Magnetic field dependence of sputtering magnetron efficiency

J. Goree and T. E. Sheridan

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

(Received

A Monte Carlo simulation of electron transport is used to predict the dependence of the ionization efficiency on the magnetic field strength of a planar magnetron. This offers insight into the operation of the magnetron, and it also provides two valuable practical results. First, the efficiency increases with field strength only up to a saturation level. Operating a magnetron with a stronger field strength would only lead to an undesirable loss of target utilization. Second, a scaling law is found that is useful for designing magnetrons of different sizes.

** **The magnetic field of a magnetron allows the operation of an intense
sputtering discharge at low neutral gas densities.^{1,2 }The magnetic
field strength is a critical parameter in a magnetron design, but it is one
that is often chosen in practice by empirical methods and guesswork. What is
needed is a model that provides not only insight into magnetron operation, but
also practical criteria for designing a magnetron.

Wendt *et al.*^{3} used a Hamiltonian model to predict that the
etch track becomes wider and the sputtering target utilization improves as the
field is made weaker, and they confirmed this prediction experimentally. ^{
}This does not mean, though, that a weaker field is always better. There
is a trade-off. A weak field provides a wide etch track and good target
utilization, while a strong field provides more effective electron confinement.
In this letter we analyze the electron confinement as a function of magnetic
field strength using a Monte Carlo simulation. Based on our results, we offer
practical criteria for designing planar magnetrons.

The device simulated here is the cylindrically symmetric planar
magnetron^{4-8} shown in Fig. 1. The magnetic field **B** is formed
by a cylindrical magnet and a concentric ring of 30 bar magnets, sandwiched
between a non-ferrous cathode and a steel pole piece. Because **B** is
highly inhomogeneous, we specify that the field is measured at the point where
it is tangential to the cathode target surface. The radius there is denoted as
*a*, and the field magnitude as *
*
*. *In most devices, the etch track is deepest at radius *a*. For
our device, *a* = 1.7 cm.

We have performed our simulations for various magnetic field strengths by
adjusting *
*
in the code. This is equivalent to adjusting the magnetization *M* of the
magnets. For comparison, Alnico 5 magnets yield *
*
= 245 G in the device shown in Fig. 1, as measured experimentally. The
magnetic field shown in Fig. 1 was computed from the magnetic
configuration.^{5} Although this field could be adapted to include the
effect of the electronic E x B drift current parallel to the cathode, here we
have not done so. The magnetic field due to this effect is at least an order
of magnitude weaker than the field produced by the permanent magnets. For this
letter, the shape of the field lines remains constant, regardless of the value
selected for *
*.

Of course one could also investigate other magnetic shapes using our method.
The shape investigated here is a "Type II unbalanced magnetron," in the
notation of Window and Savvides.^{9} This means that the far-field
dipole moment of the magnet assembly is dominated by the outer magnet ring.
Our simulations^{4} and Langmuir probe measurements^{10} show
that electrons escape from the plasma up the "chimney" along the center axis of
this device. In contrast, our earlier simulations^{11} have shown that
electrons escape radially outward in the "Type I" magnetron of Wendt* et
al*.^{3}

The electron Monte Carlo code, described in detail
previously,^{4,8,11} ran as follows. An electron starts at rest from
the cathode. Its orbit is computed by integrating the equation of motion with
a fixed time step of 12 psec, which is much smaller than the mean time between
collisions. The equation of motion includes the computed magnetic field and a
prescribed, time-independent electric field.^{12} The electric field
is one-dimensional, and it depends on the potential drop between the cathode
and the plasma. We approximate that this drop is equal to the cathode bias *
*
because the plasma potential is small in comparison. Elastic, excitation, and
ionizing collisions with neutrals occur at random intervals,^{4} and
they reduce the electron's energy and scatter its velocity
direction.^{8} The particle's orbit is terminated when its total
energy drops below the ionization potential, or when it escapes the region
where the field lines are drawn in Fig. 1. The next electron is then started
on the cathode at a radius chosen randomly in the self-consistent manner
described in Ref. 4. This procedure was repeated for an ensemble of 30 to 200
electrons.

For this letter we used the simulation to find the ionization efficiency by
cathode emission, [[eta]], which has been defined as follows.^{8} The
number of ionizations that a single electron performs, averaged over an
ensemble of electrons, is denoted by
.
We can compare
to the maximum possible number of ionizations, *
*, that can be performed by a well-confined electron in the absence of
excitation collisions.^{8} The ionization efficiency is the ratio

[[eta]] [[equivalence]]
/ *
*, (1)

and it will depend on the magnetic field, pressure, and cathode bias. The
efficiency [[eta]] lies between zero and an upper limit^{4,8} of about
0.9.

We allow ionization only by electrons born on the cathode, and ignore
ionization by electrons born in the sheath and in the main plasma. Earlier
simulations^{9} showed that this approximation causes the simulation
to underestimate [[eta]] somewhat for a magnetron with a weak field of *
*
= 104 G.

Results for the ionization efficiency at various field strengths and cathode
biases are presented in Fig. 2, which shows that [[eta]] increases with
magnetic field strength only up to a saturation level, [[eta]] = 0.9. The
simulation was run for argon at only one density (corresponding to 2 Pa at room
temperature) because we have found^{8} that the neutral density has
only a weak effect on [[eta]].

It is useful to seek a law of similarity that makes all the curves in Fig. 2
coincide. This is accomplished by constructing a dimensionless variable from
the parameters *
*
, *
*
, and *a*:

^{ }
^{. (2)}

Here *e* and *m* are the electron charge and mass. Equation 2 is
in practical units (Gauss, cm, and Volts). Figure 3 confirms that the
efficiency [[eta]] is a function of [[beta]] and that the data lie on a
similarity curve. The efficiency increases with [[beta]] until it saturates
for [[beta]] >= 15. This similarity curve and the scaling law in Eq. 2 are
the principal results of this letter.

Using a magnetic field stronger than the saturation level in Fig. 3 offers no
benefit. Indeed, it would result in an undesirable loss of etch track width
and target utilization. The model of Wendt *et al.*^{3} predicts
that the etch track width is proportional to *
*.
This scaling implies that the target utilization is [[proportional]]
,
and is thus improved by using a weaker magnetic field.

For completeness, we outline here a derivation of Eq. 2. The portion of Fig.
2 where the curves do not coincide is the transitional regime below saturation,
which arises from electrons that are scattered into unconfined orbits and are
lost before they consume their energy by ionizations.^{8} In a
Hamiltonian formalism, orbits can be predicted^{ }by the effective
potential energy surface [[Psi]].^{3,4 } Unconfined orbits are lost
through a hole in the surface, as shown in Fig. 3 of Ref. 4. ^{ }Using
a cylindrical coordinate system (*r,*[[theta]],*z*) and conservation
of canonical angular momentum,^{4} we find that for an electron born at
rest on the cathode at radius r0:

, (3)

which has units of energy. The first term of Eq. 3 is due to the magnetic field, and it prevents electrons from moving a large distance from the cathode. Here is the azimuthal component of the vector potential, defined by . The value of at the electron's birthplace is denoted . The second term is due to the electric potential [[phi]], which repels electrons from the cathode .

Now consider scaling the system: one may vary the electric potential, the
magnetization, and the magnetron size. These quantities are parameterized by *
*,
*M*, and *a,* respectively. (The gas pressure can also be altered
by a gas rarefaction effect or by the user, but the effect is negligible
because the ionization efficiency is almost independent of gas
density.^{8}) The field strength *
*
is proportional to *M. *In varying the size, the dimensions of the
magnets and the gaps between them retain the same proportions. The vector
potentials
and
scale as *a M,* while r and r0 scale as *a,* so that the first term
of Eq. 3 scales as (*a M*)^{2} . The electric potential [[phi]]
in the second term scales as *
*.
Likewise, an electron born on the cathode has an energy that scales as Vdis.
If *a, M, *and *
*
are increased while holding the ratio (*a M*)^{2}* */*
*constant, then the potential surface shape [[Psi]]* */*
*
will not change. The proportions of a hole in the surface will be unaffected
by the scaling, so that an electron born on the cathode will still be lost
after the same number of bounces. Provided that the mean free path is >=
*a, *the electron will perform no more ionizations after it escapes
through the hole.^{4,8} The ionization efficiency thus will be
unchanged by the scaling. This scaling, the ratio of the two terms in Eq. 3,
yields the dimensionless variable [[beta]] in Eq. 2.

As noted earlier, there is a design trade-off in increasing the field strength. If the field is weaker than required for saturation, one sacrifices target utilization for ionization efficiency. Beyond the saturation level, however, there is no longer a trade-off in using a field stronger than required for saturation -- things only get worse with increasing field strength. In particular, the target utilization suffers without improving the ionization efficiency. This result of our model is of practical use. It is contrary to a conventional wisdom that a stronger field is always better.

Let us now assess the range of parameters where the simulation is valid. A
wide variety of experimental data, including Langmuir probe, laser-induced
fluorescence, optical glow, and etch track profile
measurements,^{4,7,8,11} have proven the accuracy of the model for two
sets of parameters: Btan = 245 G, *a* = 1.7 cm, [[beta]] = 7, and Btan
=277 G, *a* = 5 cm, and [[beta]] = 22. So the simulation appears to be
accurate in that range of parameters. Where is it not accurate? It was
found^{11} to predict too low a value for [[eta]] for a weak field of
Btan = 104 G (due to a violation of the assumption that electrons emitted from
the cathode dominate the ionization). We also suspect that the prescribed
electric field model will fail at saturated field strengths of [[beta]]
>> 20, if the plasma potential becomes very negative. Based on these
limits, we can say that the onset of saturation (our principal result) is
probably predicted with enough accuracy, provided that Btan >> 104
G.

We summarize with two criteria for magnetron design. First, for
designs of different sizes but the same proportions, a larger magnetron should
have magnets with a weaker magnetization. This will provide a weaker *
*
in order to keep [[beta]] equal to a constant in Eq. 2. (This law is valid
provided that the magnetron radius *a* is not much larger than the mean
free path.) Second, for maximum efficiency of ionization by electrons emitted
from the cathode, the magnetic field strength should be selected to operate at
the onset of saturation in Fig. 3. This occurs at [[beta]] ~ 12 - 15 for a
cylindrically symmetric planar magnetron with the proportions shown in Fig. 1.
It would be undesirable to use a stronger field because it would decrease the
target utilization without increasing the ionization efficiency.

As a sputtering target is consumed, its surface becomes closer to the magnets
so that the magnetic field there grows in time. A magnetron designer might
wish to maximize the sputtering rate averaged over the target lifetime. In
that case, the magnetron should be designed to start at a weaker field, perhaps
[[beta]] ~ 8 - 10, when a fresh target is in place. The corresponding field
strength in Gauss can be determined from Eq. 2*. *For a magnetron of
radius *a *= 1.7 cm operated at *
*
= 400 Volts, the optimal field strength for a fresh target would be *
*
~ 310 - 395 Gauss.

**ACKNOWLEDGMENT**

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

**REFERENCES**

^{1 }John A. Thornton and Alan S. Penfold, "Cylindrical magnetron
sputtering," in Thin Film Processes, ed. by J. L. Vossen and W. Kern (Academic,
New York, 1978), p. 75.

^{2 }Robert K. Waits, in Thin Film Processes, ed. by J. L. Vossen and
W. Kern (Academic, New York, 1978), p. 131.

^{3 }A. E. Wendt, M. A. Lieberman, and H. Meuth, J. Vac. Sci. Technol.
**A6**, 1827-1831 (1988). The experiment therein was characterized by 7
< [[beta]] < 25.

^{14 }T. E. Sheridan, M. J. Goeckner and J. Goree, J. Vac. Sci.
Technol. **A8**, 30 (1990).

^{25 }T. E. Sheridan and J. Goree, J. Vac. Sci. Technol. **A7**,
1014 (1989), reports the magnetic field computed in two dimensions from a
magnetic scalar formalism. The effect of the pole piece was modeled by using
image magnets, and the 30 individual bar magnets in the outer ring were treated
as a solid annulus with a reduced magnetization to account for the gaps between
bars. This field was confirmed against experimental data.

^{36 }T. E. Sheridan, M. J. Goeckner, and J. Goree, J. Vac. Sci.
Technol. **A8**, 1623 (1990).

^{7 }M. J. Goeckner, J. Goree, and T. E. Sheridan, IEEE Trans. Plasma
Sci. (in press April 1991).

^{8 }T. E. Sheridan M. J. Goeckner, and J. Goree, Appl. Phys. Lett.
**57**, 2080 (1990).

^{9} B. Window and N. Savvides, J. Vac. Sci. Technol. **A4**, 196
(1986).

^{10 }T.E. Sheridan, M.J. Goeckner and J. Goree, J. Vac. Sci. Technol.
(in press May/June 1991).^{}

^{11 }J. E. Miranda, M. J. Goeckner, J. Goree and T. E. Sheridan, J.
Vac. Sci. Technol. **A8**, 1627 (1990).

^{412 }T. E. Sheridan and J. Goree, IEEE Trans. Plasma Sci. **17**,
884 (1989).

^{5}

FIGURE CAPTIONS

FIG. 1. Planar magnetron device. A cylindrical magnet is surrounded by a ring
of 30 bar magnets, forming the field shown here. The magnetic field strength
is characterized by its value *
*
at the radius *a* where it is tangential to the surface, as indicated by N
. In the simulation, the magnetization *M* of the magnets and hence the
value of *
*
can be adjusted without affecting the shape of the field. All dimensions are
shown in cm.

FIG. 2. Dependence of ionization efficiency, defined by Eq. 1, on the magnetic
field strength*
*.
The ionization efficiency increases with magnetic field strength only up to
saturation at [[eta]] = 0.9. The simulations assumed a neutral argon density
corresponding to a pressure of 2 Pa at room temperature. They were repeated
for the cathode biases: *
*
= 295, 400, and 505 Volts, which correspond to *
*= 13, 17, and 21. One-standard-deviation error bars are shown.

FIG. 3. Dependence of ionization efficiency [[eta]] on the dimensionless
parameter
,
where *
*
is in Gauss, *a* in cm, and *
*
in Volts. The simulation results lie on a similarity curve that saturates at
[[beta]] ~ 12 - 15. Using a magnetic field stronger than required for the
onset of saturation will not yield any increase in ionization efficiency.