**Pressure dependence of ionization efficiency in sputtering magnetrons**

T. E. Sheridan, M. J. Goeckner, and J. Goree

*Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa
52242*

(Received

Using a Monte Carlo simulation, we show how electron confinement allows sputtering magnetrons to operate at lower neutral pressures than similar unmagnetized devices. We find that at high and low pressures the ionization efficiency in a magnetron becomes constant, and it varies by only 40% between these two regimes. In contrast, the efficiency varies linearly with pressure in an unmagnetized discharge.

** **Sputtering magnetrons are plasma devices used for thin film deposition
and sputter etching. In these devices, crossed electric and magnetic fields
confine electrons in complicated orbits near a cathode target.^{1,2}
These trapped electrons ionize the neutral background gas, creating ions which
are accelerated to the cathode. Upon striking the cathode, these ions sputter
material and cause the emission of secondary electrons, which are accelerated
back into the trap where they create more ions, thus sustaining the discharge.
Collisions with neutrals eventually scatter electrons out of the
trap.^{2} Since the principal virtue of the magnetron is its ability
to operate at a low neutral pressure, it is worthwhile to study electron
transport mechanisms in the magnetron and their dependence on pressure.

In this letter we use a Monte Carlo simulation^{2,3} of the transport
of electrons emitted from the cathode to explore the ionization efficiency and
its dependence on neutral pressure.^{4} The device we simulate is our
cylindrically symmetric planar magnetron.^{5} The Monte Carlo code was
described in detail in Ref. 2, where we simulated a 1 Pa argon discharge. This
code has also been used to simulate a magnetron^{1} which has an
adjustable magnetic field.^{3}

We now briefly review this code. Single electrons are started at rest from the cathode and are followed as they move about within a cylindrical simulation box which extends 4 cm above the cathode and to a radius of 4 cm from the symmetry axis. Orbits are computed using prescribed, time-independent magnetic and electric fields. An electron's orbit is terminated when its total energy becomes too small to ionize an argon atom, or when it moves out of the simulation boundaries. The next electron is then started at a radius chosen in a manner consistent with radial distribution of previous ionization events, assuming that ions fall to the cathode without radial deflection.

The magnetic field **B **was calculated^{2} based on the actual
device, which has a central plug magnet surrounded by a ring of bar
magnets.^{5} The field is 245 G and purely tangential to the
copper cathode surface at a radius of 1.7 cm, where the etch track is deepest.
We denote this radius *a. *Even though we perform our simulation for only
this magnetron size, our results can be applied to larger or smaller devices by
using a dimensionless neutral pressure *a / mfp *(where the mean free path
is *mfp = kB* *T/[[sigma]]P,* *[[sigma]]*is the cross section*
*for momentum transfer, and *T *is room temperature).

We assume a one-dimensional electric field perpendicular to the cathode
surface.^{6} The variation of this field with distance from the
cathode was found from a numerical solution of the planar sheath
equation,^{7} assuming typical experimental parameters of 4 eV for the
electron temperature, 0.1 mm for the Debye length, and -400 Vdc for the cathode
bias. The resulting sheath width is about 3 mm. A 1.0 V / cm linear presheath
is added to this sheath solution. The same electric field is used in the
simulation here regardless of the pressure or magnetic field.

Collisions with argon neutrals are accounted for by using the total cross
sections to evaluate the probability of elastic, excitation, and ionizing
collisions at each time step. If a collision has occurred, the differential
cross sections are used to scatter the velocity direction. The electron's
energy is decreased by each collision. For ionizing collisions, we use the
energy loss data reported by Carman for M-shell ejection,^{8} which is
an improvement upon the fixed energy loss assumed in our earlier^{2,3}
simulations.

For this letter we have used the simulation to determine the average
ionization efficiency [[eta]], which is defined as follows. First, we denote
the number of ionizations that a single electron performs as *Ni*. The
average of *Ni* over an ensemble of simulated electrons is denoted by
[*Ni*. Here [*Ni* is found directly from the simulation; in general
it will increase with cathode bias and pressure. The maximum possible number
of ionizations, *[[Nu]]i max* , is defined as the number of ionizations
performed by a well-confined electron in the absence of excitation and elastic
collisions;^{9 }thus, *[[Nu]]i max* increases with cathode bias.
We can now define the ionization efficiency [[eta]] as

[[eta]] [[equivalence]] [*Ni* / *[[Nu]]i max* . (1)

Using this definition, [[eta]] is the efficiency of ionization by cathode emission at a given pressure and cathode bias. The efficiency will always lie in the range 0 < [[eta]] < 1. If [[eta]] ~ 1, electrons are efficiently confined, while if [[eta]] << 1, the confinement is ineffective and electrons escape easily from the device. In practice, [[eta]] can never reach a value of 1 because many electrons lose energy in excitation collisions. As a result, the highest possible [[eta]] is about 0.9.

We ran the simulation for a wide range of neutral pressures^{10} from
0.2 to 100 Pa.^{ }Under these conditions, the mean free path (for
electron momentum transfer in argon) ranges from 7.5 cm to 0.015 cm, assuming a
cross section^{11} of 1 x 10^{-16} cm^{2}. Note that
over this pressure range the mean free path ranges over values significantly
smaller than the system size (*a / mfp*>> 1) to those
greater than the system size (*a / mfp*<< 1). For good
statistics we used at least 300 electrons in each ensemble, and for good energy
conservation we selected a time step between 5 and 20 ps.

The principal results of this letter are shown in Fig. 1 (a), where we plot
the ionization efficiency [[eta]] against neutral pressure *P.* The
unmagnetized and magnetron discharges differ most significantly at low
pressures. Without the magnetic field **B**, we find that [[eta]] increases
nearly linearly with *P* until it saturates at *P* = 50 Pa. On the
other hand, with **B***,*** **the ionization efficiency is almost
independent of *P.* The two curves merge at about 50 Pa, where
*a / mfp *>> 1. The weak dependence of [[eta]] on
*P* demonstrates the effectiveness of the magnetron configuration in
trapping electrons. This effectiveness is the reason magnetrons can be used
for sputtering at low neutral pressures.

In Fig. 1(b) we examine in greater detail the pressure dependence of [[eta]]
for nonzero magnetic field. Note that the average number of ionizations per
electron varies by only 40% as the neutral pressure is scanned over 3 decades
from 0.2 Pa to 100 Pa. Two limiting regimes, at low and high pressures, are
evident in Fig. 1(b). At low pressures (*P* < 1 Pa), [[eta]] is
constant because the mean free path is much longer than the system size
(*a / mfp*<< 1). At high pressures (*P* > 50 Pa),
[[eta]] saturates since the mean free path is much shorter than the system
size. In this regime, collisions are frequent and electron transport becomes
diffusive, leaving no chance that an electron will escape while it still has
enough energy to ionize a neutral atom.

Even though the simulation predicts that [[eta]] becomes constant at low
pressures, real magnetron discharges extinguish at very low pressures. This
paradox is resolved by noting that our prescribed electric field omits
self-consistent effects,^{12} which play a particularly significant
role at low pressures. The electric field in a discharge self-consistently
adjusts so that electrons and ions are expelled at the same rate. At low
pressures, energetic electrons require more time to perform a given number of
ionizations. The ion-transit time, however, is relatively invariable in a
magnetron, meaning that at a sufficiently low pressure, the electrons will be
forced out of their confined orbits too quickly, and the discharge will go out.

To demonstrate electron loss at low pressures, in figure 2 we show histograms
of the number of ionizations *Ni* performed by each electron for a
magnetron at *P* = 0.2 Pa and *P* = 100 Pa. Both histograms have a
peak at 15 ionizations, representing electrons that do not escape from the
simulation before expending most of their energy. This peak has a finite width
owing to energy lost in excitation collisions, which account for about one
quarter of all inelastic collisions. The histogram for the low pressure
displays a flat tail that extends down to zero. This tail reveals that some
electrons escape carrying energy that could have caused additional ionizations.
It does not appear in the histogram for 100 Pa, where the ionization efficiency
is saturated.

In summary, we have used a Monte Carlo simulation of our magnetron device to determine the average number of ionizations per electron. We find that the ionization efficiency varies by only 40% as the neutral pressure is scanned from 0.2 to 100 Pa. This weak dependence owes to the electron confinement; even at low pressures only a few electrons are lost before consuming their energy by ionizing neutral atoms. Without the magnetic field, the ionization efficiency has a strong linear dependence on pressure.

**ACKNOWLEDGMENT**

This work was supported by the Iowa Department of Economic Development.

**REFERENCES**

^{11 }A. E. Wendt, M. A. Lieberman, and H. Meuth, "Radial current
distribution at a planar magnetron cathode," J. Vac. Sci. Technol. **A6**,
1827-1831 (1988).

^{22 }T. E. Sheridan, M. J. Goeckner and J. Goree, "Model of energetic
electron transport in magnetron discharges," J. Vac. Sci. Technol. **A8**,
30 (1990).

^{33 }J. E. Miranda, M. J. Goeckner, J. Goree, and T. E. Sheridan,
"Monte Carlo simulation of ionization in a magnetron plasma," J. Vac. Sci.
Technol. **A8**, 1627 (1990).

^{44 }S. M. Rossnagel and H. R. Kaufman, "Current-voltage relations in
magnetrons," J. Vac. Sci. Technol. **A6**, 223-229 (1988), and Ref. 3 offer
some evidence that bulk electrons also contribute to ionization, but we do not
consider them in this letter.

^{55 }T. E. Sheridan, and J. Goree, "Low frequency turbulent transport
in magnetron plasmas," J. Vac. Sci. Technol. **A7**, 1014 (1989).

^{66 }M. J. Goeckner, J. Goree, and T. E. Sheridan, "Monte Carlo
simulation of ions in a sputtering magnetron," submitted to IEEE Trans. Plasma
Sci., July 1990, reports simulations using an electric field that varies with
both radius and height above the cathode.

^{7 }T. E. Sheridan and J. Goree, "Analytic expression for the electric
potential in the plasma sheath," IEEE Trans. Plasma Sci. **17**, 884
(1989).

^{78} R. J. Carman, "A simulation of electron motion in the cathode
sheath region of a glow discharge in argon," J. Phys. D: Appl. Phys. **22**,
55 (1989).

^{9} The formal definition of *Ni max* is different from the one
we used in earlier papers^{1,6} because here we have improved upon the
simulation by not assuming a fixed energy loss in ionizing collisions.

^{10 }S. M. Rossnagel, "Gas density reduction effects in magnetrons,"
J. Vac. Sci. Technol. **A6**, 19 (1988),^{ }reported that a
sputtering wind heats and rarefies the neutrals in a sputtering discharge; we
neglect this effect.^{}

^{11 }Makato Hayashi , Nagoya Institute of Technology Report No.
IPPJ-AM-19, Research Information Center, IPP/Nagoya University, Nagoya Japan,
1981, errata 1982.

^{812 }T. E. Sheridan, M. J. Goeckner, and J. Goree, "Electron and ion
transport in magnetron plasmas," J. Vac. Sci. Technol. **A8**, 1623
(1990).

^{}

^{}FIGURE CAPTIONS

FIG. 1. (a) Pressure dependence of ionization efficiency [[eta]]
[[equivalence]] [*Ni* / *Ni max*. Without the magnetic field
**B**, [[eta]] varies almost linearly with pressure *P.* With
**B**, [[eta]] is nearly independent of pressure, allowing the magnetron to
operate at lower neutral pressures than unmagnetized devices. (b) The data
with **B** is re-plotted (on a linear scale with a suppressed zero and
one-standard-deviation error bars), revealing that [[eta]] is constant at high
and low pressure. The dimensionless pressure in the bottom scale is the ratio
of the magnetron radius *a* to the mean free path *mfp.*

FIG. 2. Histogram of ionizing collisions. At 0.2 Pa, (a) the distribution is peaked at 15 ionizations, but has a tail extending down to zero, showing that some electrons escaped while they still had the energy to create ions. At 100 Pa, (b) the distribution has no tail, due to the short mean free path.