MODEL OF ENERGETIC ELECTRON TRANSPORT

IN MAGNETRON DISCHARGES

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

Department of Physics and Astronomy

The University of Iowa

Iowa City, Iowa 52242-1410

(received

ABSTRACT

A particle model of energetic electron transport in sputtering magnetron discharges is presented. The model assumes time-independent magnetic and electric fields and supposes that scattering by neutral atoms is the dominant transport mechanism. Without scattering, we find that some orbits are confined indefinitely. Using the differential cross sections for elastic, excitation, and ionization collisions in argon, we perform a Monte Carlo simulation of the electrons emitted by ion bombardment of a planar magnetron cathode to predict the spatial distribution of ionization. We find good agreement with experimental measurements of the radial profile of ion flux to the cathode and of the axial profile of optical emission.

I. INTRODUCTION

Magnetrons are magnetized plasma devices that are widely used as sputtering
sources for thin film deposition. There are several geometrical
configurations, including cylindrical magnetrons,^{1} planar
magnetrons,^{2} and sputter guns.^{3} All of them rely on a
cathode that is bombarded by ions to produce the desired sputtering flux.

In addition to sputtering, the ion bombardment is responsible for secondary electron emission at the cathode. The resulting electrons are accelerated by an electric field and gain an energy equivalent to the cathode bias, which is typically 300 to 500 eV.

Magnetrons are all characterized by non-uniform electric and magnetic fields,
**E** and **B**, that are configured to provide confinement of electrons
in the vicinity of the cathode. The electric field is established by the
electric sheath between the plasma and the cathode,^{4} while the
magnetic field is provided externally by a set of permanent magnets or
electromagnets located behind the cathode. In all types of magnetrons, the
electrons are confined in a closed circuit in which they move in the **E** x
**B** direction. As a result of confining the electrons near the cathode,
it is possible to operate the discharge at a lower neutral pressure and a lower
discharge voltage than would be required without confinement.

An adequate picture of the confinement mechanism has been introduced by Wendt
et al.^{5} It is based on the Hamiltonian mechanics formalism of the
effective potential energy surface, which we will review in Section II.

However, an adequate model has not yet been reported for electron transport. Such a model must attribute the scattering of electrons out of the region of confinement to some mechanism. Some candidate scattering mechanisms are collisions with neutrals, Coulomb collisions with ions, and collective effects such as turbulent transport.

Other researchers proposed that turbulence is responsible for electron
transport.^{6 }Low-frequency turbulence is characterized by random
electric fields that result in random **E** x **B** velocities,
which perturb an electron out of its stable orbit. In Ref. 7, a measurement of
the turbulence level and the confinement time [[tau]] was described that showed
that low-frequency turbulent transport is negligible for low energy electrons.
For energetic electrons, it is likely that the random **E** x **B**
velocities would be the same, and that their turbulent transport would also be
negligible. Consequently, another transport mechanism must be found and
tested. One possibility is high-frequency turbulence, another is collisions.
Here, we propose that collisions with neutrals account for the transport of
energetic electrons.

Throughout this paper, we will refer to two categories of electrons, fast and bulk, according to their origin. Fast electrons originate at the cathode or in the sheath, while the bulk electrons are created in the main discharge region. We also introduce the term energetic, which refers to those electrons that have enough energy to ionize neutrals. This designation includes all of the fast electrons as well as the tail of the bulk distribution. The model developed here is applicable to the energetic electrons.

Our presentation begins in Section II by describing electron and ion orbits in the absence of collisions. In contrast to the electrons, we find that the ions are not magnetized, i.e., their orbits are not deflected significantly by the magnetic field. Elastic, excitation, and ionization electron collisions are introduced in Section III. The assumptions of our model, which is applicable to all magnetron geometries, are described in greater detail there. For the planar magnetron geometry in particular, we have developed a Monte Carlo code implementation of the model, which is described in Section IV. All the energetic electrons can be treated using this method, although in this paper we concentrate on applying it to the fast electrons that are emitted from the cathode. Simulation results are presented in Section V, where we find good agreement with experimental data.

II. COLLISIONLESS SINGLE PARTICLE ORBITS

Here we will consider the orbits of single particles, both electrons and ions, in the magnetron geometry. In this picture, the particles respond to prescribed electric and magnetic fields. Because this is a single-particle model, self-consistent plasma behavior resulting in changes in the fields is neglected.

The purpose of this section is to establish the fact that some electrons are confined. To do that, we will first introduce the concept of the effective potential and then present some illustrations of typical particle orbits. For this section, we will consider only collisionless orbits. Of course the particles actually do undergo collisions, which we will treat in Section III.

A. Effective Potential

Two approaches to understanding the motion of an electron in given electric and magnetic fields are the use of the equation of motion,

,^{ (1)}

^{}

^{and the Hamiltonian mechanics picture.8 Wendt et al.5 presented the
Hamiltonian approach as a helpful conceptual way of understanding the particle
motion in a magnetron. In this formalism, the electron moves about on an
effective potential surface in much the same way that a marble rolls about in a
bowl. The shape of the bowl determines the region that is energetically
accessible to the marble, and it will confine the marble if its surface is high
enough all around. Similarly, the effective potential surface of the magnetron
can confine an electron. }

^{ Considering the case of a planar magnetron that is cylindrically
symmetric, as shown in Fig. 1, the effective potential [[Psi]] is independent
of the azimuthal coordinate [[theta]]. Adopting a cylindrical coordinate
system (r, }[[theta]],^{ z), where z is the axial coordinate measured
from the cathode surface, the canonical momentum}

^{}

^{}
(2)

^{is conserved as the electron moves about. Here, A}[[theta]]^{ is
the magnetic vector potential, which depends on r and z. The effective
potential energy is then }

^{}

^{}
^{ , (3)}

^{}

^{}where q is the charge and ^{[[phi]] is the electric potential.
Note that [[Psi]] has both a magnetic component (the first term) and an
electric component (the second term). Since it does not depend on [[theta]],
the effective potential energy is two-dimensional, [[Psi]] = [[Psi]](r,z). For
each electron with a different }
^{,
the effective potential [[Psi]](r,z) is different.}

^{ The z dependence of the shape of [[Psi]] is sketched in Fig. 2. In the
sheath region of the discharge, the electric part q[[phi]] repels electrons
from the cathode. For larger values of z, the magnetic component of [[Psi]]
rises, preventing electrons from moving far away from the cathode. (While the
magnetic part depends on }
^{,}
^{the electric part q[[phi]] does not.) The sum of the electric and
magnetic parts of [[Psi]] form a trap, as shown by the curve for total
effective potential.}

^{ The shape of [[Psi]] shown in Fig. 2 is only a sketch of the z
-dependence. To find the exact two-dimensional effective potential surface, we
must use specific forms for the electric and magnetic components, i.e., we must
specify [[phi]] and A}[[theta]]^{. For A}[[theta]]^{, we will
use the magnetic configuration of the cylindrically symmetric planar magnetron
described in Ref. 7, which has a magnetic field with a purely radial component
of 245 G at r = 1.7 cm on the cathode surface. The computed field was
calibrated against experimental measurements.}

^{ We assume that [[phi]] depends only on z, and consists of a sheath and a
presheath. For the sheath portion we use the improved Child's-law expression
developed in Ref. 9. (This model for [[phi]](z) is appropriate for
sheaths that are collisionless and source-free. Since those two requirements
are not strictly met in magnetrons, the use of this model here is an
approximation.) The sheath is connected smoothly to a presheath with a uniform
electric field of 1 V / cm, a value which was chosen to provide a potential
drop of kTe / q over an interval of 4 cm. Here, k is Boltzmann's constant and
Te is the electron temperature. }

^{ Contour plots of [[Psi]](r,z) are presented in Fig. 3 for two different
values of P}[[theta]]^{. (By selecting a initial radius and setting
the initial velocity to zero, P}[[theta]] ^{ is determined.) The
region inside the heavy contour is accessible to an electron if the cathode is
biased at 400V. For the starting radius selected in Fig. 3 (a), a 400 eV
electron will never leave the trap as long as there is no scattering event to
change P}[[theta]]^{. This is what is meant by the term confinement.
In Fig. 3 (b), on the other hand, electrons emitted at the same electric
potential can escape through the open end at the top of the figure. }

^{ In examining these contours, one must remember that they are plots of the
effective potential energy [[Psi]] as a function of only two spatial
coordinates, r and z, since it does not depend on [[theta]]. While this
surface is two-dimensional, the motion of an electron is, of course, still
three-dimensional. The electron revolves around the z axis, and this azimuthal
rotation is superimposed on the motion in the two-dimensional potential well.
}

^{}

^{B. Orbits}

^{ Using the magnetic and electric fields from the models described above,
and integrating the equation of motion (Eq. 1) in three-dimensions, by using a
fourth-order Runge-Kutta routine,10 we produced plots of the electron and ion
orbits.}

^{ Orbits are shown in Fig. 4 for an electron that was released from the
cathode with zero velocity at the same starting radii as in Fig. 3. In the r-z
projection of Fig. 4 (a), the electron moves about in a confined region
determined by the kidney-shaped potential well. In Fig. 4 (b), the same orbit
is shown projected onto the cathode, i.e., the x-y plane. It has a starfish
shape resulting from the combination of azimuthal rotation and bouncing in the
potential well. This collisionless orbit is confined indefinitely. In
contrast, the orbit displayed in Fig. 4 (c) and (d) is unconfined. This
particle is lost through the open end of the [[Psi]](r,z) surface of Fig. 3
(b). }

^{ When an electron begins at r = 1.7 cm, which is known from experiments to
be near the center of the plasma cross section, the orbit remains on a single
surface of rotation, and takes on a cycloidal shape. The cycloid is, however,
a unique case; most trapped orbits resemble the one shown in Fig. 4 (a) and
(b).}

^{ In all cases, the paths of energetic electrons do not have the
helical shape characteristic of Larmor orbits. This is a consequence of the
gyroradius being comparable to the scale length of the magnetic field. The
orbits revolve around the z axis in the E x B direction.}

^{ Ion trajectories computed for the planar magnetron device are shown in
Fig. 5. They differ markedly from the orbits of electrons as a consequence of
their higher mass. In effect, the ions are not magnetized. They fall directly
to the cathode with very little radial displacement. This observation is used
in the physical model of energetic electron transport that we develop
next.}

^{}

^{III. TRANSPORT MODEL}

^{ In the preceding section we showed that there are electron orbits that
are confined indefinitely in a trap in the absence of some scattering
mechanism. This serves as motivation for the need to model transport. In this
section we will describe a physical model in which the energetic electrons are
scattered by collisions with neutrals.}

^{}

^{A. Assumptions}

^{ Our model portrays single electrons as if they respond to prescribed,
time-independent E and B fields and suffer collisions at random
intervals. Collective plasma effects, which would alter the fields and change
an electron's energy, are neglected. The ions are treated as if they are
unmagnetized. They fall directly from their point of origin toward the
cathode; assuming a one-dimensional electric field, they fall without any
radial deflection. When they strike the cathode, the ions create secondary
electrons that initially have a kinetic energy of typically 1 to 4 eV,11 which
we approximate to be zero.}

^{ Initially, two types of electron collisions, with neutrals and with ions,
must be evaluated. Consider a typical magnetron discharge having an ion
density of 2
1010 cm-3 and room temperature argon at a pressure of 1 Pa. For a 400 eV
electron, the perpendicular momentum deflection collision frequency is 14 MHz
for scattering by neutrals but only 250 Hz for scattering by ions.
Accordingly, we neglect Coulomb collisions in our magnetron model.}

^{}

^{B. Neutral Collisions}

^{ The history of a fast electron can be summarized as follows. After it is
created on the cathode or in the sheath, it falls down the potential hill of
the sheath, gaining energy. It slowly loses this energy due to collisions with
neutrals. Meanwhile, the orbit swirls about in the potential well, and it may
or may not be trapped. Every time a collision takes place, an electron that is
trapped may be scattered into an unconfined orbit and become lost from the
system. Thus, some fast electrons will be confined long enough to perform a
large number of ionizations, while others will be present only long enough to
perform a few or even none at all. As a result of all these ionizations, the
bulk electron population is formed.}

^{ In our treatment, three types of collisions with neutrals are taken into
account: elastic scattering, ionization, and excitation, all of which result
in scattering the direction of the electron's velocity and reducing its energy.
Several collision processes, however, are neglected. We do not allow for the
possibility of a single collision resulting in either a double ionization
(which accounts for 6 percent of the total inelastic cross section at E = 100
eV), nor do we allow the production of an excited state of an ion (which
accounts for less than one percent of the total).12 Additionally, Penning
ionization and two-step ionization are not accounted for.}

^{ The energy lost in a collision depends on its type. Ionizing events
result in the greatest energy loss, which is the ionization potential (15.8 eV
for ground state argon) plus the kinetic energy of the newly released electron.
The energy lost in an excitation collision (11.6 to 15.8 eV for argon) depends
on the energy level to which the neutral is excited. Of the three types of
collisions, an elastic scattering consumes the least energy. The fractional
loss is on the order of the mass ratio, [[Delta]]K / K = (4me/M)
sin2([[alpha]]/2), where me and K are the mass and kinetic energy of the
electron, M is the mass of the neutral, and [[alpha]] is the angle by which the
velocity is deflected.}

^{ All three types of collisions are characterized by a differential
scattering cross section
that predicts the probability per unit solid angle of scattering into an angle
[[alpha]] for a given energy [[Kappa]]. For ionization collisions in
particular, a full differential cross section would also be specified as a
function of the energy of the newly created electron.13}

^{ The angular dependence of the differential cross section
varies with energy. For most gases, the cross sections are nearly isotropic
for low electron energies, while for K > 60 eV they are generally peaked for
small values of [[alpha]], i.e., for forward scattering. }

^{ This predominance of forward scattering means that P[[theta]] and
consequently [[Psi]] are not changed greatly in most collisions. Hence, an
electron may suffer many collisions before it is scattered out of a confining
well. If the confinement is effective, energetic electrons may use up almost
all of their energy before they are lost, thereby maximizing the ion
production.}

^{}

^{IV. NUMERICAL IMPLEMENTATION}

^{ In the physical model described above, electrons move in prescribed
electric and magnetic fields, subject to collisions at random intervals. This
type of model is well suited for application in a Monte Carlo computer code.
Using such a code involves following the orbits of a large number of electrons
one at a time; keeping track of their location, velocity, and kinetic energy;
and allowing them to scatter in collisions. In order to do this, we must
provide expressions for the electric and magnetic fields, push the electrons
with an integrator based on the equation of motion, and provide accurate
differential scattering cross sections. Additionally, the use of Monte Carlo
codes requires that initial conditions for each particle be chosen in an
unbiased fashion to obtain meaningful results.}

^{ We have implemented our physical model with a fully three-dimensional
Monte Carlo program. In this section, we describe its operation and list the
approximations that it entails.}

^{}

^{A. Integrating the Orbits}

^{ The equation of motion to be integrated is Eq. 1. It is integrated in
single steps according to a fixed time step [[Delta]]t by using the same
fourth-order Runge Kutta method as in Section II. In order to follow the orbit
accurately, the value of [[Delta]]t is chosen to be small compared to both the
gyroperiod and the inverse collision frequency.}

^{ The electric and magnetic fields used here are the same as those
described in Section II. The magnetic field is recorded in a two-dimensional
grid in the r-z plane, while the electric field is gridded in one coordinate,
z. Interpolation is used to evaluate the fields between grid points.}

^{}

^{B. Collisions}

^{ At each time step during an electron orbit, we determine whether a
collision has taken place. This is done by generating a random number that is
compared to the probability per unit time of a collision. For argon, we use
the total cross sections reported in Refs. 14 and 15 and interpolate between
tabulated values.}

^{ When it has been determined that a collision takes place, another random
number is generated to determine whether it was an elastic, excitation, or
ionization event. This is evaluated according to the relative cross sections
of these processes. }

^{ According to the type of collision, the energy of the energetic electron
is reduced. In our current code we approximate that the energy lost in an
excitation or ionizing collision is always a fixed amount, 11.6 eV for
excitation and 15.8 eV for ionization of argon. By using a fixed energy loss
for ionization, we ignore the kinetic energy of the secondary electron released
in an ionization event. That energy is almost always much smaller than the
energy of the ionizing electron.13 Accordingly, we use a reduced differential
scattering cross section for ionization that does not depend on the energy of
the secondary electron. }

^{ In addition to decrementing the energy of the energetic electron when it
undergoes a collision, we randomly scatter its velocity into a new direction,
which is chosen in a manner consistent with the differential scattering cross
section. For the energy range K < 3 eV, we assume that the differential
cross sections are isotropic, while for 3 eV < K < 3 keV, we use the
normalized values of
for
elastic scattering that are tabulated in Ref. 16. We make the approximation
that these are valid for all three types of collisions. The suitability of
this simplification can be confirmed by an inspection of the shapes of
as a function of energy for the three types of collisions,13 which reveals that
they are nearly the same. }

^{}

^{C. Ensembles of Orbits}

^{An orbit must be terminated according to some criterion, such as: (1) it
leaves the simulation region, (2) a fixed maximum time has elapsed, or (3) the
energy is depleted below the level of the ionization potential. When this
happens, a new energetic electron orbit is begun from a new initial position.
This is repeated for an ensemble of electrons. After the last electron is
finished, the user can produce useful results such as a spatial profile of
ionization events. }

^{ The statistical quality of those results will be determined by both
random and systematic errors. Random errors diminish according to the square
root of the number of electrons in the ensemble. The larger the ensemble size,
the smaller the random errors will be. Of course the computer time required
increases linearly with the number of electrons, so the user cannot choose the
ensemble size to be arbitrarily large but rather chooses it to give the size of
error bars desired. }

^{ Systematic errors, on the other hand, are generally not improved upon by
increasing the ensemble size. They must be eliminated by thoughtful planning
of the code. One potential source of systematic errors that requires special
attention in all Monte Carlo codes is the choice of the initial conditions for
each electron.}

^{}

^{D. Initial Conditions}

^{ Physically, the position at which each electron starts is different.
Accordingly, in the numerical implementation, a different random starting
position must be chosen for each electron, and this must be done in a fashion
that represents the physical processes involved. The code could be used with
initial conditions chosen to simulate electrons born in the plasma and sheath
regions, but in the present paper, we will treat only the electrons originating
on the cathode due to ion bombardment. }

^{ The starting radius ri for each electron is specified in the following
convergent manner. The first electron is released from a random point.
Subsequent electrons can originate from any point on the cathode, with some
points more likely than others. To account for this, we use a radial
probability profile P(ri) to describe the probability per unit radius of an
electron being born at ri. Physically, each electron is responsible for
ionizations that would result in ions falling directly down to the cathode,
causing in the emission of new electrons at the sites of ion impact.
Accordingly, in the code, P(ri) is updated according to the history of all
previous ionizations before starting each new electron. This is done by
counting the ionizations that took place in radial bins. }

^{ The probability profile converges to a recognizable steady-state after
about thirty fast electrons have been tracked. This means that for many
applications the code must be run for an ensemble of more than thirty
electrons. }

^{}

^{V. RESULTS}

^{ As a test of the numerical implementation of our physical model, we ran
an ensemble of 600 electrons originating at the cathode. The discharge was
assumed to have the parameters of the experiment reported in Ref. 7: cathode
bias of 400 V, argon gas pressure of 1 Pa, Te = 4 eV, and density appropriate
to produce a Debye length of 0.1 mm. Under these conditions, the total
collision mean free path is 1.7 cm for a 20 eV electron and 9.4 cm for a 400 eV
electron. }

^{ The time step [[Delta]]t was selected to be 50 psec. This value was
chosen to be small enough to provide 20 integration steps per gyroperiod in a
300 G magnetic field. It also provided at least 130 steps per collision, on
average. }

^{ Orbits were followed until the electrons either (1) escaped from the
boundaries of our 4 cm by 4 cm working space on the r-z plane or (2)
until a limit of 2.5 usec had elapsed, whichever came first. The time limit
was chosen to be long enough that fewer than 0.5 percent of the remaining
electrons had enough energy to result in an ionization. For this run, we did
not stop orbits if the electron energy fell below the 15.8 eV ionization
potential of argon.}

^{ }

^{A. Collision Statistics }

^{ We recorded the location and type of each collision. Defining the
average number of ionizations per fast electron as <Ni >, our run yielded
<Ni> = 14.26
0.44 , where the uncertainty given is for a 90 percent confidence level. For
comparison, consider that a 400 eV electron can produce at most Nmax = 25
ionizations, given the ionization potential of 15.8 eV for argon. }

^{ Note that if the }discharge is to be maintained solely by ion
bombardment of the cathode, then it is necessary that <Ni> >=^{
[[gamma]] -1}, where ^{[[gamma]] }is the secondary electron
emission coefficient. Since^{ [[gamma]] -1~} 10 for many metals, our
result that <Ni> ~ 14 confirms that cathode emission alone could
sustain a magnetron plasma.^{}

^{ For the same ensemble of 600 fast electrons, there were an average of 3.65
0.16 excitation collisions per electron. This number is less than <Ni>
simply because the cross section for excitation is smaller than that for
ionization.}

^{ The most frequent type of collision is elastic, because of its large
cross section, especially for lower energies. Our ensemble experienced 48.67
elastic collisions per electron. Most of these occurred after the energy of a
trapped electron had been depleted below the 11.6 eV excitation potential,
where there is no possibility of inelastic collisions. }

^{ The distribution of the number of ionizing collisions per electron is
shown in a histogram, Fig. 6. The most probable number of ionizations per fast
electron was 20, which is at the center of a peak between 18 and 24 ionizations
per electron. (The width of this peak is accounted for by excitation
collisions.) Recalling that Nmax is 25, we see that this peak indicates that
many fast electrons remained trapped until most of their energy is lost. The
other electrons performed a smaller number of ionizations and then
escaped.}

^{ The confinement provided in an actual magnetron plasma works well,
producing almost as many ionizations as possible. This is of course the
advantage of the magnetron as a sputtering device. }

^{ The principal reason for the good confinement is the effective potential
well. Furthermore, as we discussed at the end of Section III, most
differential cross sections are peaked for forward scattering, which means that
to scatter P[[theta]] enough to detrap the electron, many collisions may be
required. To test this prediction, we performed a run for comparison where
was
isotropic and found that <Ni> was 2.52
0.15,
which is small compared to the value of 14.26 reported above. This indicates
that in contrast to forward scattering, large angle scattering leads to a loss
of confinement after fewer ionizations have occurred. Much of the magnetron's
high ionization efficiency can thus be attributed to the predominance of
forward scattering.}

^{}

^{[[Beta]]. Density of Ionizing Collisions}

^{ The density of ionization collisions [[rho]]i(r,z) is shown in Fig. 7.
Recalling that the z axis is the axis of symmetry for the planar magnetron, an
inspection of this illustration reveals that the ionization events are
concentrated in a ring located above the cathode surface. Some of the
ionization events from Fig. 7 were used to prepare Fig. 1, where the ring is
shown in a three-dimensional view. The latter diagram bears a very close
resemblance to the visual appearance of a planar magnetron discharge.}

^{ According to the physical model of electron transport, the ionization
distribution is determined by electron orbits, which are determined by the
shape of the effective potential well. It is therefore instructive to compare
[[rho]]i(r,z) in Fig. 7 to an effective potential well shape, such as Fig. 3.
The shapes are nearly the same, even though a large number of potential wells
were sampled by the electrons. This finding is consistent with the descriptive
model of electron motion presented in Section II.}

^{}

^{C. Radial Profile of Ionizations }

^{ A useful quantity that can be obtained from [[rho]]i(r,z) is the radial
ionization profile, }

^{}

^{
, (4)}

^{}

^{which represents the number of ionizations per unit area above the
cathode. Recall that in our model the ions fall directly down to the cathode
from the ionization site without radial deflection. Therefore, nr(r) can be
directly compared to experimental measurements of the radial profile of the ion
flux striking the cathode. }

^{ This ion flux is easy to measure. One indication is the depth of the
etch track in the cathode material. After a number of hours of operating our
planar magnetron in argon, we measured the etch track profile in the copper
cathode.7 It had a maximum depth of 2 mm. In Fig. 8 a plot of this
experimentally obtained profile is overlaid on a plot of the radial ionization
distribution computed from our Monte-Carlo results. The agreement between the
model and the experimental data is good. }

^{ This comparison is successful despite the fact that the etch track was
formed by operating the magnetron at a number of different pressures and
cathode biases. (Better experimental data could be obtained in the fashion of
Wendt et al. by imbedding current probes in the magnetron cathode surface.5)
}

^{ Our finding that the shape of the etch track can be predicted accurately
lends confidence to the model.}

^{}

^{D. Axial Profile of Ionizing Collisions}

^{ Inelastic collisions produce the optical emission from a magnetron
discharge. For most atomic gases, the visible light that an observer can see
originates from transitions between the excited states. Consequently, the
optical emission from a magnetron can be modeled by recording the locations of
the inelastic collisions that result in exciting the neutral to a highly
excited state. }

^{ As a proxy for excitations to those highly excited states, here we will
use the ionization events. This is a suitable approximation because the total
ionization and excitation cross sections have nearly the same dependence on
electron energy. }

^{ The axial distribution of ionizing collisions, nz(z), was computed by
integrating the ionization density radially and is shown in Fig. 9. The reader
will note the appearance in this illustration of a cathode dark space between
0 < z < 2 mm. There is also a peak in nz(z) at z = 3 mm. }

^{ This figure can be compared to an experimental measurement of axial
distribution of optical emission, which will also display a dark space as well
as a peak in the emission a few millimeters from the cathode. Such a
measurement was reported by Gu and Lieberman.17 They used an optical apparatus
that had an axial spatial resolution of 0.28 mm to detect the total visible
emission along a radial chord. Their experiment is suited for a test of our
numerical results shown in Fig. 9, although the comparison can only be
qualitative, because we used a different magnetic configuration. We do find
qualitative agreement between our simulation and their experiment.}

^{}

^{VI. CONCLUSIONS}

^{ We have presented a model of energetic electron transport in magnetron
plasmas. This single particle description is applicable to all magnetron
geometries. We have reduced the problem by making a number of approximations.
Most notably, we allow scattering of electrons by only three types of
collisions (ionization, excitation, and elastic scattering), and we track
single electron orbits in prescribed magnetic and electric fields. }

^{ We have applied the model by developing a Monte Carlo orbit code. For
the present purposes, we use it to gain an understanding of the physical
processes involved in magnetron discharges, although it may also be practical
for designing optimum magnetron configurations.}

^{ As a test of the Monte Carlo method, we simulated the electrons emitted
from the cathode of a planar magnetron. A two dimensional B field and a
one-dimensional E field were assumed. We found that cathode emission
results in an average of 14.26 ionizations per electron for a 400 V discharge
in argon at a pressure of 1 Pa. This high efficiency of ionization is
attributable to the electron confinement brought about by the magnetron's
effective potential well and by the predominance of forward scattering. The
spatial distribution of ionization predicted in the simulation was compared to
experimental measurements of the radial ion current profile and the axial
optical emission profile. These tests show good agreement between theory and
experiment, indicating that the model can accurately characterize the locations
in the discharge where ionization takes place. }

^{}

^{ACKNOWLEDGMENTS}

^{ The authors thank Mark Kushner and Amy Wendt for helpful discussions.
This work was funded by a grant from the Iowa Department of Economic
Development and an IBM Faculty Development Award. }

^{}

^{}

^{REFERENCES}

^{1} John A. Thornton and Alan S. Penfold, in __Thin Film
Processes__, edited by J. L. Vossen and W. Kern (Academic Press, New York,
1978), p. 75.

^{2} Robert K. Waits, ibid., p. 131.

^{3} David B. Fraser, ibid., p. 115.

^{4} Francis F. Chen, __Introduction to Plasma Physics and Controlled
Fusion__, 2nd ed. (Plenum Press, New York, 1984), p. 190.

^{5} A.E. Wendt, M.A. Lieberman, and H. Meuth, J. Vac. Sci. Technol.
__ A6__, 1827 (1988), and A. E. Wendt, Ph.D. thesis, University of
California at Berkeley, 1988.

^{6} S. M. Rossnagel and H. R. Kaufman, J. Vac. Sci. Technol.
__A6__, 223 (1988).

^{7} T.E. Sheridan and J. Goree, J. Vac. Sci. Technol. __A7__, 1014
(1989).

^{8 George Schmidt, Physics of High Temperature Plasmas, 2nd ed.,
(Academic Press, New York, 1979), Sec. 2-6.}

^{9 T.E. Sheridan and J. Goree, U. of Iowa Report No. 89-13, submitted to
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^{10 }C.E. Roberts, __Ordinary Differential Equations: A
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12 R.J. Carman, J. Phys. D__ 22__, 55 (1989).

^{13 }E.W. McDaniel, __Collision Phenomena in Ionized Gases__ (John
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^{}14 Makato Hayashi, Nagoya Institute of Technology Report No.
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1981, errata 1982.^{}

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__12__, 979 (1979).

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(1988).}

^{}

^{}

^{FIGURE CAPTIONS}

^{}

^{}FIG. 1 Planar magnetron. The disk shaped cathode surface shows an
etch track formed by ion bombardment sputtering. A representation of the
ionization distribution, prepared using the simulation described in this paper,
is shown by the dots. The coordinate systems used are indicated.

FIG. 2. Sketch of the effective potential [[Psi]] for an electron. The electric component of the potential appears mainly in the sheath region near the cathode. It accelerates electrons away from the cathode, giving them energy as they move into the plasma region, and it keeps electrons in the plasma from escaping to the cathode. The magnetic component keeps electrons from escaping from the vicinity of the cathode surface. This combination results in a potential well where some electrons are confined.

FIG. 3. Plot of the effective potential surface [[Psi]](r,z) for an electron
born with zero velocity on the cathode (z = 0), computed for the
planar magnetron described in Ref. 8, with a 400 V bias on the cathode. The
unit of energy is eV, as shown in the bar scale. The potentials shown were
computed for the following values of the canonical momentum
:
(a) 3.51 x 10^{-25} kg m^{2}/sec, corresponding to a starting
radius ri = 1.0 cm, and (b) 1.14 x 10^{-25} kg
m^{2}/sec, corresponding to ri = 0.5 cm. The contour shown
with a heavy line bounds the region accessible to an electron of total energy
400 eV. In (a) an electron is confined indefinitely in the absence of
scattering, while in (b) it can escape out the top.

FIG. 4. Orbits for an electron born on the surface of the cathode. The orbit began at rest at the dot located at z = 0 cm and r = 1.0 cm in (a) and (b), corresponding to the effective potential shown in Fig. 3 (a). Here, (a) is a projection of the orbit onto the r-z plane, and (b) is a projection onto the x-y plane, i.e., the cathode surface. This collisionless orbit is confined indefinitely. For (c) and (d), the orbit began at r = 0.5 cm, corresponding to Fig. 3 (b). This unconfined orbit escapes after several bounces. (These diagrams were made by integrating the equation of motion with a time step of 50 psec, and connecting the points recorded at every fourth time step.)

FIG. 5. Orbits of ions born at z = 1 cm. The paths followed by Ar^{+}
are shown for several initial radii. The ions are released at z = 1 cm with
zero initial velocity, and then the presheath electric field accelerates them
to the surface of the cathode. As they fall, they are deflected slightly in
the azimuthal direction, but only a negligible amount in the radial direction.
The electric potential is assumed here to vary with z but not r.

FIG. 6. Distribution of number of ionizing collisions for 600 electrons in
argon at a pressure of 1 Pa. This histogram has a peak centered at 20
ionizations, which corresponds to electrons that lose all their energy in
collisions without escaping from the trap. Elastic, excitation, and ionizing
collisions are taken into account. The maximum possible number of ionizations
is Nmax^{ }= 25, which is the ratio of the 400 V cathode potential to
the 15.8 eV ionization potential. The width of the peak is due to energy lost
in excitation collisions.

FIG. 7. Density of ionizing collisions, [[rho]]i(r, z), plotted as a function of r and z. The locations of the ionization events were assigned to pixels 0.5 mm by 0.5 mm in size to produce the gray-scale diagram. In pixels having less ionization than the bottom of the gray scale, the event locations are shown with dots. The region of strong ionization forms a ring separated from the cathode surface by a dark space. Note the similarity to the effective potential [[Psi]](r,z) shown in Fig. 3 (b). The event locations indicated here were used in preparing the three-dimensional view in Fig. 1.

FIG. 8. Radial profile of ionizing collisions, nr(r), from the simulation, compared to etch track depth from experimental measurements. The vertical scales for the two curves were adjusted to make the heights of their peaks match. The agreement is good.

FIG. 9. Axial profile of ionizing collisions from the model, nz(z). The cathode is located at z = 0. This can be compared to experimental measurements of the axial optical emission profile, such as the one reported in Ref. 17. Note the appearance of a cathode dark space for 0 < z < 2 mm.