A typical configuration for a PSII implant is shown in Figure 2. This shows a series of targets sitting on a base plate. The base plate is supported by a relatively thin stalk, which is attached to the chamber. As shown, the targets and base are heated on all exposed surfaces by energetic ions during the pulse and by plasma heating alone during the zero-current portion of the cycle. The plasma heating is much lower than the ion heating. The target and base plate are cooled by radiation from all exposed surfaces to the chamber walls, and the base is also cooled by conduction through the stalk to the chamber. In some instances, the base is actively cooled by a coolant flowing through the stalk. An additional heat transfer mechanism which must be modeled is conduction from the targets to the base plate. Thorough modeling of all the heat transfer mechanisms described above is truly a challenging process. For this paper, appropriate simplifications will be made in order to avoid doing a full, three-dimensional, transient analysis.
In general, prediction of PSII target temperatures requires solution of a three-dimensional, time-dependent partial differential equation with nonlinear boundary conditions (due to radiative cooling). The equation can be written:
where is the density,
is the heat capacity,
is the
temperature,
is the time, and
is the
thermal conductivity. If the thermal conductivity is uniform
throughout the target and independent of temperature, this equation
becomes:
Equations of this type must be solved simultaneously for each target in the chamber, as well as for the base plate. The equations are coupled by heat transfer between the targets and the base plate. For most applications, coupling due to radiation from target to target or from a target to the base plate can be ignored, because the targets are generally quite thin. For each partial differential equation, the boundary conditions must account for the heating (modeled here as a surface heat flux), the radiative cooling, conduction from the base plate to the chamber, and for heat transfer between the targets and the base plate.
One important assumption of this model is the treatment of the heat load due to the ion bombardment as a surface heat flux. Because the ions actually deposit their energy over a finite region, a volumetric heat generation model would be more accurate, but at the cost of increased analytical difficulty. The modeling of the heat load as a surface flux can be justified by comparing the depth of implantation (generally called the range) to the conduction depth during the pulse. Typical implantation depths are on the order of 1000 angstroms, while a typical conduction length, as deduced from a dimensional analysis of transient conduction in a semi-infinite slab, would be on the order of [4]:
where is the length over which significant temperature change
has ocurred in time
,
is the thermal diffusivity
[
)] and
is the pulse width. For typical PSII
implants, the pulse width is on the order of 10
s, which gives a
conduction length of approximately 50
m for a typical metal.
Hence typical conduction lengths are on the order of 500 times the
implantation depth and we can safely ignore the volumetric heating
effects.
The radiative cooling can be modeled with the following equation:
where is the net heat flux lost due to radiation to the
chamber,
is the form factor for radiation from the target
or base plate to the chamber,
is the emissivity of the
radiating surface (the target, in this case),
is the
Stefan-Boltzmann constant, and
is the chamber temperature.
Because the target and base plate are enclosed in the chamber and the
target area is much less than the chamber surface area, the form
factor
can be taken to be approximately 1 [5].
The conduction from both the base plate to the chamber and from the targets to the base plate can be modeled with an equivalent heat transfer coefficient. These will be described in more detail after the model has been simplified.