Controlling Czochralski Crystal Growth

Institut für Regelungs- und Steuerungstheorie

1. Project Overview	2. Process Overview	3. Control Tasks	4. Observers
5. Pulling Velocity Control	6. Temperature Control	7. References	Jobs, Student's Jobs etc.

Control Tasks

Difficulty of diameter control
Measurements and control variables
Basic physical relations
Control concepts

Difficulty of diameter control

Crystal grown with VCz-technique and
conventional PID-controller approach... :-(

Low temperature gradients have the unpleasant effect that the process reacts very sensitively to disturbances. High temperature gradients stabilize the crystal shape at the crystallization front, sure enough at the expense of crystal grid quality. Fundamental in the VCz-process is minimization of the temperature gradients. This renders diameter control a difficult task. Even weak disturbances lead to uncontrolled growth which cannot be handled by (too) slow controllers.Thus, it is not possible to control the process with conventional approaches based on linear theory. Even methods based upon newer insights, e.g. those presented in [Bardsley et.al.], [Gevelber et.al.], fail.

Measurements and control variables
The figure in the right shows the basic variables of the process. The following measurements are available:

crucible bottom temperature

main heater temperature

force acting on pulling rod

Furthermore, the following control inputs are available:

crystal pulling velocity

vertical crucible translation velocity

heater power $P_{in}$ (alternatively the crucible bottom temperature could be treated as an input, because heater power and crucible bottom temperature are "connected" by a cascade controller)

When controlling the force F acting on the pulling rod and using the heater power as the control (as has been tried so many times), the following must be taken into account:

The process has batch character. Thus, there is no set point.

Control performance is limited by the existence of a right half plane zero of the transfer function between crystal radius and force [Gevelber et.al.].
Due to the boron-oxide layer the LEC-process becomes a delay system (infinite dimensional).

Basic physical relations

Crucial for the whole process is the trend of the growth velocity

and the pulling velocity

. In order to produce a crystal of given shape both growth and pulling velocity have to follow a given trajectory. This can be seen by the following equations, which describe the behaviour of the crystal radius and its angle:

$\displaystyle \dot r_i$	$\displaystyle = v_g \tan(\alpha_i)$
$\displaystyle \dot \alpha_i$	$\displaystyle = v_g \left(\tilde c_k(r_i, \alpha_i, v_g, v_p, v_c) + \tilde c_\vartheta(r_i, \alpha_i) \tan(\alpha_i)\right),$

where $\tilde c_k(r_i, \alpha_i, v_g, v_p, v_c)$ and $c_\vartheta(r_i, \alpha_i)$ are functions of the interface radius , the growth angle $\alpha _i$ , and the velocities.

If a special crystal shape is given, e.g. in terms of a function $r : l \mapsto r_i$ with $l \in [0, L]$ and $r_i \in {\sf I\hspace{-0.25ex}R}^+$ for a crystal of length , it is possible to derive expressions for the growth velocity and the pulling velocity of the type

$\displaystyle v_g$	$\displaystyle = f_g(r_i, r_i', \alpha_i, \alpha_i', v_p, v_c)$
$\displaystyle v_p$	$\displaystyle = f_p(r_i, r_i', \alpha_i, \alpha_i', v_g, v_c).$

Here and $\alpha_i'$ stand for the first derivative of the radius and the growth angle with respect to the length , respectively.

Control concepts

As shown above one should focus attention on the correct choice of the pulling and the growth velocity.

Pulling velocity:

This case is not too difficult to handle, since the pulling velocity is available for control. It acts fast and can be used to compensate high frequency disturbances. For this purpose an observer is required. This is the matter of the next section.

Growth velocity:

Manipulation of this variable is quite a complex task because the growth velocity is not available for control. One has to take a closer look on the physical relations: From a heat balance over the crystallization front it is possible to derive the following equation describing the dependency of

on the temperature fields within melt and cyrstal:

$\displaystyle v_g = \frac{\lambda_s \nabla_n T_s - \lambda_m \nabla_n T_m}{\varrho_s \Delta H_f}$

In this equation, which is derived assuming a planar growth interface, $\Delta H_f$ stands for the specific latent heat, $\lambda_s$ , $\lambda_m$ are the heat conductivities of solid and liquid GaAs, and $\nabla_n T_s$ , $\nabla_n T_m$ are the temperature gradients within the crystal and the melt normal to the growth interface. This leads to the following

Question:: Is it possible to manipulate by the heat flow into the system in such a way that the crystal grows according to the desired shape?
Possible solution:: Modelling of the whole thermal system.
Disadvantages:

All heat flows must be taken into account (complex models);

Many uncertain parameters;

Many uncertain boundary conditions;

Thermal models are only valid for a special configuration.
Alternative:: Find basic relations describing the behaviour of the system in quality, make use of ``unconventional'' control variables (accessed by observers), make use of robust PID-approaches.

Conclusion

Two control concepts are presented:

using the pulling velocity for control $\Rightarrow$ exactly linearizing feedback controller
using the growth velocity for control $\Rightarrow$ PID controller

In both cases information about the state of the process is required $\Rightarrow$ state observer.

Last modified: Mon Jul 1 20:04:10 2002