Project Details

Semiconductor crystals like silicon (Si), silicon carbide (SiC), gallium arsenide (GaAs), and aluminum nitride (AlN) crystals have important applications in key technologies.

Industrial applications of these semiconductor crystals require the availability of large-diameter, low-defect crystals, and a high growth rate during the production process is desirable to reduce production time and costs.

• The most important industrial technique for growth of Si and GaAs bulk single crystals is the growth from the melt, especially by the Czochralski method and their variants, i.e. Liquid Encapsulated Czochralski (LEC) and Vapor pressure controlled Czochralski (VCz).

  In this kind of processes, a rotating crystal seed is dipped into the melt that is contained in a rotating crucible. The seed is then slowly pulled out of the melt, and a single crystal solidifies. Since the melt flow is turbulent, impurities may find their way into the growing crystal, and temperature oscillations with small frequencies may occur below the crystal. This should be avoided, since this would lower the quality of the grown crystal.

  To achieve this aim, one applies time dependent electromagnetic fields to the melt, which is electrically conducting so that a Lorentz force is induced into it in their presence. This Lorentz force can be used to influence the melt motion.

• The state-of-the-art procedure for the production of SiC and AlN bulk single crystals is the Physical Vapor Transport (PVT) (also called Modified Lely Method), which involves sublimation of the source material. Even though substantial progress has been made in recent years, the technological problems remain challenging, and only partial solutions exist.

This project is carried out in close cooperation with the Leibniz Institute of Crystal Growth (IKZ) in Adlershof.

• From 2002 to 2007, C9 dealt with the sublimation growth of semiconductor bulk single crystals. In this technological process, polycrystalline powder is heated via magnetic induction to temperatures up to 3000 K. The powder evaporates and is transported through a gas cavity to a cooled seed where it recrystallizes as a single crystal.

  Mathematically, one has to solve an optimal control problem with pointwise state and control constraints on a complicated, nonsmooth cylindrical domain for a PDE system featuring the following components:

  1. Maxwell's equations for the electromagnetic field
  2. the (nonlinear) heat equation, coupled to nonlocal radiation boundary conditions, where
  3. the physical coefficients are strongly temperature-dependent

  For this PDE system, necessary and sufficient optimality conditions, as well as numerical methods, were established for new classes of state-constrained optimal control problems with nonlocal radiation boundary conditions.

  These investigations were finished successfully both from the mathematical and the industrial viewpoint; the software WIAS-HiTNIHS developed in C9 was licensed to IKZ.

• In 2007, C9 shifted its efforts to model variants of the Czochralski growth of bulk semiconductor crystals from the melt. The reason for this was that the use of traveling magnetic fields (TMF), a special kind of time dependent magnetic fields to control the turbulent melt flow, has drawn increasing attention from crystal growing industry worldwide.

  The interest in these problems was triggered by the project KRISTMAG of the Technologiestiftung Berlin (TSB). In this project, a WIAS group headed by O. Klein and J. Sprekels joined forces with IKZ and two industrial partners. A technological breakthrough was achieved by demonstrating that traveling magnetic fields can be successfully applied for this task. A special heater magnet module (HMM) was developed, and a number of patent applications were filed. The groundbreaking nature of these developments was recognized by the Innovationspreis (innovation prize) Berlin-Brandenburg 2008. The software WIAS-HiTNIHS played an important role in these investigations. Presently, many companies worldwide are implementing the new technology into their growth apparatus, trying to find the optimal configurations.

  From the mathematical viewpoint, crystal growth from the melt is even more difficult to treat than sublimation growth, since, in addition to the features a.)--c.), the fluid flow has to be accounted for, modeled by the Boussinesq approximation of the Navier-Stokes equations for thermally and electrically conducting fluids.

  The mathematical problem concerns the optimal control and simulation the equations of temperature-dependent magnetohydrodynamics coupled to the energy balance, on a complicated nonsmooth domain. In addition, almost all physical coefficients are strongly temperature-dependent, the coefficient functions are discontinuous, nonlinear and nonlocal radiation boundary conditions occur. A mathematical breakthrough was achieved in the thesis by P.-E. Druet, who showed existence for this system.

In accordance with the needs of our partners, we will study optimal control problems arising from the use of traveling magnetic fields to influence melt flow during crystal growth from the melt. Issues of special interest are:

• Eddies in the melt flow should have a suitable form to keep impurities apart from the growing crystal.
• The amplitudes of temperature oscillations below the crystals should be minimized, while their frequencies must be kept sufficiently large.

Mathematically, one has to deal with new classes of control and state constrained optimization problems for the very difficult PDE system described above. The results in the thesis by P.-E. Druet have made it possible to tackle control problems.