- Email: [email protected]
Comparing modified vertical gradient freezing with rotating magnetic fields or with steady magnetic and electric fields
ARTICLE IN PRESS
Journal of Crystal Growth 287 (2006) 270–274 www.elsevier.com/locate/jcrysgro
Comparing modified vertical gradient freezing with rotating magnetic fields or with steady magnetic and electric fields X. Wanga, N. Maa,, D.F. Blissb, G.W. Iselerc, P. Beclad a
Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695, USA b US Air Force Research Laboratory, AFRL/SNHC, 80 Scott Road, Hanscom AFB, MA 01731, USA c Iseler Associates, 26 State Street, Chelmsford, MA 01824, USA d Solid State Scientific, 27-2 Wright Road, Hollis, NH 03049, USA Available online 5 January 2006
Abstract This investigation treats the flow of molten gallium-antimonide and the dopant transport during the vertical gradient freezing process using submerged heater growth. A rotating magnetic field or a combination of steady magnetic and steady electric fields is used to control the melt motion. This paper compares the effects of these externally applied fields on the transport in the melt and on the dopant segregation in the crystal. Crystal growth in a combination of steady magnetic and electric fields produces a crystal with more radial and axial dopant homogeneity than growth in a rotating magnetic field. r 2005 Elsevier B.V. All rights reserved. PACS: 44.25.+f; 44.27.+g; 41.20.Gz; 64.75.+g; 81.10.h; 81.10.Aj; 81.10.Fq Keywords: A1. Fluid flows; A1. Magnetic fields; A1. Segregation; A2. Growth from melt; A2. VGF technique
1. Introduction Semiconductor crystals can be grown from the melt by the vertical gradient freezing (VGF) process using submerged heater growth (SHG). Because molten semiconductors are excellent electrical conductors, a rotating magnetic field or a combination of externally applied steady magnetic and steady electric fields can be used to control the motion of a molten semiconductor and minimize the dopant segregation in the crystal. Ostrogorsky [1] introduced a modification of the bottom-seeded vertical gradient freezing process in which a submerged heater separates the melt into two zones, viz., a lower melt and an upper melt. The lower melt is continuously replenished with liquid from the upper melt, with a composition chosen to offset the increasing dopant level in the lower melt due to rejection along the crystal–melt interface for dopants with segregation coefficients less than unity. Corresponding author. Tel.: +1 919 515 5231; fax: +1 919 515 7968.
E-mail address: [email protected] (N. Ma). 0022-0248/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2005.11.036
A rotating magnetic field (RMF) is a periodic transverse magnetic field which rotates in the azimuthal about the centerline of the melt. An RMF is produced by a number of magnet poles which are placed at equally spaced azimuthal positions around the crystal growth furnace and connected to successive phases of a multiphase AC power source. Witkowski et al. [2] and Walker et al. [3] studied numerical modeling for melt growth in an RMF with two poles. The present study is the first investigation in which an RMF is applied during submerged heater growth. An externally applied steady radial electric field and a steady axial magnetic field can be applied during the VGFSHG process. An axial magnetic field is generated by applying a DC electric current though a copper coil. An axially downward DC electric current is applied through a long cylindrical graphite electrode at the centerline and an electrode along the periphery of the melt. Ma et al. [4] and Wang et al. [5,6] applied a combination of DC fields during the VGF-SHG process. In the present investigation, we numerically model the melt motion and dopant transport during the VGF-SHG
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
process with an RMF, and with a combination of DC fields. The purpose of this study is to compare the effects of these external fields on the transport in the melt and the dopant distribution in the crystal.
2. Problem formulation This paper treats the unsteady, axisymmetric transport of selenium in a gallium-antimonide melt during the VGFSHG process. Our dimensionless problem is sketched in Fig. 1, where br, b h and bz are the unit vectors for the cylindrical coordinate system, where all lengths are normalized by the crystal’s radius R. The depth of the lower melt is b, the radius of the inner electrode is re, the radius of the submerged heater is rh, and the length of the crystal is h. Table 1 lists these dimensionless variables. A single crystal seed, which initiates solidification, lies along the bottom of the fused-silica crucible. A graphite disc and a boron nitride disc lie below the crucible. These discs are cooled by a water-cooled hearth which removes heat along the bottom of the crucible. As the crystal solidifies at z ¼ at, the crystal–melt interface moves axially upward at the dimensionless rate a ¼ Ug/Uc, where Ug is the constant growth rate, Uc ¼ m/rR is the characteristic velocity of the melt for a uniform density r, while m is the dynamic viscosity of the melt and t is time normalized by R/Uc. The surface of the submerged heater at z ¼ at þ b moves upward at the same rate so that the depth of the melt is constant throughout growth.
Table 1 Dimensionless variables with Uc ¼ n/R
Radial coordinate Axial coordinate Rescaled axial coordinate Temperature Lower melt depth Inner electrode radius Submerged heater radius Crystal length Growth rate Velocity Pressure EM body force in RMF EM body force in DC fields Time Concentration
Submerged Fused-Silica Heater Housing Fused-Silica Crucible
ζ=+1 LOWER MELT
b
ζ=-1 z
CRYSTAL
SEED
r R=2.5 cm Fig. 1. VGF-SHG process with coordinates normalized by the crucible’s inner radius.
Dimensionless
Values
r ¼ r*/R z ¼ z*/R z ¼ 1+2(zat)/b T ¼ (T*Ts)/(ThTs) b ¼ b*/R re ¼ re*/R rh ¼ rh*/R h ¼ h*/R a ¼ Ug/Uc v ¼ v*/Uc p ¼ (p*+rgz*)/(m2/ rR2) fy ¼ fy*/(sB2ooR) fy ¼ fy*/(B0IX2pr0bR) t ¼ t*/(R/Uc) C ¼ C*/C0
0prp1 atpzpat+0.4 1pzp+1 0pTp1 0.4 0.1905 0.8 3.2 0.036255 — — — — — —
*Denotes the dimensional variables.
In a reference frame moving with the heater and the crystal–melt interface, the melt motion is steady. The equations governing the melt motion and dopant transport are: Ra ðvdrÞv ¼ rp þ Gf y b T bz þ r2 v, hþ Pr
(1a)
rdv ¼ 0,
(1b)
1 2 r T, Pr
(1c)
qC 2 þ vdrC ¼ Pe1 m r C, qt
(1d)
vdrT ¼ rh
271
where T is the deviation of the temperature from the solidification temperature Ts normalized by (DT) ¼ (ThTs) with the Boussinesq approximation, p is the deviation of the pressure from the hydrostatic pressure normalized by m2/rR2, and the melt velocity v ¼ vrbr þ vy b h þ vzbz is normalized by Uc. Here, Th is the uniform and constant temperature of the heater’s surface. The dimensionless parameters in Eqs. (1a) and (1c) are the Rayleigh number Ra ¼ gb(DT)R3/nk and the Prandtl number Pr ¼ n/k. Here, g ¼ 9.81 m2/s, n ¼ m=r and k ¼ k=rcp are the kinematic viscosity and thermal diffusivity of the melt, respectively, while k, cp, b and s are the thermal conductivity, the specific heat, the thermal volumetric expansion coefficient and the electrical conductivity of the melt, respectively. In Eq. (1a), fy is the electromagnetic (EM) body force due to either the RMF or the combination of the DC fields, while G is the characteristic ratio of the EM body force to the inertial force. Eq. (1d) is the dopant transport equation, where Pem ¼ UcR/D is the mass Pe´clet number and D is the diffusion coefficient for the dopant in the molten semiconductor.
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
We use an RMF with two poles. The magnetic flux density is Bo. An RMF actually produces a periodic, nonaxisymmetric body force in addition to the steady, axisymmetric, azimuthal body force, but the frequency of the non-axisymmetric part is two times as the axisymmetric part. The fluid cannot respond to this non-axisymmetric part so that the melt motion produced by non-axisymmetric part is negligible [2]. With electrically insulating boundaries, the EM body force fy normalized by sBo2oR is [3] 1 J 1 ðln rÞ cosh 12 ln bz r X . (2) f y ðr; zÞ ¼ 2 n¼1 ðln 2 1ÞJ 1 ðln Þ cosh 12 ln b Here, o is the circular frequency of the AC electric power source, ln are the roots of ln J 0 ðln Þ J 1 ðln Þ ¼ 0, and Ji is the Bessel function of the first kind and ith order. The rescaled axial co-ordinate is z ¼ 1+2(zat)/b so that 1pzp+1. For an RMF, in Eq. (1a), G ¼ soBo2R4/rn2, which is also known as the magnetic Taylor number. We present the contours of fy for b ¼ 0.4 in Fig. 2, which drives flow in the azimuthal direction. The maximum value of fy is at r ¼ 1 and z ¼ 0. We use a combination of a steady axial magnetic field B0 and a steady electric current I. The electric current flows radially outward and axially downward toward the outer graphite electrode. The induced electric field is negligible compared with the static electric field [4], and the EM body force fy normalized by B0 I=2pr0 bR is 1 cosh wn2 b ðz þ 1Þ X f y ðr; zÞ ¼ , (3a) An Fwn J 1 ðwn rÞ cosh wn b n¼1 " F¼b
1 X n¼1
#1 sinhðln bÞ An J 1 ðwn r0 Þ , coshðln bÞ
(3b)
where wn are the roots of J0(wn) ¼ 0. For a combination of DC fields, in Eq. (1a), G ¼ B0 J c R3 =rn2 . Here, J c ¼ I=ð2pr0 bRÞ is the characteristic electric current density, where r0 is a radial position that can be used to provide an approximate estimate of the radial electric current density. This solution given by Eq. (3a) is only valid for weak magnetic fields for which the induced electric field is
1 -0.1 -2
-0.35 -1
ζ
272
-0.4
0
-0.5
-0.7 -0.6
-0.35 -0.5 -0.6 -0.4
-1 0
0.2
0.4
0.6
0.8
1
r Fig. 3. Dimensionless, steady, azimuthal, EM body force fy in DC fields with b ¼ 0.4 and re ¼ 0.1905.
negligible. Wang et al. [6] found that growth in weak fields produces better crystals with less segregation. In Fig. 3, we present the contours of fy for b ¼ 0:4 and re ¼ 0:1905. The maximum magnitude of the body force occurs at the periphery of the inner electrode at the top of the melt, which is much larger than that produced by the RMF in Fig. 2. We use no-slip and no-penetration conditions along the crucible, on the surface of the submerged heater. We assume a simple velocity profile vz ¼ a½1 þ k1 ð1 rÞ þ k2 ð1 r2 Þ in the gap [4], where k1 ¼ 3ð1 þ r2h Þ=ð1 rh Þ3 and k2 ¼ 2ð1 þ rh þ r2h Þ=ð1 rh Þ3 ð1 þ rh Þ. The thermal boundary conditions are T ¼ 0 at z ¼ 1, T ¼ 1 at z ¼ þ1 and qT=qr ¼ 0 at r ¼ 1. Before solidification begins, the dopant is at the initial uniform concentration Co, which is used to normalize the concentration in the melt and crystal, so that Cðr; z; t ¼ 0Þ ¼ 1. The boundary condition along the crystal–melt interface is ð2=bÞqC=qz ¼ Peg ðks 1ÞC, where Peg ¼ U g R=D ¼ aPem is the growth Pe´clet number and ks ¼ 0:1 is the segregation coefficient for the dopant in the molten semiconductor. The boundary condition at the gap is C ¼ ks. The surfaces of the crucible and heater are impermeable. The concentration in the crystal Cs(r,z) ¼ ksC(r,z ¼ 1, t ¼ z/a). We introduce a Stokes streamfunction for the radial and axial velocities, vr ¼ ð1=rÞ@c=@z and vz ¼ ð1=rÞqc=qr, respectively, which identically satisfies conservation of mass Eq. (1b). We use Chebyshev spectral method with Gauss–Lobatto collocation points to numerically approximate the equations.
1
ζ
3. Results 0
0.0001 0.0003
0.001
0.003
0.01
0.03
0.07
-1 0
0.2
0.4
0.6
0.8
1
r Fig. 2. Dimensionless, steady, azimuthal, EM body force fy in an RMF with b ¼ 0.4.
For the present process, b ¼ 0:4, re ¼ 0:1905, rh ¼ 0:8, and h ¼ 3:2. With the thermophysical properties of molten gallium-antimonide, Pr ¼ 0:0443, Ra ¼ 88; 641:4, Pem ¼ 19:1542 and Uc ¼ 1.532 105 m/s. For a growth rate Ug ¼ 2 mm/hr, Peg ¼ 0.6944, a ¼ 0.036255 and the dimensionless time to grow the crystal is h/a ¼ 88.2627. We present results for growth in an RMF with Bo ¼ 1 mT, for which the magnetic Taylor number is G ¼ 166,411.9. The isotherms are horizontal so that there is
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
273
1
ζ
3
0.5
0
2
5
10
20
30
40 0.0155
2.5
-1 0
0.2
0.4
0.6
0.8
2
1
0.0165
r
(a)
z
1
1.5 0.01
0.0005 0.002 ζ
0.0145
0.03
0.0172
0.06
0
1 -0.001
-0.01
0.018
-0.04 0.5
-1 0
0.2
0.4
0.6
0.8
0.025
1
0.04
r
(b)
Fig. 4. Steady state melt motion in an RMF: (a) azimuthal velocity vy(r,z), (b) meridional streamfunction c(r,z).
0.07
0 0
0.5 r
1
Fig. 6. Constant-concentration curves in the crystal with an RMF Cs(r,z).
1 0.11 0.12 ζ
0.13 0.14
0 0.15 0.16 0.17 -1 0
0.2
0.4
0.6
0.8
1
r Fig. 5. Constant-concentration curves in the melt in RMF after steady state, C(r,z,t ¼ 39.9389).
no thermal buoyant convection. Figs. 4(a) and (b) present for the steady melt motion. The azimuthal EM body force drives a forced azimuthal flow shown in Fig. 4a, where the maximum value of the azimuthal velocity is vy,max ¼ 43.6809. In Fig. 4b, the meridional melt motion is split into two circulations, i.e., an upper counterclockwise circulation and a lower clockwise one. The gap flow due to the replenishment reflected in the c ¼ 0.0005 and c ¼ 0.002 contours. The maximum and minimum values of the meridional streamfunction are cmin ¼ 0.04740 and cmax ¼ 0.06964, respectively. Initially, the melt’s concentration is uniform at C ¼ 1. At early stages of growth, the concentration along z ¼ 1 rises to C41 due to rejection along the crystal–melt interface. As growth progresses, the meridional melt motion provides a significant convective dopant transport, and the concentration is Co1 everywhere. When 45.25% of the crystal has grown at t ¼ 39.9389, the dopant transport reaches a steady state
as shown in Fig. 5, where the maximum value is Cmax ¼ 0.1751. The minimum value is always equal to 0.1 because of the replenishment in the gap. After steady state dopant transport has been achieved, the crystal solidifies with the same radial distribution. In this section of the crystal, the difference between the dopant concentration at the centerline and at the periphery is D ¼ 0.003370. The crystal’s dopant distribution is presented in Fig. 6. In order to compare the effects of RMF and a combination of DC fields, we treat growth in the DC fields with the same value of G ¼ 166,411.9, which corresponds to B0Jc ¼ 9.4248 T A/m2. The isotherms are horizontal. We present the steady melt motion in Fig. 7. The EM body force produces a much stronger azimuthal flow compared with the RMF, as reflected in Fig. 7a, where the minimum value of the azimuthal velocity is vy,min ¼ 707.085. This stronger azimuthal flow drives a stronger secondary meridional flow, as shown in Fig. 7b. The minimum and maximum values of the meridional streamfunction are cmin ¼ 6.4529 and cmax ¼ 6.7415, respectively. This stronger EM stirring provides a much stronger convective dopant transport, so that the dopant transport reaches a steady state at an earlier time t ¼ 11.0328 when only 12.5% of the crystal has grown, which is presented in Fig. 8, where the maximum value is Cmax ¼ 0.1084. In the section of the crystal that has solidified after steady state, the radial segregation is D ¼ 0.0001599, which is more than an order of magnitude smaller than that for a crystal grown in an RMF.
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
274
1
-300 ζ
the VGF-SHG process. Both cases provide a strong EM stirring and, therefore, provide a strong convective dopant transport. A combination of DC fields produces a crystal with more axial and radial uniformity than an RMF with the equivalent parameters.
-50
0
-600 -500
-570
-570 -600
-1 0
0.2
0.4
0.6
0.8
1
r
(a) 1
This research was supported by the US Air Force Office of Scientific Research under Grant FA9550-04-1-0249. The calculations were performed on the Cray X1 provided by the DoD High Performance Computing Modernization Program under Grant AFSNH2487.
0.5 3
6
-0.5 ζ
Acknowledgments
0 -3 -6 -1 0
0.2
0.4
0.6
0.8
1
r
(b)
Fig. 7. Steady state melt motion with DC fields: (a) azimuthal velocity vy(r,z), (b) meridional streamfunction c(r,z).
1 0.1045 ζ
0.105 0
0.1065 0.106
-1
0.107 0
0.2
0.4
0.6
0.8
1
r Fig. 8. Constant-concentration curves in the melt in DC fields after steady state, C(r,z,t ¼ 11.0328).
4. Conclusions We have compared the effects of either an RMF or a combination of DC fields on the dopant transport during
References [1] A.G. Ostrogorsky, J. Crystal Growth 104 (1990) 233. [2] L.M. Witkowski, J.S. Walker, P. Marty, Phys. Fluid. 11 (1999) 1821. [3] J.S. Walker, M.P. Volz, K. Mazuruk, Int. J. Heat Mass Transfer 47 (2004) 1877. [4] N. Ma, D.F. Bliss, G.W. Iseler, J. Crystal Growth 259 (2003) 26. [5] X. Wang, N. Ma, D.F. Bliss, G.W. Iseler, J. Thermophys. Heat Transfer 19 (2005) 95. [6] X. Wang, N. Ma, D.F. Bliss, G.W. Iseler, Int. J. Eng. Sci. 43 (2005) 908.
Journal of Crystal Growth 287 (2006) 270–274 www.elsevier.com/locate/jcrysgro
Comparing modified vertical gradient freezing with rotating magnetic fields or with steady magnetic and electric fields X. Wanga, N. Maa,, D.F. Blissb, G.W. Iselerc, P. Beclad a
Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695, USA b US Air Force Research Laboratory, AFRL/SNHC, 80 Scott Road, Hanscom AFB, MA 01731, USA c Iseler Associates, 26 State Street, Chelmsford, MA 01824, USA d Solid State Scientific, 27-2 Wright Road, Hollis, NH 03049, USA Available online 5 January 2006
Abstract This investigation treats the flow of molten gallium-antimonide and the dopant transport during the vertical gradient freezing process using submerged heater growth. A rotating magnetic field or a combination of steady magnetic and steady electric fields is used to control the melt motion. This paper compares the effects of these externally applied fields on the transport in the melt and on the dopant segregation in the crystal. Crystal growth in a combination of steady magnetic and electric fields produces a crystal with more radial and axial dopant homogeneity than growth in a rotating magnetic field. r 2005 Elsevier B.V. All rights reserved. PACS: 44.25.+f; 44.27.+g; 41.20.Gz; 64.75.+g; 81.10.h; 81.10.Aj; 81.10.Fq Keywords: A1. Fluid flows; A1. Magnetic fields; A1. Segregation; A2. Growth from melt; A2. VGF technique
1. Introduction Semiconductor crystals can be grown from the melt by the vertical gradient freezing (VGF) process using submerged heater growth (SHG). Because molten semiconductors are excellent electrical conductors, a rotating magnetic field or a combination of externally applied steady magnetic and steady electric fields can be used to control the motion of a molten semiconductor and minimize the dopant segregation in the crystal. Ostrogorsky [1] introduced a modification of the bottom-seeded vertical gradient freezing process in which a submerged heater separates the melt into two zones, viz., a lower melt and an upper melt. The lower melt is continuously replenished with liquid from the upper melt, with a composition chosen to offset the increasing dopant level in the lower melt due to rejection along the crystal–melt interface for dopants with segregation coefficients less than unity. Corresponding author. Tel.: +1 919 515 5231; fax: +1 919 515 7968.
E-mail address: [email protected] (N. Ma). 0022-0248/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2005.11.036
A rotating magnetic field (RMF) is a periodic transverse magnetic field which rotates in the azimuthal about the centerline of the melt. An RMF is produced by a number of magnet poles which are placed at equally spaced azimuthal positions around the crystal growth furnace and connected to successive phases of a multiphase AC power source. Witkowski et al. [2] and Walker et al. [3] studied numerical modeling for melt growth in an RMF with two poles. The present study is the first investigation in which an RMF is applied during submerged heater growth. An externally applied steady radial electric field and a steady axial magnetic field can be applied during the VGFSHG process. An axial magnetic field is generated by applying a DC electric current though a copper coil. An axially downward DC electric current is applied through a long cylindrical graphite electrode at the centerline and an electrode along the periphery of the melt. Ma et al. [4] and Wang et al. [5,6] applied a combination of DC fields during the VGF-SHG process. In the present investigation, we numerically model the melt motion and dopant transport during the VGF-SHG
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
process with an RMF, and with a combination of DC fields. The purpose of this study is to compare the effects of these external fields on the transport in the melt and the dopant distribution in the crystal.
2. Problem formulation This paper treats the unsteady, axisymmetric transport of selenium in a gallium-antimonide melt during the VGFSHG process. Our dimensionless problem is sketched in Fig. 1, where br, b h and bz are the unit vectors for the cylindrical coordinate system, where all lengths are normalized by the crystal’s radius R. The depth of the lower melt is b, the radius of the inner electrode is re, the radius of the submerged heater is rh, and the length of the crystal is h. Table 1 lists these dimensionless variables. A single crystal seed, which initiates solidification, lies along the bottom of the fused-silica crucible. A graphite disc and a boron nitride disc lie below the crucible. These discs are cooled by a water-cooled hearth which removes heat along the bottom of the crucible. As the crystal solidifies at z ¼ at, the crystal–melt interface moves axially upward at the dimensionless rate a ¼ Ug/Uc, where Ug is the constant growth rate, Uc ¼ m/rR is the characteristic velocity of the melt for a uniform density r, while m is the dynamic viscosity of the melt and t is time normalized by R/Uc. The surface of the submerged heater at z ¼ at þ b moves upward at the same rate so that the depth of the melt is constant throughout growth.
Table 1 Dimensionless variables with Uc ¼ n/R
Radial coordinate Axial coordinate Rescaled axial coordinate Temperature Lower melt depth Inner electrode radius Submerged heater radius Crystal length Growth rate Velocity Pressure EM body force in RMF EM body force in DC fields Time Concentration
Submerged Fused-Silica Heater Housing Fused-Silica Crucible
ζ=+1 LOWER MELT
b
ζ=-1 z
CRYSTAL
SEED
r R=2.5 cm Fig. 1. VGF-SHG process with coordinates normalized by the crucible’s inner radius.
Dimensionless
Values
r ¼ r*/R z ¼ z*/R z ¼ 1+2(zat)/b T ¼ (T*Ts)/(ThTs) b ¼ b*/R re ¼ re*/R rh ¼ rh*/R h ¼ h*/R a ¼ Ug/Uc v ¼ v*/Uc p ¼ (p*+rgz*)/(m2/ rR2) fy ¼ fy*/(sB2ooR) fy ¼ fy*/(B0IX2pr0bR) t ¼ t*/(R/Uc) C ¼ C*/C0
0prp1 atpzpat+0.4 1pzp+1 0pTp1 0.4 0.1905 0.8 3.2 0.036255 — — — — — —
*Denotes the dimensional variables.
In a reference frame moving with the heater and the crystal–melt interface, the melt motion is steady. The equations governing the melt motion and dopant transport are: Ra ðvdrÞv ¼ rp þ Gf y b T bz þ r2 v, hþ Pr
(1a)
rdv ¼ 0,
(1b)
1 2 r T, Pr
(1c)
qC 2 þ vdrC ¼ Pe1 m r C, qt
(1d)
vdrT ¼ rh
271
where T is the deviation of the temperature from the solidification temperature Ts normalized by (DT) ¼ (ThTs) with the Boussinesq approximation, p is the deviation of the pressure from the hydrostatic pressure normalized by m2/rR2, and the melt velocity v ¼ vrbr þ vy b h þ vzbz is normalized by Uc. Here, Th is the uniform and constant temperature of the heater’s surface. The dimensionless parameters in Eqs. (1a) and (1c) are the Rayleigh number Ra ¼ gb(DT)R3/nk and the Prandtl number Pr ¼ n/k. Here, g ¼ 9.81 m2/s, n ¼ m=r and k ¼ k=rcp are the kinematic viscosity and thermal diffusivity of the melt, respectively, while k, cp, b and s are the thermal conductivity, the specific heat, the thermal volumetric expansion coefficient and the electrical conductivity of the melt, respectively. In Eq. (1a), fy is the electromagnetic (EM) body force due to either the RMF or the combination of the DC fields, while G is the characteristic ratio of the EM body force to the inertial force. Eq. (1d) is the dopant transport equation, where Pem ¼ UcR/D is the mass Pe´clet number and D is the diffusion coefficient for the dopant in the molten semiconductor.
ARTICLE IN PRESS X. Wang et al. / Journal of Crystal Growth 287 (2006) 270–274
We use an RMF with two poles. The magnetic flux density is Bo. An RMF actually produces a periodic, nonaxisymmetric body force in addition to the steady, axisymmetric, azimuthal body force, but the frequency of the non-axisymmetric part is two times as the axisymmetric part. The fluid cannot respond to this non-axisymmetric part so that the melt motion produced by non-axisymmetric part is negligible [2]. With electrically insulating boundaries, the EM body force fy normalized by sBo2oR is [3] 1 J 1 ðln rÞ cosh 12 ln bz r X . (2) f y ðr; zÞ ¼ 2 n¼1 ðln 2 1ÞJ 1 ðln Þ cosh 12 ln b Here, o is the circular frequency of the AC electric power source, ln are the roots of ln J 0 ðln Þ J 1 ðln Þ ¼ 0, and Ji is the Bessel function of the first kind and ith order. The rescaled axial co-ordinate is z ¼ 1+2(zat)/b so that 1pzp+1. For an RMF, in Eq. (1a), G ¼ soBo2R4/rn2, which is also known as the magnetic Taylor number. We present the contours of fy for b ¼ 0.4 in Fig. 2, which drives flow in the azimuthal direction. The maximum value of fy is at r ¼ 1 and z ¼ 0. We use a combination of a steady axial magnetic field B0 and a steady electric current I. The electric current flows radially outward and axially downward toward the outer graphite electrode. The induced electric field is negligible compared with the static electric field [4], and the EM body force fy normalized by B0 I=2pr0 bR is 1 cosh wn2 b ðz þ 1Þ X f y ðr; zÞ ¼ , (3a) An Fwn J 1 ðwn rÞ cosh wn b n¼1 " F¼b
1 X n¼1
#1 sinhðln bÞ An J 1 ðwn r0 Þ , coshðln bÞ
(3b)
where wn are the roots of J0(wn) ¼ 0. For a combination of DC fields, in Eq. (1a), G ¼ B0 J c R3 =rn2 . Here, J c ¼ I=ð2pr0 bRÞ is the characteristic electric current density, where r0 is a radial position that can be used to provide an approximate estimate of the radial electric current density. This solution given by Eq. (3a) is only valid for weak magnetic fields for which the induced electric field is
1 -0.1 -2
-0.35 -1
ζ
272
-0.4
0
-0.5
-0.7 -0.6
-0.35 -0.5 -0.6 -0.4
-1 0
0.2
0.4
0.6
0.8
1
r Fig. 3. Dimensionless, steady, azimuthal, EM body force fy in DC fields with b ¼ 0.4 and re ¼ 0.1905.
negligible. Wang et al. [6] found that growth in weak fields produces better crystals with less segregation. In Fig. 3, we present the contours of fy for b ¼ 0:4 and re ¼ 0:1905. The maximum magnitude of the body force occurs at the periphery of the inner electrode at the top of the melt, which is much larger than that produced by the RMF in Fig. 2. We use no-slip and no-penetration conditions along the crucible, on the surface of the submerged heater. We assume a simple velocity profile vz ¼ a½1 þ k1 ð1 rÞ þ k2 ð1 r2 Þ in the gap [4], where k1 ¼ 3ð1 þ r2h Þ=ð1 rh Þ3 and k2 ¼ 2ð1 þ rh þ r2h Þ=ð1 rh Þ3 ð1 þ rh Þ. The thermal boundary conditions are T ¼ 0 at z ¼ 1, T ¼ 1 at z ¼ þ1 and qT=qr ¼ 0 at r ¼ 1. Before solidification begins, the dopant is at the initial uniform concentration Co, which is used to normalize the concentration in the melt and crystal, so that Cðr; z; t ¼ 0Þ ¼ 1. The boundary condition along the crystal–melt interface is ð2=bÞqC=qz ¼ Peg ðks 1ÞC, where Peg ¼ U g R=D ¼ aPem is the growth Pe´clet number and ks ¼ 0:1 is the segregation coefficient for the dopant in the molten semiconductor. The boundary condition at the gap is C ¼ ks. The surfaces of the crucible and heater are impermeable. The concentration in the crystal Cs(r,z) ¼ ksC(r,z ¼ 1, t ¼ z/a). We introduce a Stokes streamfunction for the radial and axial velocities, vr ¼ ð1=rÞ@c=@z and vz ¼ ð1=rÞqc=qr, respectively, which identically satisfies conservation of mass Eq. (1b). We use Chebyshev spectral method with Gauss–Lobatto collocation points to numerically approximate the equations.
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For the present process, b ¼ 0:4, re ¼ 0:1905, rh ¼ 0:8, and h ¼ 3:2. With the thermophysical properties of molten gallium-antimonide, Pr ¼ 0:0443, Ra ¼ 88; 641:4, Pem ¼ 19:1542 and Uc ¼ 1.532 105 m/s. For a growth rate Ug ¼ 2 mm/hr, Peg ¼ 0.6944, a ¼ 0.036255 and the dimensionless time to grow the crystal is h/a ¼ 88.2627. We present results for growth in an RMF with Bo ¼ 1 mT, for which the magnetic Taylor number is G ¼ 166,411.9. The isotherms are horizontal so that there is
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Fig. 6. Constant-concentration curves in the crystal with an RMF Cs(r,z).
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r Fig. 5. Constant-concentration curves in the melt in RMF after steady state, C(r,z,t ¼ 39.9389).
no thermal buoyant convection. Figs. 4(a) and (b) present for the steady melt motion. The azimuthal EM body force drives a forced azimuthal flow shown in Fig. 4a, where the maximum value of the azimuthal velocity is vy,max ¼ 43.6809. In Fig. 4b, the meridional melt motion is split into two circulations, i.e., an upper counterclockwise circulation and a lower clockwise one. The gap flow due to the replenishment reflected in the c ¼ 0.0005 and c ¼ 0.002 contours. The maximum and minimum values of the meridional streamfunction are cmin ¼ 0.04740 and cmax ¼ 0.06964, respectively. Initially, the melt’s concentration is uniform at C ¼ 1. At early stages of growth, the concentration along z ¼ 1 rises to C41 due to rejection along the crystal–melt interface. As growth progresses, the meridional melt motion provides a significant convective dopant transport, and the concentration is Co1 everywhere. When 45.25% of the crystal has grown at t ¼ 39.9389, the dopant transport reaches a steady state
as shown in Fig. 5, where the maximum value is Cmax ¼ 0.1751. The minimum value is always equal to 0.1 because of the replenishment in the gap. After steady state dopant transport has been achieved, the crystal solidifies with the same radial distribution. In this section of the crystal, the difference between the dopant concentration at the centerline and at the periphery is D ¼ 0.003370. The crystal’s dopant distribution is presented in Fig. 6. In order to compare the effects of RMF and a combination of DC fields, we treat growth in the DC fields with the same value of G ¼ 166,411.9, which corresponds to B0Jc ¼ 9.4248 T A/m2. The isotherms are horizontal. We present the steady melt motion in Fig. 7. The EM body force produces a much stronger azimuthal flow compared with the RMF, as reflected in Fig. 7a, where the minimum value of the azimuthal velocity is vy,min ¼ 707.085. This stronger azimuthal flow drives a stronger secondary meridional flow, as shown in Fig. 7b. The minimum and maximum values of the meridional streamfunction are cmin ¼ 6.4529 and cmax ¼ 6.7415, respectively. This stronger EM stirring provides a much stronger convective dopant transport, so that the dopant transport reaches a steady state at an earlier time t ¼ 11.0328 when only 12.5% of the crystal has grown, which is presented in Fig. 8, where the maximum value is Cmax ¼ 0.1084. In the section of the crystal that has solidified after steady state, the radial segregation is D ¼ 0.0001599, which is more than an order of magnitude smaller than that for a crystal grown in an RMF.
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the VGF-SHG process. Both cases provide a strong EM stirring and, therefore, provide a strong convective dopant transport. A combination of DC fields produces a crystal with more axial and radial uniformity than an RMF with the equivalent parameters.
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This research was supported by the US Air Force Office of Scientific Research under Grant FA9550-04-1-0249. The calculations were performed on the Cray X1 provided by the DoD High Performance Computing Modernization Program under Grant AFSNH2487.
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Acknowledgments
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4. Conclusions We have compared the effects of either an RMF or a combination of DC fields on the dopant transport during
References [1] A.G. Ostrogorsky, J. Crystal Growth 104 (1990) 233. [2] L.M. Witkowski, J.S. Walker, P. Marty, Phys. Fluid. 11 (1999) 1821. [3] J.S. Walker, M.P. Volz, K. Mazuruk, Int. J. Heat Mass Transfer 47 (2004) 1877. [4] N. Ma, D.F. Bliss, G.W. Iseler, J. Crystal Growth 259 (2003) 26. [5] X. Wang, N. Ma, D.F. Bliss, G.W. Iseler, J. Thermophys. Heat Transfer 19 (2005) 95. [6] X. Wang, N. Ma, D.F. Bliss, G.W. Iseler, Int. J. Eng. Sci. 43 (2005) 908.