GaAs Vertical Gradient Freeze Process Intensification - American

Article pubs.acs.org/crystal

GaAs Vertical Gradient Freeze Process Intensification Natasha Dropka,* Alexander Glacki, and Christiane Frank-Rotsch Leibniz-Institute for Crystal Growth (IKZ), Max-Born-Strasse 2, 12489 Berlin, Germany ABSTRACT: The GaAs vertical gradient freeze (VGF) crystal growth process can be intensified in various ways, e.g., utilizing external fields, scaling up, numbering up the crucibles, etc. Successful application of traveling magnetic fields (TMFs) in 4 in. VGF GaAs growth for the flow control and process acceleration encouraged us to search for synergistic conceptions. Pros and cons of process scale up and numbering up under TMFs were addressed using threedimensional numerical simulations. The comparison of concepts was focused on the control of solid/liquid interface morphology and energy balance. A novel multicrucible furnace design was proposed and compared with a single-crucible design, both based on KRISTMAG technology. The simulation results showed the clear superiority of the numbering-up concept; e.g., the total energy consumption per run with equal yield was reduced to 32% of the value for the standard process. Moreover, a beneficial interface morphology was achievable without a trade-off with growth rates.

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INTRODUCTION To meet the contemporary needs of high-frequency microelectronic and optoelectronic applications, high-yield highperfection VGF GaAs crystals are required at low process costs. The severe requirements imposed on the wafer with regard to the homogeneity of dislocation density, the high thermal stability of the structural defects, and the high mechanical strength should suppress degradation of produced devices.1,2 In comparison to other industrially applied bulk growth methods such as liquid encapsulation Czochralski (LEC), vaporpressure-controlled Czochralski (VCz), and vertical Bridgman (VB), VGF is characterized by low investment and operating costs but also by low productivity.2 The reason for the latter lies mainly in different growth velocities; i.e., in the LEC process, the crystals can be grown fairly rapidly at 7−10 mm/h, while in the VGF process, crystal growth is considerably slower (∼3 mm/h).2 Since the first attempts of Gault et al. in the 1980s to grow GaAs by the VGF method,3 this process has matured remarkably, particularly because of the systematic optimization of the furnace setup by numerical modeling. The temperature field at all process stages was improved by optimizing the design of the crucible, crucible support, and heaters and coolers as well as process parameters.4−7 Nevertheless, the low productivity of the VGF process remained an issue. One way to enhance the yield is to perform process intensification utilizing, e.g., external fields (e.g., magnetic fields, vibration, and ultrasound), scaling up the crucibles, numbering up the crucibles, etc. The front-runner of industrial VGF process intensification is the scale-up approach. It refers to an increase in the crystal load, i.e., crystal diameter and/or crystal length. So far, GaAs crystals up to 8 in. in diameter have been reported.8 Structural and electrical features of wafers made from 8 in. crystals were similar to those known for state-of-the-art 6 in. SI GaAs VGF/ © 2014 American Chemical Society

VB crystals, but only in the case of short ingots with a maximal length of 14 cm. With longer ingots, their quality abruptly decreased while the average EPD value increased. An increase in crystal size reduces the surface to volume ratio of GaAs and thereby inhibits the exchange of heat with the surroundings. Moreover, more latent heat is generated at the solid/liquid (s/ l) interface that has to be removed. As a consequence, the interface shape near the crystal periphery easily becomes concave, increasing the local thermal stress that causes an increase in the dislocation densities, their accumulation, and finally polycrystalline growth.2,9−11 The increased risk of nucleation of polycrystallinity is the main constraint that hampers scaling up. The numbering-up approach describes parallel repetitions and utilization of growth units to achieve higher total throughputs. This already well-established method in chemical industry12 reached for the first time industrial maturity in multicrucible VGF furnaces.13,14 The proposed designs consisted of several crucibles arranged side by side and a common resistance heating device surrounding them. The s/l interface shape in each melt was solely determined by the temporal change of the heater power distribution, i.e., by the cooling program. No additional tools for the control of the melt flow and interface morphology were available. To the best of our knowledge, neither experimental nor numerical studies of the transport phenomena in multicrucible VGF furnaces have appeared in the literature. In the past few years, successful control of the melt motion by TMF particularly boosted further development of VGF single-crystal growth.15−19 TMF and its Lorentz forces alter the flow and the convective heat transport, particularly in the Received: June 4, 2014 Revised: August 28, 2014 Published: September 5, 2014 5122

dx.doi.org/10.1021/cg500814m | Cryst. Growth Des. 2014, 14, 5122−5130

Crystal Growth & Design

Article

Figure 1. Crucial parts of VGF furnace geometry: (a) side HMM with three heating coils in a single-crucible design,25 (b) side HMM with three heating coils in a multicrucible (four) furnace design,26 and (c) 4 in. GaAs ingot, pBN crucible, and B2O3.

vicinity of the growing interface. The changed flow pattern affects the interface morphology by reducing the level of interface bending, as shown experimentally for Ge and GaAs and numerically for GaAs, Ge, Si, CdTe, and BaF2.17−24 We also obtained promising numerical results of interface shaping by downward TMF during accelerated growth of 4 in. VGF GaAs over the range of crystal growth rates from 3 to 9 mm/ h.16 Although a number of articles appeared in the literature about the numerical investigations of VGF processes, they were mostly based on quasi-steady axisymmetrical (2D) models of lab-size equipment with a simplified cylindrical crucible neglecting a conical base. Bearing in mind the sensitivity of single-crystal growth to any disturbance in thermal and magnetic fields and from another side a challenge to ensure their rotational symmetry, we find there is still a great need for full three-dimensional (3D) numerical studies on an industrial scale that will identify the critical parameters of the process and allow its improvement. To overcome the drawbacks of the existing concepts and to emphasize their advantages, we applied a synergy principle combining process scale up and numbering up with TMF. For this purpose, we derived a novel multicrucible furnace design and compared multicrucible VGF furnaces with a singlecrucible design for 4 in. ingots of various weights.25,26 Both considered furnaces were equipped with an internal side KRISTMAG heater-magnet module (HMM) with graphite heating coils positioned on each other and supplied with a combination of dc and out-of-phase ac.27 Such a design allows the simultaneous generation of heat and TMF in the melt. User-defined Lorentz forces of required direction, magnitude, and spatial distribution were induced in the melt by the variation of the ac/dc ratio, frequency, and phase shift among the HMM coils. For the first time, we performed systematic close-to-real transient 3D CFD and electromagnetic numerical studies with the furnaces mentioned above. The obtained numerical results allowed us to address pros and cons of synergic process intensification concepts with respect to the control of s/l interface morphology and energy balance.

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model for multicrucible furnace comprised another side KRISTMAG HMM (Figure 1b) with three heating coils that encircled four 4 in. crucibles of various loadings, analogous to the single-crucible case. The material data of GaAs used in this study are listed in Table 1.

Table 1. Material Properties of GaAs Used in This Study20,28 solid GaAs λ (W m−1 K−1) ρ (kg/m3) Cp (J kg−1 K−1) ΔHl,s (J/kg) σel (Ω−1 m−1) ν (Pa s)

melted GaAs

7.12 5170 420

17.8 5720 434 7.26 × 105

1.0 × 10 −

7.9 × 105 2.79 × 10−3

The transport phenomena were governed by the continuity, Navier−Stokes, energy, and induction equations together with Ohms law and Lorentz force density (eqs 1−8). Lorentz force density FL was defined as a force per unit volume acting on a melt. ∇·u ⃗ = 0

(1)

⎡ ∂u ⃗ ⎤ ρ⎢ + (u ⃗ ·∇)u ⃗ ⎥ = ρg ⃗ − ∇p + μ∇2 u ⃗ + FL⃗ ⎣ ∂t ⎦

(2)

ρg = ρref (1 − βTref )g

(3)

⎡ ∂T ⎤ ρCp⎢ + (u ⃗ · ∇)T ⎥ = λ∇2 T ⎣ ∂t ⎦

(4)

∂B ⃗ 1 =− (∇2 B ⃗ ) + [∇ × (u ⃗ × B ⃗ )] ∂t μσel

(5)

∇·B ⃗ = 0

(6)

j ⃗ = σel[E ⃗ + (u ⃗ × ⇀ B )] ≈ σelE ⃗

(7)

FL⃗ = j ⃗ × B ⃗

(8)

Because of the complexity of a 3D model, we applied a stepwise methodology, performing global simulations for the whole furnace where melt convection was neglected and local simulations for the GaAs melt and crystal, including the influence of buoyancy and TMF. The global 3D thermal analysis provided time-dependent thermal boundary conditions (BCs) for the local 3D simulation. Dirichlet (temperature) BCs were used at the crystal boundaries and Neumann (heat flux) BCs at the melt boundaries. The flux BCs allowed the study of an impact of the buoyancy and Lorentz forces on the peripheral melt flow and the temperature gradients. The local simulations were additionally split into electromagnetic (EM) and hydrodynamic parts that were solved consequently using the commercial software ANSYS

MODEL AND METHODOLOGY

The model for a single-crucible furnace described GaAs growth under downward TMF that was generated by a side KRISTMAG HMM with three heating coils (Figure 1a). Supplementary heat originated from top and bottom common resistance heaters. A cylindrical 4 in. crucible with a conical bottom made of pyrolytic boron nitride (pBN) was loaded with 5−9 kg of GaAs and covered with B2O3 (Figure 1a,c). The 5123



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Figure 2. Maximal Lorentz force density as a function of (a) frequency f, (b) ac amplitude Io, and (c) phase shift Δϕ. CFX 14.0 and Ansys Emag, i.e., externally coupled. The s/l interface position was determined iteratively using the following approach. (i) The initial Lorentz force density distribution in the melt was calculated for the initial interface position obtained from the global simulations. (ii) The Lorentz force density was used as a mechanical momentum source in the local flow simulations. (iii) At the end of the local CFD simulations, the new obtained interface position, i.e., the new melt geometry, was used for Lorentz force recalculation. (iv) The process was repeated several times until agreement between the interface and isotherm shapes in the melt was reached. In all EM simulations, melt was considered as a solid body. A similar approach was successfully applied in the numerical modeling of free surface dynamics in the magnetic field.29 The meshes for global simulation in the case of one- and fourcrucible furnaces consisted of 1.5 × 107 and 1.95 × 107 control volumes, respectively. The mesh for local simulation of transport phenomena taking place in, e.g., 7 kg of GaAs consisted of 5.8 × 106 control volumes. All used meshes were of the hybrid type with refinement at the walls. Radiation in the furnace was described by the discrete transfer radiation method. The power distribution in heaters mimicked the real process parameters, and their temporal change provided a growth velocity of ∼2 mm/h. The flow regime in the melt was considered laminar. In the transient local calculations, the time step was equal to 0.5 s. Pure buoyancy-driven melt flow was used for initiation of the magnetically driven flows and as a benchmark. A single-crucible VGF furnace based on KRISTMAG technology loaded with 5.5 kg of GaAs was used as a benchmark in a discussion about overall energy consumption. The transport phenomena were discussed using several dimensionless quantities such as Reynolds Re (eq 9), Rayleigh Ra (eq 10), Stefan Ste (eq 11), and forcing number F (eq 12) that give a ratio of inertial to viscous forces, buoyancy to viscous forces times the ratio of momentum and thermal diffusivities, sensible heat to latent heat, and magnetic to viscous forces, respectively. Skin depth δ of TMF is defined in eq 13.

Remax =

umax dρ μ

Ra = Gr × Pr =

λs Ste =

F=

δ=

βg ΔTrad,avd3 (10)

να

( ddTz )s − λl( ddTz )l rgrowthρs ΔHl,s

2σelπfkBmax 2 HR4 4ρν 2

1 μμ σ πf r 0 el

,k=

(11) φ A TMF

(12)

(13)

The obtained velocity and Lorentz force density fields were visualized using surface and 3D streamlines. The streamlines of any vector quantity (e.g., velocity and Lorentz force density) are a family of curves that are instantaneously tangent, i.e., parallel to the vector field. The interface deflection Δz was measured at the melt symmetry axis with respect to the three-phase junction (melt−crystal−crucible) and varied between concave (Δz < 0) and convex (Δz > 0). Searching for a suitable TMF for interface shaping, we studied the influence of magnetic parameters frequency, phase shift, and ac magnitude on the Lorentz force field with respect to the field intensity, special distribution, and direction. Moreover, we also addressed the influence of the amount of load and the progress of crystallization front on the TMF.

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RESULTS AND DISCUSSION Lorentz Force Density. The influence of electromagnetic parameters, GaAs loading, and process time on Lorentz force density magnitude, direction, and distribution in the melt was studied in a single-crucible furnace. The results showed that the magnitude of FL depends linearly on frequency, quadratically on ac magnitude,15 and alternately on phase shift, with maxima depending on the frequency and minimum at Δϕ = 0° (Figure 2). The reason for the decrease in FL with a decrease in |Δϕ| lies in a shift of a locus of the maximal FL downward to the crucible cone, where the distance between the melt and the HMM coils becomes

(9) 5124



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large. In directional solidification of Si in alternating magnetic field in rectangular crucibles, such behavior was not observed;30 i.e., when Δϕ = 0°, FL reached a maximum. The findings revealed the crucial role of frequency in the direction and spatial distribution of TMF (Figure 3). The

Figure 4. Simulated Lorentz force density in GaAs as a function of phase shift Δϕ in the case of downward TMF with f = 200 Hz, Io = 16 A, and a GaAs load of 5 kg: (left) 3D streamlines, (middle) 2D contours in the midvertical plane, and (right) 2D streamlines in the midvertical plane. Figure 3. Simulated Lorentz force density in GaAs as a function of frequency in the case of downward TMF with Δϕ = −100°, Io = 16 A, and a GaAs load of 5 kg: (left) 3D streamlines, (middle) 2D contours in the midvertical plane, and (right) 2D streamlines in the midvertical plane.

higher the frequency, the steeper the FL vectors. Moreover, with an increase in frequency f from 10 to 600 Hz, TMF penetration depth δ decreased from 179 to 23 mm. Full penetration of a 4 in. melt was achieved for f ≤ 124 Hz. Therefore, the TMF with a lower f that provides FL with pronounced meridional direction and higher δ has a higher s/l interface shaping potential. Furthermore, the spatial distribution of FL vectors showed rotational asymmetry caused by the asymmetry of the HMM setup caused by, e.g., the position of the power connectors. Figure 4 shows FL vectors in a GaAs melt for a Δϕ range from −60° to −120° and for constant values of Io and f. With an increase in the negative Δϕ, the direction and spatial distribution of FL changed thoroughly. The highest feasibility for interface shaping for the selected f was observed for Δϕ ∼ −100°. As shown in our recent paper, similar effects may be obtained for various combinations of magnetic parameters if the criterion F/(Gr × Ste) = 2.4−3.1 is fullfilled.16 A change in Io had no impact on the spatial distribution and direction of FL. With the progress of the crystallization front and for constant magnetic parameters, the distribution of FL changed. The magnitude of FL slightly increased, while its spatial distribution remained similar in a “compressed” form (Figure 5). When the s/l interface reached the cylindrical part of the crucible, FL streamlines were almost exclusively directed downward. Before that stage, i.e., in the crucible cone, the horizontal component of FL dominated. Bearing in mind that only the vertical component of FL has a shaping potential, we found this result explained why the prevention of nucleation of polycrystalinity by TMFs is a challenge from the beginning to the end of the solidification process.

Figure 5. Simulated Lorentz force density in GaAs as a function of time, i.e., progress of the crystallization front for downward TMF with f = 100 Hz, Io = 16 A, Δϕ = −100°, and a GaAs load of 5 kg: (left) 3D streamlines, (middle) 2D contours in the midvertical plane, and (right) 2D streamlines in the midvertical plane.

The scale up of the GaAs load significantly influenced TMF even if magnetic parameters were not changed. An increase in the weight of GaAs from 5 to 9 kg resulted in an increase in the magnitude of FL by almost one-quarter, while the spatial distribution remained similar in a “stretched” form (Figure 6). However, it can be also noticed some TMF weakening in the melt top region by higher loads. Such a behavior occurred because of the position of the melt free surface relative to the side HMM from where TMF originated; i.e., the larger the load, the weaker the impact of TMF from the top of the melt. 5125



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symmetry. This minor asymmetry in the B field resulted from a slit in the topology of windings that were surrounding one crucible. It was an inevitable trade-off for simplicity of HMM design. If the magnitudes of FL distributions in the single-crucible and multicrucible VGF furnaces were compared for the same magnetic parameters Io, f, and Δϕ, the results revealed the stronger FL in the former setup. In other words, an equally strong FL is obtained if Io increases with the number of crucibles while f and Δϕ are kept constant. Flow and Thermal Fields. Simulation results for a temperature distribution in a single-crucible VGF furnace equipped with HMM are shown in Figure 8. This snapshot corresponded to the beginning of the crystallization, when 7 kg of the GaAs melt was exposed to the axial temperature gradient of 2.9 K/cm. The temperature distribution in the GaAs melt at various points in time during the crystallization phase for loads of 7 and 9 kg is shown in Figure 9. Results were obtained from global simulation neglecting convection, for a crystal growth rate of ∼2 mm/h in the cylindrical part of the crucible. This first approximation of the real temperature field already pointed out the critical issues of VGF GaAs growth. With the progress of crystallization, axial temperature gradients decreased in both the crystal and the melt. Because the heat conductivity of the GaAs crystal is lower than that of the melt and because of the high latent heat and conical bottom ingot geometry with a reduced surface for the heat transfer, the axial heat exchange is retarded. Therefore, the heat accumulates in front of the growing interface, promoting its concavity. Similar results were obtained for VB GaAs growth.32,33 If the comparison is performed at the same moment of crystallization, an increase in GaAs load will cause an increase in axial temperature gradients and a decrease in growth rate. In practice, such a deceleration of the growth process is tackled by an adjustment of the cooling program. In our study if loads increase from 7 to 9 kg at the same moment of crystallization corresponding to panels c and f of Figure 9, temperature gradients will increase by factors of 1.2 in the melt and 1.1 in the crystal, respectively. If buoyancy convection were added to the model, the resulting flow regime would be laminar as assumed with, e.g., Remax = 2.5 × 103 and Ra = 9.3 × 105 at 37 vol % solidified GaAs with a 9 kg load. The observations showed a complex 3D flow pattern with opposite slow rotating toroidal vortices in the lower zone of the melt cylinder and fast rotating toroidal vortices at the melt upper rim.16 The lower vortex rotating upward along the crucible wall is responsible for the concavity of the s/l interface. The melt bulk was predominantly stagnant with slow rotating axial vortices. Obtained results underlined a need for 3D modeling to understand the real melt flow patterns. If downward TMF with maximal magnetic flux B = 6.8 mT was imposed on 9 kg of GaAs, the buoyancy flow structure significantly changed, causing an alignment of the velocity vectors with Lorentz forces in the vicinity of a three-phase junction and pronounced intensification of the flow in the melt bulk and near the s/l interface. As a consequence, the interface rim was shifted downward and the crystal morphology improved; i.e., interface deflection switched from concave (Δz = −2.7 mm) to convex (Δz = 1.9 mm) (Figure 10). As described in the literature, the interface shape may become asymmetric when a TMF of higher frequencies is

Figure 6. Simulated Lorentz force density in a melt as a function of GaAs weight (5−9 kg) for Io = 85.5 A, f = 10 Hz, and Δϕ = −100°: (left) 3D streamlines, (middle) 2D contours in the midvertical plane, and (right) 2D streamlines in the midvertical plane.

Numbering up the crucibles within the HMM imposed an interesting design challenge on the coil shapes, because an axial symmetry of the temperature and magnetic field in each crucible has to be ensured. Our novel HMM design with three coils positioned on each other where each coil closely encircled all four crucibles (Figure 1b) met the requirements and avoided unnecessary complexity. One example of Lorentz force density distribution in four GaAs melts for downward TMF is shown in Figure 7. No significant difference in their distributions was observed among the melts. However, a 4° axial tilt of the locus of minima in B and consequently inclination of FL isosurfaces in each single melt were noticed. The locus of maxima in B was positioned in each melt outward relative to the furnace axis of

Figure 7. Simulated Lorentz force density in GaAs melts in the multicrucible furnace (a snapshot for downward TMF with f = 10 Hz, Δϕ = −100°, Io = 150 A, and a load of 9.3 kg/crucible). 5126



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Figure 8. Temperature distribution in a single-crucible VGF furnace equipped with HMM at the beginning of the crystallization of 7 kg of GaAs: (a) whole furnace, (b) HMM, and (c) GaAs.

asymmetric current supplies. In our study, such an effect was not observed because of low used frequency (f) of 10 Hz. Magnetically driven flow remained in the laminar regime, characterized by, e.g., Remax = 1.5 × 103, Ra = 5.4 × 105, and F = 3.3 × 105 at 37 vol % solidified GaAs. Simulation results for a temperature distribution in a multicrucible VGF furnace equipped with HMM are shown in Figure 11. The axial tilt of 5° of the normal vector to the temperature isosurfaces relative to the furnace central axis (Figure 11c) was offset in the presence of the oppositely inclined downward TMF (Figure 12). Observed inclinations originated from rotational asymmetry in a design of HMM coils and in a position of corresponding power connectors. The resulting axial tilt of only 1.5° in the multicrucible setup correlated well with the tilt of 1.2° observed in the singlecrucible furnace. Therewith, the proposed HMM design for the numbering-up concept fulfilled the imposed magnetic and thermal symmetry requirements. Hence, the interface shaping potential of Lorentz forces was nearly identical in both studied VGF furnaces; i.e., it was independent of the number of crucibles. Comparison of Process Intensification Concepts. The most important criteria for judging the success of VGF GaAs growth in general are the process productivity, yield, and energy consumption. The productivity is commonly defined as a crystal mass grown per unit time, while the yield is defined as a fraction of crystals that meet the customer’s specifications. Energy consumption is usually related to the time and the amount of crystals produced. A comparison of numbering-up and scale-up process intensification concepts with respect to energy consumption is shown in Figure 13. A change in total heating power during the run, being one of the most intensive of all operating costs, pointed out the main differences between the concepts. Integration of the curves for the specific power over time gave total energy consumption for numbering-up, standard, and scale-up processes 85, 270, and 226 kWh/kg of GaAs, respectively. The numbering up of crucibles to four within the HMM as shown in this study reduced the total energy consumption per run to 32%, while other operating costs, e.g., labor, gas, and water consumption, decreased to 25% of the value for the single-crucible process. The scale up of ingot length prolonged the crystallization phase and therewith the corresponding energy consumption (Figure 13b), while heating and cooling phases were similar in duration and power consumption. Other operational costs, e.g., labor, gas, and cooling water, increased during scale up by the

Figure 9. Snaphots of the simulated temperature distribution without TMF in (a−c) 7 kg of GaAs and (d−f) 9 kg of GaAs grown in a singlecrucible furnace using the same cooling program.

Figure 10. Snaphots of velocity streamlines and temperatures in a single-crucible furnace loaded with 9 kg of GaAs for (a−c) buoyancydriven flow and (d−f) flow driven by downward TMF with f = 10 Hz, Δϕ = −100°, and Io = 173 A.

used.31 The asymmetry in thermal and magnetic fields originated from the steplike HMM coil windings and 5127



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Figure 11. Simulated temperature distribution in a multicrucible VGF furnace equipped with HMM at the beginning of the crystallization of 7 kg of GaAs: (a) whole furnace, (b) HMM, and (c) GaAs.

the 5.5 kg loading. In the four-crucible setup with a 5.5 kg load, the process lasted 23 fewer hours than in the standard singlecrucible furnace. With regard to productivity, the numbering-up concept ranked first with 0.11 kg/h, followed by the scale-up concept with 0.031 kg/h and finally the standard process with 0.025 kg/h; i.e., the productivity of the numbering-up process was 4.5 and 3.6 times higher than the productivity of the standard and scale-up processes, respectively. To achieve the high yields, arsenic losses and interface morphology are the main constraints. Evaporation of arsenic and degradation of the crystal surface after leaving the boron oxide melt occur if the axial temperature gradient is kept constant in GaAs during the scaling up. This approach preserves the ratio of axially transported sensible heat through the s/l interface to the latent heat (the Ste number is kept constant) that is crucial for the interface morphology but leads to the overheating of the melt. On the other hand, heavy constraints on the maximal temperature in the melt may cause morphological instabilities at the s/l interface because of the low-melt temperature gradients. Such an undesired scenario may be avoided if a cooling program is changed so that the lowest temperature in the crystal decreases. As a consequence, axial temperature gradients will increase, but the growth rate will decrease along with productivity. Thereby, the Ste number will increase and the interface flatten. Even preserving the same cooling program during scaling up retards the growth and decreases the productivity. For example, in our simulations of the growth of 7 kg of GaAs in the single-crucible furnace corresponding to the condition shown in Figure 8c, temperature extrema were 1316 and 1530 K, the growth rate (rgrowth)

Figure 12. Snaphots of velocity streamlines in a GaAs melt in a multicrucible furnace for (a and b) buoyancy-driven flow and (c and d) flow driven by downward TMF with f = 10 Hz, Δϕ = −100°, and Io = 381 A. Each crucible was loaded with 7 kg of GaAs.

same factor as the total process time. The scale up of the loading had no influence on the fixed costs for the equipment except the crucible. The total cycle time for solidification of 7 kg of GaAs in our single-crucible furnace lasted 11.3 h longer than in the case of

Figure 13. Simulated temporal change in total energy consumption for various process intensification concepts for a growth rate of ∼2 mm/h: (a) numbering up (one crucible vs four crucibles) and (b) scale up in a 4 in. crucible. 5128



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was 2.4 mm/h, and Ste equalled 3.6 at 70 vol % solidified. In the case of 9 kg GaAs loads and the same cooling program at the same moment in the crystallization process, temperature extrema were 1309 and 1534 K, the Ste number increased to 4.4, and the growth rate decreased to 1.9 mm/h (Figure 9f). As previously mentioned, the interface shaping potential of Lorentz forces was independent of the number of crucibles for the same ingot length. With the progress of crystallization and lengthening of the ingot, the interface concavity of buoyancydriven flow increased. The easiest way to compensate for increased deflection is to apply downward TMF with Lorentz force density vectors of the same direction and spatial distribution, but with increased magnitude. To ensure that, the frequency and phase shift of ac were kept constant while the ac amplitude, i.e., the ac/dc ratio, was varied. The total sum of ac and dc that was responsible for the side heat supply did not change. If buoyancy deflection reaches Δzbuo = 0.1R, only a reduction in concavity, but no planar or slightly convex morphology, may be achieved using TMF.16 The only way to tackle such a problem is to increase the axial temperature gradients and at the same time lower the growth rate and therewith the generated latent heat. Thereby, crystal quality increases, but productivity decreases. Despite many disadvantages, long crystals still have one advantage compared to short ones: a higher fraction of the usable ingot for wafering, i.e., the ratio of cylindrical to the conical ingot weight increased with scaling up. For example, if the GaAs load increased from 5 to 9 kg in a 4 in. crucible, the fraction of conical waste decreased from 19 to 11 wt %. These results show that the proposed innovative principle of numbering up in TMF brings the most benefits in terms of process efficiency and crystal quality by optimizing driving forces and maximizing specific areas to which those driving forces apply, i.e., by maximizing the synergistic effects from crystal growth processes and magnetically driven flows.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Part of this work was partially supported by the German Federal State of Berlin in the framework of the ‘“Zukunftsfonds Berlin”’ and the Technology Foundation Innovation center Berlin (TSB). It was cofinanced by the European Union within the European Regional Development Fund (EFRE). Further support was given by the Leibniz Association within the “Pact of Research and Innovation”. Fruitful discussions with Robert Menzel are kindly acknowledged.

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NOMENCLATURE ATMF distance between two neighboring magnetic coils (m) B magnetic induction (T) Cp specific heat (J kg−1 K−1) d crucible diameter (m) F forcing number f frequency (Hz) FL Lorentz force density (N/m3) g gravity acceleration (m/s2) Gr Grashof number (dimensionless) ΔHl,s latent heat of solidification (J/m3) h ingot height (m) H melt height (m) I current (A) j current density (A/m2) k TMF wavenumber (rad/m) p pressure (Pa) Pr Prandtl number (dimensionless) rgrowth crystal growth velocity (m/s) R radius of crucible (m) Ra Rayleigh number (−) Re Reynolds number (dimensionless) Ste Stefan number (dimensionless) T temperature (K) t time (s) Tamag magnetic Taylor number (dimensionless) u melt velocity (m/s) w ingot weight (kg) z axial coordinate (m) Δz interface deflection (m)

CONCLUSIONS

We numerically compared scale-up and numbering-up concepts for 4 in. VGF GaAs growth under TMF with respect to the control of interface morphology and energy balance, using a VGF growth of 5.5 kg of GaAs under TMF in a single crucible as a reference. The numbering up of 4 in. crucibles showed clear superiority in comparison to other two concepts, particularly concerning total energy consumption. The scale up of ingot length was characterized by longer crystallization times and by similar heating and cooling times. The reduction of the negative interface deflection by Lorentz forces was feasible in all considered concepts. Nevertheless, a planar or slightly convex interface morphology may be a challenge to achieve in long scaled-up ingots without decreasing the growth rate. The temporal adjustment of TMF intensity was required if Gr and Ste numbers changed with the progress of crystallization. We strongly believe that a further synergy of process intensification concepts, e.g., numbering up of crucibles with an increased diameter of 6 in. for VGF growth under TMF, has great potential for industrial applications. Verification of this hypothesis will be the topic of our future work. At the same time, the first growth experiments proposed here in a 4 in. multicrucible setup have been started.22

Greek Symbols

α β δ Δϕ λ μ μ0 ν ρ σel

thermal diffusivity (m2/s) thermal expansion coefficient (K−1) skin depth (m) phase shift (deg) thermal conductivity (W m−1 K−1) dynamic viscosity (Pa s); relative magnetic permeability of material (dimensionless) magnetic permeability of free space (H/m) kinematic viscosity (m2/s) density (kg/m3) electric conductivity (A V−1 m−1)

Subscripts

av average buo buoyancy l liquid (melt) 5129


Crystal Growth & Design s mag max r rad ref

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(31) Kasjanow, H.; Nacke, B.; Eichler, St.; Jockel, D.; Frank-Rotsch, Ch.; Lange, P.; Kießling, F. M.; Rudolph, P. J. Cryst. Growth 2008, 310, 1540−1545. (32) Volz, M. P.; Mazuruk, K.; Aggarwal, M. D.; Cröll, A. J. Cryst. Growth 2009, 311, 2321−2326. (33) Koai, K.; Sonnenberg, K.; Wenzl, H. J. Cryst. Growth 1994, 137, 59−63.

solid (crystal) magnetic maximal relative radial reference

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