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On the high growth rates in electroepitaxial growth of bulk semiconductor crystals in magnetic field
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Journal of Crystal Growth 275 (2005) e1–e6 www.elsevier.com/locate/jcrysgro
On the high growth rates in electroepitaxial growth of bulk semiconductor crystals in magnetic field Sadik Dost, Hamdi Sheibani, Yongcai Liu, Brian Lent Crystal Growth Laboratory, Department of Mechanical Engineering, University of Victoria, Victoria, BC, Canada, V8W 3P6 Available online 7 December 2004
Abstract Liquid phase electroepitaxial (LPEE) growth experiments conducted under a static magnetic field (aligned perfectly with the vertical symmetry axis of the growth system) show that the growth rate is proportional to the applied magnetic field intensity, and increases with the field intensity level, but does not depend on the field direction. The relationship between growth rate and magnetic field intensity is almost linear. The increase in growth rate is significant, for instance, about more than ten times under a 4.5 kG field level at 3 A/cm2 electric current density. A model that introduces a new mobility coefficient (electromagnetic mobility) is proposed to predict such high growth rates. The model also predicts growth interface shapes closely. r 2004 Elsevier B.V. All rights reserved. PACS: 81.10.Dn; 81.10.h; 83.10.Ff; 81.05.Ea Keywords: A1. Solidification; A2. Single crystal growth; A2. Electroepitaxy under magnetic field; B1. Gallium compounds
1. Introduction Liquid phase electroepitaxy (LPEE) is of significant technological interest in growth of bulk single crystals of alloy semiconductors. In LPEE, growth is achieved by passing an electric current through the growth cell while the overall furnace temperature is kept constant during the entire growth period. The applied electric current is the Corresponding
author. Tel.: +1 250 721 8898; +1 250 721 6294. E-mail address: [email protected] (S. Dost).
fax:
sole driving force for growth, and gives rise to the electromigration of species in the liquid solution that sustains a controlled-growth [1–4]. Although the furnace temperature is constant in LPEE, the combined effect of the Joule heating in the solid crystals and the Peltier heating/cooling at the liquid/solid interfaces leads to relatively strong convection in the solution. LPEE has a number of advantages over bulk growth techniques, including controlled growth, uniform crystal composition, low growth temperature, and very low temperature gradients in the solution zone, etc. In spite of these, the natural
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.10.117
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convection occurring in the solution zone may still adversely affect the quality of grown crystals and lead to growth instabilities. The application of a magnetic field is an option in suppressing natural convection. For instance, a static magnetic field aligned perfectly with the axis of the growth cell gives rise to two magnetic body force components in the horizontal plane; one in the radial direction inward, and the other one in the circumferential direction. They act on the points of the liquid solution, and their magnitudes depend on the square of the field intensity and the velocity components of fluid at that point. By increasing the field intensity, the magnitudes of these magnetic body forces can be increased. Under the effect of these two varying magnetic force components and the constant gravitational body force in the vertical direction (three orthogonal forces), the velocity of particles in the liquid solution varies. Increase in the magnetic field intensity decreases the maximum velocity in the solution and, consequently, suppresses the convection in the liquid solution (see for instance Refs. [5–18] for the use of applied magnetic field). Three-dimensional numerical simulations have shown that the application of a static magnetic
field may indeed suppress convection [17,18]. However, there is a critical magnetic field intensity level below which the maximum velocity in the liquid solution decreases with increasing field intensity, but above this field intensity level the maximum velocity increases drastically with increasing field intensity. It appears that the magnetic body forces reduce the effect of gravity up to a point, above which the delicate balance established between the magnetic and gravitational body forces becomes unfavorable, and convection becomes very strong; even stronger than that of no magnetic field [17,18]. Such observations have also been made experimentally; shown that crystals grown at higher field intensity levels were damaged and uneven, and growth interfaces lost stability [15]. In LPEE, the growth rate is proportional to the applied electric current density (see Refs. [1–4]). In addition, the presence of an applied magnetic field in LPEE increases the mass transport in the liquid solution tremendously; leading to flat and thick crystals with very high growth rates [14,15]. For instance, a large number of bulk GaAs and InGaAs single crystals with 25 mm diameter and up to 9 mm thicknesses have been grown [14] with
Fig. 1. View of sample GaAs crystals grown at various magnetic field levels. All are 25 mm in diameters, (a) and (b) are about 4.5 mm in thickness while (c) and (d) are proportionally thinner since they were grown under weaker field levels. Except (a), in the rest growth was stopped before the source was depleted. Thus, there was a secondary, poly crystalline growth during the cooling period. Interfaces between the single and poly sections are visible.
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and without the application of a strong magnetic field (samples of crystals grown with and without the application of magnetic field are shown in Fig. 1). For instance the growth rate at J ¼ 3 A=cm2 electric current density was more than ten times higher for a 4.5 kG magnetic field level than that of no magnetic field [14]. In crystal growth, the use of linear constitutive equations is generally sufficient for most modeling purposes (see, for instance, Refs. [12,13,16]). However, when a growth system is subjected to a strong applied magnetic field, some higher order (nonlinear) constitutive coefficients (for example, the cross interactions of magnetic field with temperature and concentration gradients, electric field and fluid flow) may have significant effects on the growth process. In the absence of such effects in modelling, some experimental results cannot be predicted accurately. For instance, high growth rates observed in LPEE growth of bulk crystals under magnetic field [14] cannot be predicted from the models based only on linear constitutive coefficients [12,15]. The general nonlinear continuum model developed for a binary system (this work is presently under review) may be used for the LPEE growth of GaAs to predict accurately not only these high growth rates but also the experimental growth interface shapes [17]. This is the objective of this article.
2. Experimental procedure In order to determine whether the growth rate is also dependent on the magnetic field direction, new experiments were conducted for the present work. Two sets of experiments (four experiments in each set for the growth of GaAs) were conducted at B ¼ 1 and 2 kG magnetic field intensity levels by applying the magnetic field vector B both upward and downward. All the experiments were successful and the grown GaAs crystals were single crystals. The growth rates in these experiments were the same whether B was up or down, and the average values are presented in Table 1 (third and fourth columns) along with the results of Ref. [14] for the sake of completeness (the second and fifth columns). This showed that
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Table 1 Summary of experimental results at growth temperature of 850 1C Magnetic field B (kGauss) Electric current (A/cm2) Experimental growth rate (mm/day) 70.01 Reference source
0.0 3.0 0.50
1.0 3.0 1.62
2.0 3.0 2.35
[14]
4.5 3.0 6.10 [14]
the mass transport due to electromigration depends only on the magnetic field intensity.
3. A nonlinear model for the mass transport in LPEE growth of GaAs under magnetic field General constitutive equations of a binary liquid phase can be simplified under the following assumptions for the growth of GaAs crystals by LPEE under a static applied magnetic field: I. Boussinesq approximation holds. II. The electric and magnetic fields are aligned vertically with the symmetry axis of the growth crucible; the LPEE growth system was designed to achieve this condition. III. The electric and magnetic fields are constant; do not vary in time and space. Indeed, the magnetic field measurements made in the absence of growth crucible show that the magnetic field is almost uniform in the space where the growth cell is located [14]. The LPEE growth crucible was designed so as to have an almost uniform electric current distribution in the liquid zone. In addition, the computed electric field in Refs. [12,13] was almost uniform. IV. The induced magnetic field due to the applied electric current is small, so is neglected. V. The contribution of Joule heating in the liquid zone is neglected since the liquid Ga–As solution is a good conductor. In addition, the contribution of ðv BÞ; where v is the velocity vector, in the mass flux and also in the electric current to the electric current (i.e. J ¼ rE E where rE is the electric conductivity) can also be neglected since, as shown numerically
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in Refs. [12,13], that the contribution of ðv BÞ with respect to the electric field E is very small. Under the above assumptions the nonlinear constitutive equation for the mass flux i takes the following form: i ¼ DC rC þ ðDEC þ DECB BÞCðE þ v BÞ;
(3.1)
where DC is the well-known diffusion coefficient, and DEC and DECB are the second- and third-order material coefficients representing, respectively, the contributions of electric and magnetic fields to the mass flux. C is the mass concentration of Arsenic (As) in the Ga-rich liquid solution, and r stands for the gradient operator. Under the above assumptions, the use of Eq. (3.1) in the mass transport equation yields DC r2 C þ ðDEC þ DECB BÞðE þ v BÞ qC þ v rC: ð3:2Þ rC ¼ qt The linear constitutive equations for heat flux, electric current and stress, and the other associated field equations are not presented here for the sake of space (they can be found in Refs. [16–18]).
4. Total mobility—effect of applied magnetic field The first term in the mass transport equation, Eq. (3.2), is the well-known diffusion term due to Fick’s law. The second term, i.e. ðDEC þ DECB BÞðE þ v BÞ rC represents the total contribution of applied electric current density to mass transport under the effect of a static external magnetic field. This effect is known as electromigration, and its coefficient, which is called here the ‘‘total mobility’’, is written as mt ¼ DEC þ DECB B mE þ mEB B; where the material constant mE (a second-order material coefficient) is the classical electric mobility of the solute (As) in the liquid solution (Ga–As solution) due to the applied electric current in the absence of an applied magnetic field. The constant mEB is a third-order material coefficient that represents the contribution of the applied magnetic field intensity to the electromigration of species. This term is
new, defined for the first time by the authors, and called the electromagnetic mobility. Below, we present a short discussion on the relative contributions of each term in the mass transport under the effect of applied magnetic field, and then give an estimate for the numerical value of the electromagnetic mobility using the experimental results given in Table 1. The diffusion (the first term on the left-hand side of Eq. (3.1), i.e. DC r2 C) and also the natural convection (the last term on the right-hand side, i.e.v rC) contribute to mass transport in LPEE [12–16]. However, experiments and modeling studies show that their contributions in LPEE are relatively small compared with that of electromigration. In the second term of Eq. (3.1), i.e. ðmE þ mEB BÞðE þ v BÞ rC; the term ðv BÞ rC is the contribution of the applied magnetic field due to the motion of the fluid particles (coupling term). We have examined its contribution and found that it is very small compared with the contribution of E rC (in the order of 3% based on a maximum velocity of 0.01 m/s and a 10 kG field level) [12,13], and is then neglected in the present model. Then, we assume that the growth rate is proportional to the second term in Eq. (3.1). Under these conditions, using the experimental results, we evaluated the value of mE in the Ga–As solution in the absence of applied magnetic field (see Table 1) as mE ¼ 0:7 105 m2 =Vs: The numerical simulations based on this value verify the experimental growth rates at all three electric current levels (J ¼ 3; 5, and 7 A=cm2 ) (see Refs. [12–16]). The numerical values of the total mobility mt ¼ mE þ mEB B were calculated using the results of new experiments in which the magnetic field vector B was used both upward and downward. The growth rates in these experiments were almost the same whether B was up or down. In other words, the mass transport due to electromigration depends only on the magnetic field intensity, not on its direction. Using the measured growth rates given in Table 1, the total magnetic mobility values were calculated and given in Table 2, from which the electromagnetic mobility coefficient was calculated as mEB ¼ 1:4 105 m2 =VsðkGÞ: In order to see
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the relationship between the field intensity and the total mobility, using these constant values, a dimensionless total mobility is defined as m ¼ mt =mE 1 þ ðmEB =mE ÞB ffi 1 þ 2B:
netic field. This point deserves attention for future research. The mass transport equation, Eq. (3.2), then becomes
(4.1)
As can be seen the contribution of magnetic field intensity to the total mobility (i.e. total mass transport by electromigration) is almost linear (within the limits of experimental measurements), and is very significant. Electromigration is believed to be the result of momentum exchange of species in metallic liquids under the effect of an applied electric current, and the mobility of species (As in this case) is proportional to the electric current density with a proportionality constant (material constant) mE [1,2]. As mentioned earlier, mE is the electric mobility of a species, and has been calculated for various metallic solutions. What is happening to this mobility under an applied magnetic field? This question was answered in this work by introducing a mathematical model: assuming that the mobility of species in the bulk is affected by the applied magnetic field. Physically, this means that the momentum exchange becomes much easier under the effect of magnetic field; making the movement of species much easier and faster along the magnetic field lines due to less resistance (less momentum loss) to the mobility of species. Since the mass transport is enhanced drastically, this approach of looking first at mass transport in the bulk was the logical way to go about it. This physical concept was reflected mathematically through Eq. (4.1) which assumes that the mobility of species under magnetic field increases linearly with the field intensity, leading to higher mass transport rates as observed in experiments. It is possible that some other either bulk- or surfaceeffects may be responsible and/or contributing to the enhancement of mass transport under mag-
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DC r2 C þ mt E z
qC qC ¼ þ v rC qz qt
(4.2)
and the growth may then be calculated by r qC 1 V g ¼ L DC þ mt E z C ; qn CS C rS
(4.3)
Fig. 2. Evolution of growth interfaces: solid lines and dashed lines show interfaces computed using only the electric mobility under, respectively, B ¼ 0 and 1.0 kG field levels, and the dotted lines represent growth under 1.0 kG field using the total mobility. As can be seen, the growth rates using only the electric mobility (with and without magnetic field) are almost the same, which are in disagreement with experiments. The interface shapes are also not as flat as of experiments (see Fig. 1). The computed growth interfaces under a 1.0 kG magnetic field level using the total mobility (dotted lines) are in agreement with the experimental values.
Table 2 Calculated mobility values Magnetic field B (kG) at J ¼ 3 A=cm2 Experimental growth rate (mm/day)
0.0 0.50
1.0 1.62
2.0 2.35
4.5 6.10
Total mobility mt ¼ mE þ mEB B (m2/Vs)
0.7 105
2.3 105
3.4 105
7.1 105
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where rL ; rS and C S are, respectively, the density of solution, density of solid crystal, and the solid composition. The evolution of growth interfaces, computed by using Eq. (4.3), is shown in Fig. 2. In the figure, the solid and dashed lines represent the computed growth interfaces using only the electric mobility, mE under, respectively, B ¼ 0 and 1.0 kG field levels [12,13]. As can be seen, the growth rates in both cases (with and without magnetic field) are almost the same, which are in disagreement with the experimental results of [14] and this work. The interface shapes are also not as flat as those of experiments (see the sample crystals in Fig. 1). The dotted lines represent the evolution of computed growth interfaces under magnetic field of 1.0 kG using the total mobility mt : These growth rates are in agreement with the experimental values. This shows the significance of inclusion of the electromagnetic mobility in the model.
5. Conclusions LPEE experiments show that the growth rate is proportional to the applied magnetic field and increases with the field intensity level. But the growth rate does only depend on the intensity of the applied magnetic field; not on its direction. The relationship between growth rate and magnetic field intensity is almost linear. A model that introduces a new mobility coefficient (electromagnetic mobility) predicts such high growth rates. The model also predicts flatter growth interface shapes. Results of numerical simulations performed for a binary system (GaAs) using the total mobility are in agreement with experiments.
Acknowledgment The financial support provided by the Microgravity Science Program of the Canadian Space Agency (CSA), and the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. References [1] L. Jastrzebski, H.C. Gatos, A.F. Witt, J. Electrochem. Soc. 123 (1976) 1121. [2] L. Jastrzebski, Y. Imamura, H.C. Gatos, J. Electrochem. Soc. 125 (1978) 1140. [3] K. Nakajima, J. Appl. Phys. 61 (9) (1987) 4626. [4] K. Nakajima, J. Crystal Growth 98 (1989) 329. [5] R.W. Series, D.T.J. Hurle, J. Crystal Growth 113 (1991) 305. [6] D.T.J. Hurle (Ed.), Handbook of Crystal Growth 2, Bulk Crystal Growth, Part B: Growth Mechanisms and Dynamics, North-Holland, Amsterdam, 1994. [7] J. Baumgartl, A. Hubert, G. Muller, Phys. Fluids A 5 (1993) 3280. [8] M. Salk, M. Fiederle, K.W. Benz, A.S. Senchenkov, A.V. Egorov, D.G. Matioukhin, J. Crystal Growth 138 (1994) 161. [9] S. Dost, Z. Qin, J. Crystal Growth 153 (1995) 123. [10] S. Senchenkov, I.V. Barmin, A.S. Tomson, V.V. Krapukhin, J. Crystal Growth 197 (1997) 552. [11] K. Ghaddar, C.K. Lee, S. Motakef, D.C. Gillies, J. Crystal Growth 205 (1999) 97. [12] S. Dost, Y.C. Liu, B. Lent, J. Crystal Growth 240 (2002) 39. [13] Y.C. Liu, Y. Okano, S. Dost, J. Crystal Growth 244 (2002) 12. [14] H. Sheibani, S. Dost, S. Sakai, B. Lent, J. Crystal Growth 258 (3–4) (2003) 283. [15] H. Sheibani, Y.C. Liu, S. Sakai, B. Lent, S. Dost, Int. J. Eng. Sci. 41 (2003) 401. [16] Y. Okano, H. Kondo, S. Dost, Int. J. Appl. Electromagn. Mech. 18 (4) (2003) 217. [17] Y.C. Liu, S. Dost, H. Sheibani, Int. J. Transport Phenom. 6 (2004) 51. [18] S. Dost, B. Lent, H. Sheibani, Y.C. Liu, C. R. Mecanique 332 (5–6) (2004) 413.
Journal of Crystal Growth 275 (2005) e1–e6 www.elsevier.com/locate/jcrysgro
On the high growth rates in electroepitaxial growth of bulk semiconductor crystals in magnetic field Sadik Dost, Hamdi Sheibani, Yongcai Liu, Brian Lent Crystal Growth Laboratory, Department of Mechanical Engineering, University of Victoria, Victoria, BC, Canada, V8W 3P6 Available online 7 December 2004
Abstract Liquid phase electroepitaxial (LPEE) growth experiments conducted under a static magnetic field (aligned perfectly with the vertical symmetry axis of the growth system) show that the growth rate is proportional to the applied magnetic field intensity, and increases with the field intensity level, but does not depend on the field direction. The relationship between growth rate and magnetic field intensity is almost linear. The increase in growth rate is significant, for instance, about more than ten times under a 4.5 kG field level at 3 A/cm2 electric current density. A model that introduces a new mobility coefficient (electromagnetic mobility) is proposed to predict such high growth rates. The model also predicts growth interface shapes closely. r 2004 Elsevier B.V. All rights reserved. PACS: 81.10.Dn; 81.10.h; 83.10.Ff; 81.05.Ea Keywords: A1. Solidification; A2. Single crystal growth; A2. Electroepitaxy under magnetic field; B1. Gallium compounds
1. Introduction Liquid phase electroepitaxy (LPEE) is of significant technological interest in growth of bulk single crystals of alloy semiconductors. In LPEE, growth is achieved by passing an electric current through the growth cell while the overall furnace temperature is kept constant during the entire growth period. The applied electric current is the Corresponding
author. Tel.: +1 250 721 8898; +1 250 721 6294. E-mail address: [email protected] (S. Dost).
fax:
sole driving force for growth, and gives rise to the electromigration of species in the liquid solution that sustains a controlled-growth [1–4]. Although the furnace temperature is constant in LPEE, the combined effect of the Joule heating in the solid crystals and the Peltier heating/cooling at the liquid/solid interfaces leads to relatively strong convection in the solution. LPEE has a number of advantages over bulk growth techniques, including controlled growth, uniform crystal composition, low growth temperature, and very low temperature gradients in the solution zone, etc. In spite of these, the natural
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.10.117
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convection occurring in the solution zone may still adversely affect the quality of grown crystals and lead to growth instabilities. The application of a magnetic field is an option in suppressing natural convection. For instance, a static magnetic field aligned perfectly with the axis of the growth cell gives rise to two magnetic body force components in the horizontal plane; one in the radial direction inward, and the other one in the circumferential direction. They act on the points of the liquid solution, and their magnitudes depend on the square of the field intensity and the velocity components of fluid at that point. By increasing the field intensity, the magnitudes of these magnetic body forces can be increased. Under the effect of these two varying magnetic force components and the constant gravitational body force in the vertical direction (three orthogonal forces), the velocity of particles in the liquid solution varies. Increase in the magnetic field intensity decreases the maximum velocity in the solution and, consequently, suppresses the convection in the liquid solution (see for instance Refs. [5–18] for the use of applied magnetic field). Three-dimensional numerical simulations have shown that the application of a static magnetic
field may indeed suppress convection [17,18]. However, there is a critical magnetic field intensity level below which the maximum velocity in the liquid solution decreases with increasing field intensity, but above this field intensity level the maximum velocity increases drastically with increasing field intensity. It appears that the magnetic body forces reduce the effect of gravity up to a point, above which the delicate balance established between the magnetic and gravitational body forces becomes unfavorable, and convection becomes very strong; even stronger than that of no magnetic field [17,18]. Such observations have also been made experimentally; shown that crystals grown at higher field intensity levels were damaged and uneven, and growth interfaces lost stability [15]. In LPEE, the growth rate is proportional to the applied electric current density (see Refs. [1–4]). In addition, the presence of an applied magnetic field in LPEE increases the mass transport in the liquid solution tremendously; leading to flat and thick crystals with very high growth rates [14,15]. For instance, a large number of bulk GaAs and InGaAs single crystals with 25 mm diameter and up to 9 mm thicknesses have been grown [14] with
Fig. 1. View of sample GaAs crystals grown at various magnetic field levels. All are 25 mm in diameters, (a) and (b) are about 4.5 mm in thickness while (c) and (d) are proportionally thinner since they were grown under weaker field levels. Except (a), in the rest growth was stopped before the source was depleted. Thus, there was a secondary, poly crystalline growth during the cooling period. Interfaces between the single and poly sections are visible.
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and without the application of a strong magnetic field (samples of crystals grown with and without the application of magnetic field are shown in Fig. 1). For instance the growth rate at J ¼ 3 A=cm2 electric current density was more than ten times higher for a 4.5 kG magnetic field level than that of no magnetic field [14]. In crystal growth, the use of linear constitutive equations is generally sufficient for most modeling purposes (see, for instance, Refs. [12,13,16]). However, when a growth system is subjected to a strong applied magnetic field, some higher order (nonlinear) constitutive coefficients (for example, the cross interactions of magnetic field with temperature and concentration gradients, electric field and fluid flow) may have significant effects on the growth process. In the absence of such effects in modelling, some experimental results cannot be predicted accurately. For instance, high growth rates observed in LPEE growth of bulk crystals under magnetic field [14] cannot be predicted from the models based only on linear constitutive coefficients [12,15]. The general nonlinear continuum model developed for a binary system (this work is presently under review) may be used for the LPEE growth of GaAs to predict accurately not only these high growth rates but also the experimental growth interface shapes [17]. This is the objective of this article.
2. Experimental procedure In order to determine whether the growth rate is also dependent on the magnetic field direction, new experiments were conducted for the present work. Two sets of experiments (four experiments in each set for the growth of GaAs) were conducted at B ¼ 1 and 2 kG magnetic field intensity levels by applying the magnetic field vector B both upward and downward. All the experiments were successful and the grown GaAs crystals were single crystals. The growth rates in these experiments were the same whether B was up or down, and the average values are presented in Table 1 (third and fourth columns) along with the results of Ref. [14] for the sake of completeness (the second and fifth columns). This showed that
e3
Table 1 Summary of experimental results at growth temperature of 850 1C Magnetic field B (kGauss) Electric current (A/cm2) Experimental growth rate (mm/day) 70.01 Reference source
0.0 3.0 0.50
1.0 3.0 1.62
2.0 3.0 2.35
[14]
4.5 3.0 6.10 [14]
the mass transport due to electromigration depends only on the magnetic field intensity.
3. A nonlinear model for the mass transport in LPEE growth of GaAs under magnetic field General constitutive equations of a binary liquid phase can be simplified under the following assumptions for the growth of GaAs crystals by LPEE under a static applied magnetic field: I. Boussinesq approximation holds. II. The electric and magnetic fields are aligned vertically with the symmetry axis of the growth crucible; the LPEE growth system was designed to achieve this condition. III. The electric and magnetic fields are constant; do not vary in time and space. Indeed, the magnetic field measurements made in the absence of growth crucible show that the magnetic field is almost uniform in the space where the growth cell is located [14]. The LPEE growth crucible was designed so as to have an almost uniform electric current distribution in the liquid zone. In addition, the computed electric field in Refs. [12,13] was almost uniform. IV. The induced magnetic field due to the applied electric current is small, so is neglected. V. The contribution of Joule heating in the liquid zone is neglected since the liquid Ga–As solution is a good conductor. In addition, the contribution of ðv BÞ; where v is the velocity vector, in the mass flux and also in the electric current to the electric current (i.e. J ¼ rE E where rE is the electric conductivity) can also be neglected since, as shown numerically
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in Refs. [12,13], that the contribution of ðv BÞ with respect to the electric field E is very small. Under the above assumptions the nonlinear constitutive equation for the mass flux i takes the following form: i ¼ DC rC þ ðDEC þ DECB BÞCðE þ v BÞ;
(3.1)
where DC is the well-known diffusion coefficient, and DEC and DECB are the second- and third-order material coefficients representing, respectively, the contributions of electric and magnetic fields to the mass flux. C is the mass concentration of Arsenic (As) in the Ga-rich liquid solution, and r stands for the gradient operator. Under the above assumptions, the use of Eq. (3.1) in the mass transport equation yields DC r2 C þ ðDEC þ DECB BÞðE þ v BÞ qC þ v rC: ð3:2Þ rC ¼ qt The linear constitutive equations for heat flux, electric current and stress, and the other associated field equations are not presented here for the sake of space (they can be found in Refs. [16–18]).
4. Total mobility—effect of applied magnetic field The first term in the mass transport equation, Eq. (3.2), is the well-known diffusion term due to Fick’s law. The second term, i.e. ðDEC þ DECB BÞðE þ v BÞ rC represents the total contribution of applied electric current density to mass transport under the effect of a static external magnetic field. This effect is known as electromigration, and its coefficient, which is called here the ‘‘total mobility’’, is written as mt ¼ DEC þ DECB B mE þ mEB B; where the material constant mE (a second-order material coefficient) is the classical electric mobility of the solute (As) in the liquid solution (Ga–As solution) due to the applied electric current in the absence of an applied magnetic field. The constant mEB is a third-order material coefficient that represents the contribution of the applied magnetic field intensity to the electromigration of species. This term is
new, defined for the first time by the authors, and called the electromagnetic mobility. Below, we present a short discussion on the relative contributions of each term in the mass transport under the effect of applied magnetic field, and then give an estimate for the numerical value of the electromagnetic mobility using the experimental results given in Table 1. The diffusion (the first term on the left-hand side of Eq. (3.1), i.e. DC r2 C) and also the natural convection (the last term on the right-hand side, i.e.v rC) contribute to mass transport in LPEE [12–16]. However, experiments and modeling studies show that their contributions in LPEE are relatively small compared with that of electromigration. In the second term of Eq. (3.1), i.e. ðmE þ mEB BÞðE þ v BÞ rC; the term ðv BÞ rC is the contribution of the applied magnetic field due to the motion of the fluid particles (coupling term). We have examined its contribution and found that it is very small compared with the contribution of E rC (in the order of 3% based on a maximum velocity of 0.01 m/s and a 10 kG field level) [12,13], and is then neglected in the present model. Then, we assume that the growth rate is proportional to the second term in Eq. (3.1). Under these conditions, using the experimental results, we evaluated the value of mE in the Ga–As solution in the absence of applied magnetic field (see Table 1) as mE ¼ 0:7 105 m2 =Vs: The numerical simulations based on this value verify the experimental growth rates at all three electric current levels (J ¼ 3; 5, and 7 A=cm2 ) (see Refs. [12–16]). The numerical values of the total mobility mt ¼ mE þ mEB B were calculated using the results of new experiments in which the magnetic field vector B was used both upward and downward. The growth rates in these experiments were almost the same whether B was up or down. In other words, the mass transport due to electromigration depends only on the magnetic field intensity, not on its direction. Using the measured growth rates given in Table 1, the total magnetic mobility values were calculated and given in Table 2, from which the electromagnetic mobility coefficient was calculated as mEB ¼ 1:4 105 m2 =VsðkGÞ: In order to see
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the relationship between the field intensity and the total mobility, using these constant values, a dimensionless total mobility is defined as m ¼ mt =mE 1 þ ðmEB =mE ÞB ffi 1 þ 2B:
netic field. This point deserves attention for future research. The mass transport equation, Eq. (3.2), then becomes
(4.1)
As can be seen the contribution of magnetic field intensity to the total mobility (i.e. total mass transport by electromigration) is almost linear (within the limits of experimental measurements), and is very significant. Electromigration is believed to be the result of momentum exchange of species in metallic liquids under the effect of an applied electric current, and the mobility of species (As in this case) is proportional to the electric current density with a proportionality constant (material constant) mE [1,2]. As mentioned earlier, mE is the electric mobility of a species, and has been calculated for various metallic solutions. What is happening to this mobility under an applied magnetic field? This question was answered in this work by introducing a mathematical model: assuming that the mobility of species in the bulk is affected by the applied magnetic field. Physically, this means that the momentum exchange becomes much easier under the effect of magnetic field; making the movement of species much easier and faster along the magnetic field lines due to less resistance (less momentum loss) to the mobility of species. Since the mass transport is enhanced drastically, this approach of looking first at mass transport in the bulk was the logical way to go about it. This physical concept was reflected mathematically through Eq. (4.1) which assumes that the mobility of species under magnetic field increases linearly with the field intensity, leading to higher mass transport rates as observed in experiments. It is possible that some other either bulk- or surfaceeffects may be responsible and/or contributing to the enhancement of mass transport under mag-
e5
DC r2 C þ mt E z
qC qC ¼ þ v rC qz qt
(4.2)
and the growth may then be calculated by r qC 1 V g ¼ L DC þ mt E z C ; qn CS C rS
(4.3)
Fig. 2. Evolution of growth interfaces: solid lines and dashed lines show interfaces computed using only the electric mobility under, respectively, B ¼ 0 and 1.0 kG field levels, and the dotted lines represent growth under 1.0 kG field using the total mobility. As can be seen, the growth rates using only the electric mobility (with and without magnetic field) are almost the same, which are in disagreement with experiments. The interface shapes are also not as flat as of experiments (see Fig. 1). The computed growth interfaces under a 1.0 kG magnetic field level using the total mobility (dotted lines) are in agreement with the experimental values.
Table 2 Calculated mobility values Magnetic field B (kG) at J ¼ 3 A=cm2 Experimental growth rate (mm/day)
0.0 0.50
1.0 1.62
2.0 2.35
4.5 6.10
Total mobility mt ¼ mE þ mEB B (m2/Vs)
0.7 105
2.3 105
3.4 105
7.1 105
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S. Dost et al. / Journal of Crystal Growth 275 (2005) e1–e6
where rL ; rS and C S are, respectively, the density of solution, density of solid crystal, and the solid composition. The evolution of growth interfaces, computed by using Eq. (4.3), is shown in Fig. 2. In the figure, the solid and dashed lines represent the computed growth interfaces using only the electric mobility, mE under, respectively, B ¼ 0 and 1.0 kG field levels [12,13]. As can be seen, the growth rates in both cases (with and without magnetic field) are almost the same, which are in disagreement with the experimental results of [14] and this work. The interface shapes are also not as flat as those of experiments (see the sample crystals in Fig. 1). The dotted lines represent the evolution of computed growth interfaces under magnetic field of 1.0 kG using the total mobility mt : These growth rates are in agreement with the experimental values. This shows the significance of inclusion of the electromagnetic mobility in the model.
5. Conclusions LPEE experiments show that the growth rate is proportional to the applied magnetic field and increases with the field intensity level. But the growth rate does only depend on the intensity of the applied magnetic field; not on its direction. The relationship between growth rate and magnetic field intensity is almost linear. A model that introduces a new mobility coefficient (electromagnetic mobility) predicts such high growth rates. The model also predicts flatter growth interface shapes. Results of numerical simulations performed for a binary system (GaAs) using the total mobility are in agreement with experiments.
Acknowledgment The financial support provided by the Microgravity Science Program of the Canadian Space Agency (CSA), and the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. References [1] L. Jastrzebski, H.C. Gatos, A.F. Witt, J. Electrochem. Soc. 123 (1976) 1121. [2] L. Jastrzebski, Y. Imamura, H.C. Gatos, J. Electrochem. Soc. 125 (1978) 1140. [3] K. Nakajima, J. Appl. Phys. 61 (9) (1987) 4626. [4] K. Nakajima, J. Crystal Growth 98 (1989) 329. [5] R.W. Series, D.T.J. Hurle, J. Crystal Growth 113 (1991) 305. [6] D.T.J. Hurle (Ed.), Handbook of Crystal Growth 2, Bulk Crystal Growth, Part B: Growth Mechanisms and Dynamics, North-Holland, Amsterdam, 1994. [7] J. Baumgartl, A. Hubert, G. Muller, Phys. Fluids A 5 (1993) 3280. [8] M. Salk, M. Fiederle, K.W. Benz, A.S. Senchenkov, A.V. Egorov, D.G. Matioukhin, J. Crystal Growth 138 (1994) 161. [9] S. Dost, Z. Qin, J. Crystal Growth 153 (1995) 123. [10] S. Senchenkov, I.V. Barmin, A.S. Tomson, V.V. Krapukhin, J. Crystal Growth 197 (1997) 552. [11] K. Ghaddar, C.K. Lee, S. Motakef, D.C. Gillies, J. Crystal Growth 205 (1999) 97. [12] S. Dost, Y.C. Liu, B. Lent, J. Crystal Growth 240 (2002) 39. [13] Y.C. Liu, Y. Okano, S. Dost, J. Crystal Growth 244 (2002) 12. [14] H. Sheibani, S. Dost, S. Sakai, B. Lent, J. Crystal Growth 258 (3–4) (2003) 283. [15] H. Sheibani, Y.C. Liu, S. Sakai, B. Lent, S. Dost, Int. J. Eng. Sci. 41 (2003) 401. [16] Y. Okano, H. Kondo, S. Dost, Int. J. Appl. Electromagn. Mech. 18 (4) (2003) 217. [17] Y.C. Liu, S. Dost, H. Sheibani, Int. J. Transport Phenom. 6 (2004) 51. [18] S. Dost, B. Lent, H. Sheibani, Y.C. Liu, C. R. Mecanique 332 (5–6) (2004) 413.