength) do not only result in further suppression of the convective flow in the liquid, but also lead to a progressive change of the overall structure of the flow followed by the appearance of Hartmann layers in the vicinity of the rigid walls.
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They have also demonstrated that when Ha is large enough (Ha > 10) the decrease of the strength of the convective flow asymptotically approaches a power-law dependence in Ha; and the flow in the central region becomes more and more unidirectional. Some recent studies [32–34] have focused on the influence of thermal-electromagnetic convection (TEMC) [35] that is generated by an induced magnetic force measured by the product of the electromotive force and the local temperature gradient in the presence of an applied magnetic field. As mentioned in these studies, the TEMC may become significant under certain conditions, for instance, when the solution and solid have different thermoelectric power and the growth interface is not isothermal (e.g. GeSi and HgCdTe), or when the growth interface is faceted where the local temperature gradient is large due to interface undercooling, or when the thermoelectric power in solution is a function of a non-uniform liquid composition. Note that most of the TEMC studies are related to reduced gravity or microgravity conditions [34] under which the natural convection is weaker. To the best of our knowledge, the literature on the 3D modeling of crystal growth under an applied magnetic field is not so rich. We would like to cite here some of the related 3D studies. For instance, 3-D models have been developed for the Czochralski growth process to examine the oxygen transport and fluid flow in silicon melt under vertical [35–37], horizontal [36,38], and cusp [38,39] magnetic fields. Numerical simulations for the LiNbO3 melt flow under a uniform vertical magnetic field were carried out in Ref. [40]. Simulations showed that the spoke pattern flow that appeared at azimuthal planes was due to the Marangoni instability, and the rotating motion was driven by the magnetic body force that was induced by the interaction between the vertical magnetic field and the radial motion [23,24]. A study for temperature contours under a transverse magnetic field for a Czochralski system was given in Ref. [41], where a quite peculiar convection pattern under a certain magnetic field was obtained; velocity components were all downwards in a vertical cross section parallel to the magnetic field, while those in a plane perpendi-
cular to the magnetic field were natural convection dominant, which indicates the possible complexity of flow field and its dependence on certain growth conditions in the presence of a magnetic field. Other 3D simulation studies, under the presence of a magnetic field and often without a particular growth system, mainly focus on the flow field in a certain geometrical system. The natural convection in a differentially heated cubical box with three different orientations of the magnetic field along the coordinate axes was investigated numerically in Ref. [42]. It was found that the applied magnetic field suppressed the natural convection most efficiently when the magnetic field is imposed perpendicular to the heated vertical walls. Numerical results in a cylindrical geometry under a constant magnetic field were compared with experiments in Ref. [43], and used towards the improvement of the numerical model introduced for better predictions. The numerical simulation of the flow of liquid gallium melt in a cylindrical container in Ref. [44] showed varying temperature fluctuations of nearly sinusoidal form when the temperature difference is 0.65 K and the applied vertical magnetic field is 20 G. Results were verified by experiments. The suppression of such temperature fluctuations by increasing the magnetic intensity was also demonstrated. Furthermore, through numerical simulation, they found evidence that the induced magnetic field follows the oscillatory convection and temperature variation. The effect of a constant magnetic field on electrically conducting liquid-metal flows has been numerically (in three dimensions) investigated in Ref. [45]. Results have again, similar to their 2D analysis [31], showed that when Ha reaches a certain critical value, which is found to depend on the direction of the applied magnetic field, the decrease of the flow intensity takes on an asymptotic power-law form, and the flow patterns have significant changes. They have separated the flow structure into three regions: the core flow, Hartman layers which develop in the immediate vicinity of the rigid horizontal boundaries or at the end walls, and parallel layers appearing in the vicinity of the sidewalls. Flow transitions in isothermal metallic liquids confined in cylindrical cavities under a rotating magnetic field were also
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
simulated in Ref. [46]. A parametric study for the change of aspect ratio of the geometry, Hartman number, and rotating Reynolds number was provided. For the sake of brevity, a review of experimental investigations on the effect of an external magnetic field on flow patterns and crystal growth processes is not given here. The reader is referred to Refs. [23,24,47]. It is however important to point out that both experimental and numerical studies carried out to study the effect of a magnetic field in crystal growth are often related to unsteady convection, especially when free surfaces exist in the growth system, since one of the advantages of applying a magnetic field is to suppress such unsteady flows. However, if the magnetic body force becomes much larger than other flow driving body forces (for instance due to buoyancy and surface tension), it will likely result in a stronger convection than the one with no magnetic field effect. Some studies have indeed demonstrated this point [48,49]. Our experimental interest is focused on the LPEE growth of a ternary In0.04Ga0.96As system under a static vertical magnetic field. In the present article, however, since the emphasis is on the variations of flow patterns and intensities, the present 3D simulation study is carried out for the growth of a binary system (GaAs) for computational simplicity. Simulation results are compared with other studies. The effect of magnetic field non-uniformity is also investigated.
2. The simulation model The LPEE growth process under an applied magnetic field can be influenced by many factors, such as fluid flow, heat and mass transfer, electric and magnetic fields, various thermoelectric effects and their interactions in the solution, and heat and electric conduction with various thermoelectric effects in the solid. For a typical schematic LPEE growth configuration, see Fig. 1 and Refs. [14,20]. The development of the fundamental equations of the LPEE growth process for binary and ternary systems can be found in Refs. [8,14,19,50].
15
2.1. The governing equations 3D time-dependent governing equations describing the fluid flow, heat and mass transport in the liquid solution (binary) are given as follows: Continuity: 1q 1 qv qw ðruÞ þ þ ¼ 0: ð1Þ r qr r qj qz Momentum: qu qu v qu qu v2 þu þ þw qt qr r qj qz r u 2 qv 1 qp FrM 2 þ ; ¼n r u 2 2 r r qj r qr r qv qv v qv qv uv þu þ þw þ qt qr r qj qz r v 2 qv 1 qp FjM þ ; ¼ n r2 v 2 þ 2 r r qj r qj r qw qw v qw qw þu þ þw qt qr r qj qz 1 qp FzM þ ¼ nr2 w r qz r gbt ðT T0 Þ þ gbc ðC C0 Þ: Mass transport: qC qC v qC qC þu þ þw qt qr r qj qz qC qC þ mEz ¼ Dr2 C: þ mEr qr qz Energy: qT qT v qT qT þu þ þw ¼ ar2 T qt qr r qj qz
ð2Þ
ð3Þ
ð4Þ
ð5Þ
ð6Þ
with the Laplacian operator 1q q 1 q2 q2 r r2 ¼ þ 2 2 þ 2; r qr qr r qj qz where u; v and w are, respectively, the velocity components in the radial ðrÞ; circumferential (j) and vertical (z) directions. n is the kinematic viscosity, p is the pressure, T is the temperature, C is the arsenic (As) concentration in the Ga–As solution, r is the solution density, bt and bc denote, respectively, the thermal and solutal expansion coefficients, T0 and C0 represent the
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
16
Fig. 1. Schematic view of the LPEE growth crucible, and the coordinate system.
reference temperature and concentration, respectively, g is the gravitational constant, a is the thermal diffusivity. m and D denote, respectively, the solute mobility and the diffusion coefficient. FrB ; FzB ; and FjB are the magnetic body force components, respectively, along the r; z and j directions. These magnetic force components are introduced in detail in the next section. Er and Ez are the electric field intensities in the r and z directions, which are obtained by solving an axisymmetric quasi-steady electric field (f) equation: q2 f 1 qf q2 f þ ¼0 þ qr2 r qr qz qf qf with Er ¼ ; Ez ¼ : qr qz
ð7Þ
In the derivation of the governing equations it was assumed that the solution is incompressible and Newtonian, and the well-known Boussinesq approximation—assuming a constant fluid density in all equations except in the body force term due to buoyancy—holds. The boundary conditions associated with the governing Eqs. (1)–(7) are as follows: Along the vertical wall: u ¼ 0; v ¼ 0; w ¼ 0; z z qf qC 0 T ¼ Tg ¼ 0; ¼ 0: DT; qr qr H
ð8Þ
Along the growth interface: u ¼ 0; v ¼ 0; w ¼ 0; qT qT kl ¼ pJ; ks qz qz C ¼ C1 :
s
qf ¼ J; qn ð9Þ
Along the dissolution interface: u ¼ 0; v ¼ 0; w ¼ 0; qT qT ks ¼ þpJ; f ¼ 0; C1 ¼ C2 : kl qz qz
ð10Þ
In the above equations p is the Peltier coefficient, J is the electric current density, s is the electrical conductivity of the liquid, H is the height of the solution zone along the vertical axis, ks and kl are the thermal conductivities of the solid and liquid, respectively. Also, C1 and C2 are, respectively, the solute concentrations at the growth and dissolution interfaces, which are determined by the interfacial equilibrium conditions (see Ref. [51]). DT is used to denote the temperature drop due to the heat loss when heat is transferred from the outside wall of the furnace to the vertical wall of the cell. Its value is estimated by considering the heat transfer through the whole LPEE system that was computed in our previous articles [51,52]. It was assumed that the contribution of latent heat is negligible since the growth rate is very low.
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The initial conditions are C ¼ C0 ; u ¼ 0; v ¼ 0; w ¼ 0 and T ¼ Tg at t ¼ 0:
ð11Þ
2.2. Magnetic body force components The applied static magnetic field is considered to be perfectly aligned with the vertical axis of the growth system B ¼ Bez ;
ð12Þ
where B is the magnitude of the applied magnetic field. The magnetic body force acting on the points of the metallic liquid solution is then expressed by FM ¼ J B
ð13Þ
and the current density vector J is assumed to be given by Ohm’s law as follows: J ¼ sðE þ v BÞ;
ð14Þ
where E and v represent, respectively, the electrical field and velocity vectors acting on fluid points. The induced electric field due to the applied magnetic field and the fluid motion was neglected (see Ref. [24] for justification). The solution of Eq. (7) together with the corresponding boundary conditions lead to the following expression for E: qf J E ¼ E z ez ¼ ð15Þ ez ¼ ez : qz s Eq. (15) implies that the electrical field intensity is zero along the r and j directions, and is constant along the z-direction. The use of Eq. (15) in Eq. (14) leads to J J ¼ sðBver Buej þ ez Þ ð16Þ s and then the magnetic body force in Eq. (13) becomes M M FM ¼ FM r e r þ F j ej þ F z ez
¼ sB2 ðuer þ vej Þ;
ð17Þ
where the components of the magnetic body force, used in Eqs. (2)–(4), are then given by FrM ¼ sB2 u; FjM ¼ sB2 v; FzM ¼ 0:
ð18Þ
17
Deviation from axisymmetry: Our measurements show that the magnetic field distribution in the magnet opening where the growth crucible is to be located is almost uniform. However, when the growth system is lowered into the magnet opening, it is not possible to measure the field distribution, and it is likely that the field uniformity would be altered. Furthermore, the center of the growth system may not be aligned perfectly with the vertical axis due to possible unintentional experimental errors. This will result in further deviations from axisymmetry. This in turn may affect the growth adversely and lead to unsatisfactory growth. It was therefore desirable to investigate the influence of such asymmetries. In order to accommodate them in the model, it was assumed that the center of the magnetic field is offset with respect to the symmetry axis, and is located at P0 ðR0 ; y0 ; zÞ in the horizontal plane (see Fig. 1). The magnitude of the magnetic field is assumed to be in the form of B ¼ B0 ð1 þ AR0 Þ with R0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðR sin y R0 sin y0 Þ2 þ ðR cos y R0 cos y0 Þ2 ; ð19Þ where A is the coefficient measuring the nonuniformity of the magnetic field, B0 is the magnitude of the applied magnetic field at the field center. R0 is the distance of a point PðR; y; zÞ in the r j plane to the center of the magnetic field at P0 ; R is the distance of the point P to the center of the circular plane Oð0; 0; zÞ; and R0 is the distance of the point P to point O.
3. Numerical technique The commercial CFX software is used to solve the governing equations of the model. The computation mesh in the liquid is 120 40 80 in the r; j; and z directions, respectively, which is demonstrated to be sufficient for an accurate and stable solution. Since the focus was on the flow field only, the evolution of the growth and dissolution interfaces was not included in our computations. The mass transport equation in the solution was solved simultaneously
18
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
in order to take into account the influence of concentration field on the flow field (through solutal Grashof number); however, the concentration field is not presented in this article. Transient terms are considered in the heat, mass and momentum transfer equations in order to account for possible unsteady flows and their influence. The simulation results are presented at t ¼ 1:0 h since certainly the flow field has fully developed by that time.
4. Results and discussion The physical properties and operating parameters of the binary GaAs system are the same as those used in [51,52]. The physical and geometrical data used are summarized in Table 1 for the sake of completeness. Values of the dimensionless numbers, namely the thermal Grashof number GrT ; the solutal Grashof number GrC ; the Prandtl number Pr; and the Hartmann number Ha; i.e., bt gDTH 3 b gDCH 3 ; GrC ¼ c 2 ; 2 n n rffiffiffiffiffi n s ; Pr ¼ ; Ha ¼ BH a rn GrT ¼
Table 1 Physical and growth parameters of the LPEE GaAs system Parameter
Symbol
Value
Growth temperature Electrical current density Peltier coefficient Electrical conductivity Thermal conductivity of the solution Thermal diffusivity Solutal diffusivity Solutal viscosity Solution density Solutal mobility Thermal expansion coefficient Solutal expansion coefficient Crystal radius Solution height Magnitude of magnetic field Non-uniformity coefficient Thermal Grashof number Solutal Grashof number Hartmann number Prandtl number
Tg J
1073 K 3 104 A/m2
p s kl
0.3 V 2.5 106 O–1 m1 52.6 W/mK
a D n r m bt
0.3 104 m2/s 4 109 m2/s 1.21 107 m2/s 5.63 103 kg/m3 0.027 104 m2/V s 9.85 105 K1
bc
84.0 kg/m3
Rc H B
0.0120 m 0.0103 m 0–12 kG
A
0–4
GrT
6.84 104
GrC Ha Pr
7.10 103 0–871.5 4 103
ð20Þ 4.1. Effect of applied magnetic field strength
are also given in the table. It must be mentioned that all our simulations are carried out in half of the cylindrical cell for computational efficiency. However, to make sure that the half domain solution represents fully the flow structure of the full domain, a full domain solution was carried out for a case that asymmetry was assured (under a large magnetic field, B ¼ 4 kG). Results demonstrated that the halfplane treatment is reliable. Computed temperature distributions in the horizontal plane near the growth interface and also in the vertical plane at j ¼ 0 are given in Fig. 2. Temperature distributions agree with our earlier 2D solutions. Since electric current is passing through the source material the structure of isotherms indicate that the shape of the growth interface will be single-humped (concave towards the crystal) as expected [51,52].
To have a better sense for the flow field in the growth cell, the computed simulation results are presented in three distinct planes, the vertical (r z) plane at j ¼ 0 that represents typical flow structures along the growth direction, the horizontal (r j) plane at the middle of the growth cell (z ¼ 5:15 mm) where the changes in the flow field will be more prominent for radial and circumferential velocity components representing mixing, and finally the horizontal (r j) plane near the growth interface (z ¼ 0:4 mm) where the flow field is often closely related to the quality of the grown crystals. Fig. 3 summarizes the computed results for the flow field that presented in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) for four levels of magnetic field strengths (B ¼ 0:0; 0.5, 1.0,
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
19
Fig. 2. Temperature distribution in K: (a) near the growth interface in the horizontal plane, (b) in the vertical plane at j ¼ 0:
Fig. 3. Flow field in the horizontal plane in the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) at: (a) B ¼ 0:0 kG, (b) B ¼ 0:5 kG, (c) B ¼ 1:0 kG, and (d) B ¼ 2:0 kG.
and 2.0 kG), while Fig. 4 show the results for the flow field in the horizontal plane near the growth interface (left column) and in the vertical plane at j ¼ 0 (right column) for two levels of magnetic field intensities (B ¼ 2:5 and 3.0 kG), and in the
horizontal plane at the middle of the growth cell (left column) and in the vertical plane at j ¼ 0 (right column) for B ¼ 4:0 and 8:0 kG. Flow intensities are computed in m/s, and the scales of flow strengths are shown in each figure.
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Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
Fig. 4. Flow field in the horizontal plane near the growth interface at z ¼ 0:4 mm (left column) and in the vertical plane at j ¼ 0 (right column) for (a) B ¼ 2:5 kG, and (b) B ¼ 3:0 kG, and in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane (right column) for (c) B ¼ 4:0 kG, and (d) B ¼ 8:0 kG.
Results shown for the vertical plane along the growth direction appear more complex than those of the 2D results of [51,52] given for the same growth system (Fig. 3a). This is assumed to be mainly due to the inclusion of the contribution of the circumferential velocity component. In addition, the size of the growth crucible used in this study is twice the one used in [47,48]; this might have also contributed towards these differences. Such differences in the flow field of 2D and 3D models have also been observed in [20,31,46]. The strongest flow is seen in the lower part of the crucible cell, near the centre of the half domain along the r-direction. This is similar to what was observed in Ref. [20]. A weak flow cell forms just
above the strongest flow cell. The flow in the horizontal planes is nearly homocentric. The flow in the horizontal plane near the growth interface is stronger in the middle region along the radial direction and becomes weaker and weaker near to the growth cell wall or to the axial center (Fig. 3a—left column). On the other hand, in the horizontal plane at the middle of the growth cell (Fig. 3a—left column) there are two maximum points for the flow intensity along the radial direction, with a quite strong flow in the central region. Notice that the flow field in Fig. 3a is not strictly axisymmetric, especially in the middle of the growth cell, although the flow is stable. This result suggests that our system is near the
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
axisymmetry breaking of the flow [53], although the Grashof number is a little bit lower than the critical Grashof number (2.5 105) under the same conditions (Prandtl number and geometry aspect ratio) given in Ref. [53] in the analysis of axisymmetry breaking of natural convection in a vertical Bridgman growth configuration. Figs. 3b–d and Figs. 4a and b show the influence of the applied magnetic field on the flow field. Results are also summarized in Figs. 5 and 6 for various aspects of the flow field. Simulation results show three distinct characteristics depending on the level of applied magnetic field: (a) the ‘‘weak’’ magnetic field, with intensities from 0.0 to 2.0 kG,
21
(b) the ‘‘intermediate’’ magnetic field, with levels from 2.0 to 3.0 kG, and (c) the ‘‘high’’ magnetic field, with intensities above 3.0 kG. Flow characteristics are quite different at each of these field levels. Let us first focus on Figs. 3b–d which represent results for magnetic field levels from 0.5 to 2.0 kG. In this category, the weak magnetic field level, and an increase in the applied magnetic field intensity, results in significant reduction in the flow intensity, which is, in general, desirable for a stable and controlled crystal growth. The flow cells are strongest near the vertical wall and forms the socalled Hartmann layer [25,45–47]. At higher
Fig. 5. Variation of the flow strength U ¼ ðu2 þ v2 þ w2 Þ1=2 along the radial direction at j ¼ p=2 under various magnetic field strengths: (a) B ¼ 0:0 kG, (b) B ¼ 1:0 kG, (c) B ¼ 2:0 kG, and (d) B ¼ 3:0 kG. Solid lines at z ¼ 5:05 mm (in the middle of the growth cell), and dashed lines at z ¼ 0:4 mm (near the growth interface).
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Fig. 6. Variation of the maximum velocity with magnetic field intensity: three distinct regions of stability of the flow field are obvious. The flow field is stable up to Ha ¼ 150; the region between Ha ¼ 150 and 220 the flow is transitional, and above Ha ¼ 220; the flow is unstable.
magnetic field strengths, the relative intensity of these flow cells becomes further stronger compared with the flow in the rest of the growth cell domain, and the Hartmann layer becomes thinner which is in accordance with the scaling analysis in Ref. [31]. Furthermore, the strongest flow cells appearing in the lower part of the growth cell (Fig. 3a) move further down towards the growth interface with increasing intensity, and new flow cells form near the dissolution interface. These strong flow cells form very visible boundary layers near the growth and dissolution interfaces (the so-called ‘‘end layers’’ [31,47]), and hence give rise to strong vertical velocity gradients in the vicinity of the growth interface, which may have an adverse effect on the crystal growth process. One may conclude that the application of a magnetic field may not always be beneficial for the growth process [25,30]. It was shown in Ref. [25] that the radial nonuniformity in vertical Bridgman is the most significant at the intermediate levels of magnetic field strength. Flow fields become perfectly homocentric (and hence axisymmetric) with the increase of magnetic
field strength, and hence the flow intensity decreases. The flow velocities in the central region of the cell (both in the r and z directions) become more uniform at higher magnetic field levels, and form a core region of uniform flow field [25,45], leading to a domain in which the flow is suppressed with the application of an applied magnetic field. Indeed the applied magnetic field at the weak intensity levels can suppress fluid flows in a growth system, as shown in Fig. 3. It must be mentioned here that a magnetic field level higher than 8.0 kG is very strong for an LPEE application. It is not possible for us in this study to present any correlation (or a comparison) between the present numerical results and our ongoing experiments. Only one thing can be said, though, that thus far it was not possible to exceed the level of 4.5 kG in experiments. It led to unsatisfactory growth due to strong convection in the liquid zones. However, the growth rate was increased about four times under a magnetic field of 4.5 kG (experimental results will be reported soon). Fig. 4 represents the simulation results for the intermediate magnetic intensity levels selected as B ¼ 2:5; 3.0, 4.0, and 8.0 kG. In this case, flow intensity increases with magnetic field strength. This result has not been widely reported in the relevant literature; however, it is reasonable due to the following reasoning. The magnetic body force and the body force due to buoyancy (maybe others such as surface tension) acting on the points of the liquid, although act in different directions, balance each other in a close container (growth cell for instance). Up to a certain level of magnetic body force, the magnetic body force suppresses the fluid flow by counterbalancing the buoyancy force. However, when the level of magnetic field exceeds a certain value, it surpasses the buoyancy force and becomes an excessive force, feeding the convection in the liquid. Thus, the flow strength increases further. Some experimental and numerical studies indeed demonstrated the enhancement of heat transfer (and hence flow strength) in a melt under a stationary magnetic field [48,49,54]. As can be seen from Fig. 4, the flow patterns show dramatic changes, two flow cells were formed in each half of the vertical plane, with
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
the upper cells getting larger and lower cells getting smaller with the increasing magnetic field intensity. In the vertical plane, some strong unidirectional flows appear in the middle region along the r-direction, and some with very weak intensity in the middle forming a small cylindrical region where the flow is nearly stationary. The flow fields are no longer axisymmetric and homocentric. Such an axisymmetry breaking may be caused by the unsteadiness of the flow field according to Ref. [53]. Although it is not presented here, the flow stability was also examined (at B ¼ 2:5 and 3.0 kG levels) comparing the results at different times, and found that the flow field is essentially stable in spite of some small changes with time. It is then speculated that the reason of the axisymmetry breaking of the flow field at these magnetic field levels may be due to a very delicate balance between the buoyancy and magnetic forces even though the flow still remains stable. The variation of the flow strength along the radial direction for B ¼ 0:0; 1.0, 2.0, and 3.0 kG levels were shown in Fig. 5. Solid and dashed lines represent, respectively, the values at the middle and the lower regions of the growth cell. As can be seen, the flow strength fluctuates and shows differences in these regions. However, at higher magnetic field levels, the difference becomes less obvious. The variation of the flow strength in the radial direction near the growth interface is almost symmetric in the absence of a magnetic field. However, this symmetry is broken with the increasing magnetic field intensity. Although higher magnetic field intensity levels, higher than 4.5 kG, appears to be not practical for the LPEE growth process, to the best of our knowledge at the moment of course, for the sake of completeness of the present numerical study and also for the purpose of comparison with other studies, higher magnetic field intensity levels were also examined. The levels of B ¼ 4:0 kG and B ¼ 8:0 kG were selected. The flow patterns become dramatically different and show large fluctuations with time, and temperature distributions show asymmetric behavior. These results are not presented here for the sake of space. However, the maximum velocity values
23
are presented in Fig. 6 for all magnetic field levels. In Fig. 6, the three regions of flow stability are clearly presented. The flow velocity decreases with increasing magnetic field in the stable region (up to Ha ¼ 150), but in the intermediate and unstable regions (between Ha ¼ 150 and 220, and above Ha ¼ 220; respectively) the flow velocity increases. If one examines the logarithmic plot of the maximum velocity (Umax ) of the flow field as a function of the Hartmann number, given in Fig. 6, can see that within the stable region, the relationship between the flow intensity and the Hartmann number obeys a power law of Umax pHa5=4 ;
ð21Þ
which has been demonstrated by many authors [25,31,45–47], although the index of the power law is slightly different due to different system parameters and conditions of the present study. In the unstable region, this relationship becomes Umax pHa5=2 :
ð22Þ
In the transitional (intermediate) region, on the other hand, as one expects, the change of the maximum velocity (Umax ) of the flow field with the Hartmann number is so dramatic that the power law is not suitable. It is important to point out that the results given here for the intermediate and unstable regions are new and, to the best of our knowledge, were not encountered in the literature. 4.2. Influence of magnetic field non-uniformity As mentioned in Section 1, in order to see the effect of field non-uniformities, certain non-uniformity levels were incorporated into the simulation model. Selecting y0 ¼ 0 and R0 ¼ 0:008 m, which, together with the crystal radius of R ¼ 0:012 m, gives the maximum difference of the magnetic field as 2A% (see Eq. (19), for instance A ¼ 1:0 and 3.0, correspond to the nonuniformity levels of 2.0% and 6.0%, respectively. These non-uniformity levels were examined for the magnetic field level of B ¼ 1:0 kG. The results of numerical simulations were presented in Fig. 7.
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Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
Fig. 7. Flow field under B ¼ 1:0 kG and three different non-uniformity values: (a) A ¼ 0; (b) A ¼ 1:0; and (c) A ¼ 3:0: The left column describes the flow patterns in the horizontal plane near the growth interface at z ¼ 0:4 mm, and the right column shows the flow patterns in the vertical plane at j ¼ 0:
Results show that even for a 6% the nonuniformity changes in the flow field are small, indicating that the natural convection becomes weaker under the application of an external magnetic field, and the influence of such nonuniformities at this level of external magnetic field is not significant. However, the flow field becomes asymmetric with increasing non-uniformity (see Fig. 7c). As one expects, at higher levels of applied magnetic field such non-uniformities may have significant effects on the flow field; however, due to the practical insignificance, the simulation results conducted for B ¼ 8:0 kG are not presented here.
5. Conclusion A 3D numerical simulation model for the liquid phase electroepitaxial (LPEE) growth of GaAs
under a vertical stationary magnetic field was presented. Computations were carried out using the commercial CFX software package. The effect of magnetic field intensity and non-uniformity on the flow field in the liquid solution was investigated. Numerical results show that the flow patterns exhibit three distinct stability characteristics: a stable flow field up to a magnetic field level of Ha ¼ 150; a transitional flow between Ha ¼ 150 and 220; and an unstable flow above Ha ¼ 220: In the stable region, the applied magnetic field suppresses the flow field, and the flow intensity decreases with increasing magnetic field exhibiting a power law Umax pHa5=4 relationship for the maximum velocity (Umax ). The flow in the horizontal plane is perfectly homocentric, and the flow in the central region of the growth cell becomes more uniform and weaker with increasing magnetic field levels. Boundary layers appear in
Y.C. Liu et al. / Journal of Crystal Growth 244 (2002) 12–26
the vicinity of the growth and dissolution interfaces, and a Hartmann layer forms along the cell wall and becomes thinner with the increasing magnetic field strength. In the transitional region, the flow intensity increases dramatically with the increase in magnetic field strength. The flow patterns are significantly different than those in the stable region. The flow field is no longer axisymmetric but still stable. In the unstable region, the flow structure and intensity change with time. Under a strong magnetic field, the flows cells are confined to the vicinity of the vertical wall and exhibit significant non-uniformity near the growth interface. Such strong flow fluctuations and non-uniformities near the growth interface may have an adverse effect on the growth process and lead to an unsatisfactory growth. In this region, the maximum velocity (Umax ) obeys approximately a power law of Umax pHa5=2 : Results show that for a successful growth the effect of applied magnetic field must be optimized. The numerical study of the effect of nonuniformity in the applied magnetic field shows that at low magnetic field intensities (less then 150 Ha), the influence of non-uniformities on the flow field is insignificant. Under a strong magnetic field, however, such non-uniformities may become significant and affect the growth process adversely. A comparison with 2D simulation results for the same growth system shows that the flow patterns of the 3D model seem much complex. Observations in the flow structures at high magnetic field strengths could not be observed through the 2D models, partially due to the omission of the velocity and magnetic force components along the circumferential direction. Acknowledgements The financial support provided by the Canadian Space Agency Microgravity Science Program is gratefully acknowledged. References [1] L. Jastrzebski, H.C. Gatos, A.F. Witt, J. Electrochem. Soc. 123 (1976) 1121.
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