Bohun, C. S.

Dellar, P.J.

Hjorth, P.G.

Ockendon, J.R.

English

Mathematics in Industry Information Service Eprints Archive

SAE Magnetics Ltd. produces read/write heads for hard disk drives. These tiny devices are mounted on
arms inside the hard disk drive, and are designed to fly a few micro metres (/-LID) above the spinning disk
surface. The head converts electrical signals to and from magnetic fields that interact with the magnetic
material of the disk surface, reading or writing data onto the surface. The distance between the read/write
head and the disk surface must remain constant for reliable operation. The heads are therefore designed
with a specific aerodynamic profile that must be etched out of the material making up the heads. The
heads are fabricated in large numbers within a wafer of magnetic material, similar to the manufacture of
integrated circuits from silicon wafers, before the wafer is broken up into individual heads.
The aerodynamic profiles are etched into the wafers inside an ion milling machine. Etching takes
place by bombarding the surface of wafer with ions of a gas, typically argon or oxygen. The ions are
accelerated to very high velocities by electrostatic forces. Prior to etching, the wafer surface is partially
coated with a layer of material called a resist that protects some areas of the wafer while exposing others.
After milling the resist is stripped off, leaving a pattern of eroded grooves in the wafer material.
Milling is a "one shot" process. The wafers are placed at one end of a vacuum chamber (see figure 1).
Up to four wafers may be processed simultaneously by placing them on ajig that slowly rotates, exposing
each wafer to the ion beam in turn. The etching depth is controlled simply by the amount of time spent
by each wafer in the ion beam before the wafers are retrieved from the chamber. The wafers are then
tested, and any wafers not etched to the desired depth must be discarded.
Efficient production therefore requires an accurate prediction of the etching rate. Wafers of different
materials will be processed in a working day, using different protective etch resists, different gases,
different accelerating voltages, and different inclinations between the wafers and the ion beam. Between
milling runs, the machine is idle for a certain time, known as the "stop time". There is evidence for an
increasingly erratic etching rate when the milling machine is restarted after longer stop times (see figure
2). The data supplied by the company suggest a time scale on the order of 20 minutes characterises the
onset of the problem.
II Department of Mathematics, Imperial College London. London SW7 2AZ. UK
** Department of Mathematics. DTU, DK-2800 Kgs. Lyngby. Denmark
tt Department of Mathematics. University of Ontario Institute of Technology. Ontario L IH 7K4. Canada
HOCIAM. Mathematical Institute. 24-29 St. Giles', Oxford OX I 3LB, UK
" I" t,
,OAr
: 0 0,, 0,,
••, Ar+,,,,,
"" I\.' t
Figure I: A schematic of the ion milling machine. Neutral gas introduced into the chamber is ionized at the hot
tungsten filament, then accelerated by electrostatic forces through a screen towards the work piece at the far left.
The ions blast into the material. milling away tracks and grooves wherever the surface of the work piece is not
protected by a coating of etch resist.
• Dee12 --Poly. (De c)
fS 10 Ic~ 8 I~ •5 - I~ I-0 4, -- -I4l ~ T :~ 2 •./ • • • I0
00:00 00 :07 00: 14 00 :21 00: 28 00:36 00 43
stop time
Figure 2: Onset of unpredictable etching rates occurs after a characteristic stop time on the order of 20 minutes,
and the scatter seems to grow worse with increasing stop time. Figure supplied by SAE Magnetics (H.K.) Ltd.
We begin by noting some orders of magnitudes for parameters and relevant physical processes inside
the milling machine. Our discussions were informed by a laboratory manual for ion milling machines
provided by the University of California at Berkeley. [1]
The chamber operates at room temperature (roughly 300 K) except for the tungsten filament, which
has a temperature on the order of 1000 K.
Ion milling machines run at very low pressures, a typical pressure being p = 2 X 10-5 torr. [1 ] One
torr is the pressure required to SUppOlt one millimetre of mercury,
1
1 torr = 760 atm rv 133 Pa.
The torr is the conventional unit of pressure in high vacuum applications, and the standard atmosphere is
defined to be exactly 760 torr. The pressure inside the milling machine in SI units is thus of order
The last conversion reflects the relation 1 atm ;::::;105 Pa. At this pressure the mean free path), (the typical
distance which a gas molecule travels before colliding with another gas molecule) at room temperature
is
substantially larger than the size of the vacuum chamber. A randomly moving particle therefore typi-
cally crosses the inside of the milling chamber several times, colliding with the walls each time, before
colliding with another particle.
The typical particle velocity is a function of temperature only, about 400 metres per second for a
monatomic gas at room temperature.
The ions are accelerated by potential differences between the grid and the wafer on the order of one
kilovolt. The velocities achieved through electrostatic acceleration are very much larger than the thermal
velocities. The conversion between electron volts and temperature is [2]
using Boltzmann's constant k ;::::;1.38 X 10-23 Joules per Kelvin to relate energy with temperature.
In more familiar terms, room temperature corresponds to an energy of 1/40 of an electron volt per
particle,
1 .40 eV per particle;::::;290 K. (2.5)
The electrostatic energies in the milling machine are on the order of a electron kilovolt, some 4 x 104
higher than the thermal energies at room temperature. Since energy is proportional to velocity squared,
the velocity imposed by electrostatic acceleration is thus some 200 times the typical thermal veloc-
ity.
The ions move coherently in the beam, not because ion-ion scollisions are effective at smoothing
out thermal variations in their individual velocities, but because the imposed electrostatic acceleration
completely dominates the random thermal motion of the incoming gas.
Whether or not the ion bombardment produces etching depends on whether the energy per particle in
the beam exceeds the binding energy between atoms in the target wafer, and whether the ions effectively
exchange energy with the wafer when they collide.
Each successful ion-wafer collision knocks out one or more atoms of the wafer. These atoms form a mist
of debris or "sub-product" that fills the chamber. Some of the debris will be pumped out of the chamber
by the vacuum pump, but some debris presumably deposits itself onto the walls of the chamber, or on
working parts like the accelerator grid.
That the debris concentration hinders the milling rate can be inferred from the data in figure 3) sup-
plied by SAE Magnetics. The etching rate is observed to down when more wafers are present, presum-
ably because the debris concentration somehow disturbs the flux of ions onto the wafers. We assume that
the debris concentration is proportional to the area of unprotected wafer, and that this area is proportional
to the number of wafers in the chamber.
50
ID
U 40~
ID
H
ID 30~~·M 20'"d
ID
+-' 10o:q
l:l:::
0
0 2 3
Jig quantity
Figure 3: Differences in milling rate for different numbers of wafers present in the chamber simultaneously.
Figure supplied by SAE Magnetics (H.K.) Ltd.
We consider the debris to be present in the chamber in two states: (l): bound debris, which is loosely
(re)combined debris binding to the walls or other solid surfaces of the chamber. Denote the thickness of
the bound layer of debris by the variable 9 which is a function of time. (2): free debris, which has just
been knocked out of the target matrix or out of the bound debris state, and is free streaming about in the
chamber. This state is characterized by a number density n, measured in number per volume.
In this model the two debris states have quite different temperatures. The bound debris is in thermal
equilibrium with the chamber walls and thus is at room temperature, whereas the free debris may be
expected to inherit the very much larger velocities associated with the ion beam.
When the ionising coil is switched off, we take the debris thickness 9 to be governed by a crude
number balance equation:
dg 1
- = -(gO - g)
dt T
where go is proportional to n, the number density of free debris.
Debris build-up bears some resemblence to the process of epitaxy.
Epitaxy is the process by which material (molecules) is added to a crystalline substrate slowly building
up molecular layers. This can take place both in solutions and in gas, the latter known as vapor phase
epitaxy.
Vapour phase epitaxy experiments normally takes place at considerably higher pressures, 50 to 500
X 10~3 torr (about 1000 times higher than the milling chamber pressure, though still a tiny fraction of
atmospheric pressure), and also at slightly higher temperatures, 450 K instead of 300 K. On the other
hand, if the debris atoms leave the wafer with velocities comparable to the incoming ions, the debris
atoms velocities are hundreds of times higher than typical thermal velocities in vapour phase epitaxy.
Moreover, at the lower pressure all the debris atoms may be expected to collide with the walls, since the
mean free path is much larger than the distance to the walls. The mass flux of debris atoms colliding
with the chamber walls may therefore not be too dissimilar to that seen in epitaxy experiments.
If we assume that n is roughly constant during a stop period, the value of 9 governed by (3.1) rises
first linearly, then levels off at the value 90 (see Figure (4).)
When the ionising coil is switched back on after a stop period, radiation from the coil might be
expected to heat loosely bound debris on the walls, causing it to fly back into the chamber and disrupt
the milling. Additionally, the ions from the beam may also loosen wall bound debris, adding further to
the debris density.
If we take the data from Figure (3) to indicate a linear decrease of (the average) milling rate r with
increasing number density n,
and further assume that a higher value of the thickness 9 leads to a higher value of n immediately after a
stop period, we get a rate differences as a function of stop times as depicted in Figure (5).
20
(1)
U
15::::
(1)
••••
~ 104-i.....•
"'0
(1)
5~ro••••
0
0 2 3 4
stop time
Figure 5: Absolute value of the rate difference as a function of stop time, for the model where number density
following a stop time is proportional to the solution of equation (3.1).
This behavior is in qualitative agreement with the SAE Magnetics data in Figure 2.
In this model, longer stop times will always lead to increased rate differences, because longer stop
times lead to more debris build-up; the only way to control the problem is to inhibit the debris build-up,
by modifying the rate constant T in equation (3.1).
One way of doing this is to consider the temperature dependence.
The known ([3], [4], [5]) temperature dependence of vapor phase epitaxy growth rate is sketched in
Figure 6.
Figure 6: Epitaxial growth rate as a functionof (inverse) temperature of bound substrate. Typicalgrowth rates are
I to to nanometres per second.
For sufficiently high temperatures, the growth rate is nearly constant, limited only by the mass trans-
fer from gas to solid, i.e., the rate at which gas molecules can be made to impact on the walls.
On the other hand, the lower the temperature of the wall, the better the wall 'repels' an impact from
a free flying debris atom, and the slower the build-up of debris on the wall. At low temperatures, we
expect the build-up rate to be limited, with a Boltzmann factor that depends on temperature as
1 -E/kT
- cv e
T
In the Ion Milling Machine this means that for given stop times, the lower the wall temparature, the less
build-up of residue will take place during the stop time.
An epitaxial-like process of debris build-up both during operation and during a stop time may be expected
in the ion milling machine.
If the behavior of Figure 2 is caused by release of debris from the walls, the observation of more
instability for longer stop time is consistent with debris building up over time.
One way to reduce the problem is to inhibit the build-up of debris. From Figure 6 one can see that a
reduction in temperature should decreases the rate of debris build-up.
Based on the possible model analogy with epitaxy, we propose then that a way to suppress build-up
of debris layers on the walls during stops is to cool the walls.
[I] Microfabrication Laboratory Manual, University of California at Berkeley, chapter 7.14 - Ion Beam
Milling.
[2] M. Danos and J. Rafelski, eds., Pocketbook of Mathematical Functions (Verlag Harri Deutsch,
Frankfurt, 1984).
[3] G. Stringfellow, Organometallic Vapor-Phase Epitaxy, Theory and Practice (Academic Press, Inc.,
San Diego, USA, 1990).
[4] D. Hurle, ed., Handbook of crystal growth. Vol. 3: Thinjilms and epitaxy (North-Holland Elsevier
Science Publishers, Amsterdam, 1994).
[5] K. Jensen, D. Fotiadis, and T. Mountziaris, Detailed models of the MOVPE process, Journal of
Crystal Growth 107(1), I (1991).


Etch rate prediction for iron milling machines

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