Backstepping-based tracking control of the vertical gradient freeze crystal growth process

TL;DR

The paper tackles tracking the moving solid–liquid interface in a two-phase Stefan problem modeling vertical gradient freeze crystal growth, where the interface obeys a Stefan condition $\rho_m q_p \dot{\gamma} = \kappa_1 \partial_z T_1(\gamma,t) - \kappa_2 \partial_z T_2(\gamma,t)$. It combines a flatness-based feedforward design with a multi-step backstepping approach to handle time-varying PDE-ODE dynamics and to track reference trajectories for both the interface position and the gradient at the interface, using an extended plant for gradient tracking. The contributions include domain fixing, linearisation around a reference, decoupling via an auxiliary $\boldsymbol{N}(z,t)$, a Volterra backstepping transformation with kernel equations, and a numerical demonstration on GaAs crystal growth, showing convergence of the tracking errors. The work provides a systematic control framework for precise interface tracking and gradient control in gradient-freezing crystal growth, with potential improvements in crystal quality and process robustness.

Abstract

The vertical gradient freeze crystal growth process is the main technique for the production of high quality compound semiconductors that are vital for today's electronic applications. A simplified model of this process consists of two 1D diffusion equations with free boundaries for the temperatures in crystal and melt. Both phases are coupled via an ordinary differential equation that describes the evolution of the moving solid/liquid interface. The control of the resulting two-phase Stefan problem is the focus of this contribution. A flatness-based feedforward design is combined with a multi-step backstepping approach to obtain a controller that tracks a reference trajectory for the position of the phase boundary. Specifically, based on some preliminary transformations to map the model into a time-variant PDE-ODE system, consecutive decoupling and backstepping transformations are shown to yield a stable closed loop. The tracking controller is validated in a simulation that considers the actual growth of a Gallium arsenide single crystal.

Figures (4)

• Figure 1: Cross section of a typical plant (without jacket heaters).
• Figure 2: Solid lines represent the planned reference trajectories of the interface position $\gamma _{\mathrm{r}}^{}$ and the temperature gradient $\partial_z T_{1,\text{r}}( \gamma _{\mathrm{r}}^{} ,t)$ at the interface. Based on that, dashed lines indicate calculated trajectories for the growth rate $\dot\gamma_{\mathrm{r}}(t)$ of the crystal and the gradient $\partial_z T_{2,\text{r}}( \gamma _{\mathrm{r}}^{} ,t)$ in the melt (implied by \ref{['eq:stefan_cond_sep']}).
• Figure 3: Errors $\Delta \gamma _{\mathrm{}}^{} = \gamma _{\mathrm{}}^{} - \gamma _{\mathrm{r}}^{}$ and $\partial_{ z }\Delta T_1( \gamma _{\mathrm{}}^{} ,t)=\partial_{ z }T_1( \gamma _{\mathrm{}}^{} ,t)-\partial_{ z }T_{1,\text{r}}( \gamma _{\mathrm{r}}^{} ,t)$ of the flat output components resulting from the feedforward controller (dashed) and the tracking controller (solid).
• Figure 4: Logarithmic error $\varepsilon_i( z ,t)\coloneqq\log\left|\frac{T_i(z,t)-T_{i,\text{r}}(z,t)}{T_{i,\text{r}}(z,t)}\right|$ of the joint temperature profiles $T_i(z,t)$ of crystal ($i=1$) and melt ($i=2$), relative to their references $T_{i,\text{r}}(z,t)$. Herein, both phases are separated by $\gamma _{\mathrm{}}^{}$.