**Electron and ion transport in magnetron plasmas**

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

*Department of Physics and Astronomy*

*The University of Iowa*

*Iowa City, Iowa 52242*

(Received

We demonstrate experimentally that there is a strong link between electron transport and ion transport in sputtering magnetrons plasmas. Electron densities and discharge currents for He, Ne, Ar, Kr, and Xe discharges are measured and used to infer the charged particle confinement time. We find that the confinement time increases with ion mass. We find fair agreement between the experimental results and a model which assumes that ion motion is collisionless and unmagnetized, and that the spatial dependence of the electric potential and ion source are similar in the different discharges.

**I. INTRODUCTION**

Sputtering magnetrons are widely used for thin film deposition and sputter
etching.^{1} In these devices electrons are confined by the electric
sheath at the cathode and an externally imposed magnetic field. The magnetic
field near the cathode is typically weak enough that electrons are magnetized,
but ions, because of their much larger mass Mi, are not. The combination of
the electric potential and the magnetic field forms an effective potential
well,^{2,3} or trap region. In the absence of collisions many
electrons are unable to escape from the trap region to the anode. In contrast,
the travel of ions to the cathode is not impeded by the magnetic
field.^{3} Maintenance of charge neutrality demands that electrons and
ions leave the trap region at the same rate.^{4} In this report we
clarify the mechanisms responsible for ion transport to the cathode and the
concomitant escape of electrons from the trap region to the anode.

Previous authors^{5} speculated that low frequency turbulence is the
mechanism responsible for electron transport. By the measuring the bulk
electron confinement time [[tau]] and comparing it to the energy density in the
turbulent waves at different discharge currents, Sheridan and Goree^{6}
demonstrated that low frequency turbulence is probably not responsible for the
observed transport.

In previous work^{3 }we proposed a model where collisions with neutral
gas particles play an important role in electron transport in a sputtering
magnetron. Comparison with experiment confirmed that the model accurately
predicts the spatial distribution of ionization events.^{7} However,
since the model deals only with the motion of single electrons in static
electric and magnetic fields and does not include the interplay between
electron and ion transport, it cannot predict the magnitude of the discharge
current or plasma density.

A complete model for the sputtering magnetron cannot neglect ion transport,
since in equilibrium one electron must escape from the plasma for each ion
which is expelled from the plasma. To achieve this balance, the electric field
in the plasma **E** (i.e., the negative gradient of the plasma potential)
adjusts self-consistently until it has a magnitude and shape such that ions are
lost to the cathode, and electrons to the anode, at equal rates. This
self-consistent **E** field abets the collisional escape of electrons from
the trap region. Since electrons and ions are created at nearly the same rate,
the confinement time for electrons and ions must be nearly the same. Because
heavier ions experience less acceleration in a given electric field, we expect
that the charged particle confinement time and hence the plasma density will
increase with ion mass.

In the next section we describe an experiment which demonstrates that electron confinement and ion confinement are inextricably linked, as discussed above. In Sec. III we present a simple model and compare it to the experimental results. We draw conclusions from this work in Sec. IV.

**II. EXPERIMENT**

For the experiment reported here, we have measured the scaling
of the charged particle confinement time [[tau]] with ion mass Mi in our
sputtering magnetron. We used either He, Ne, Ar, Kr, or Xe as the working gas.
Before describing the particulars of the experiment, we need to explain how we
determine the charged particle confinement time.

Here we show that the confinement time for charged particles is proportional to the ratio of the plasma density n, measured at a given location, to the discharge current Idis. Assume that there is some volume V enclosing a plasma source g, given as the number of ions created per unit time and volume. Since the plasma must be quasineutral, we let n = ne = ni, where ne and ni are the electron and ion densities respectively. The number of ions or electrons in V, which we call N, is then given by

^{
, (1)}

where the integral is over V. If the plasma is in equilibrium then the number of ions created in V per unit time [the integral in Eq. (1)] must equal the number of ions lost from V per unit time. This loss rate is Idis/e. Consequently,

[[tau]] = eN / Idis . (2)

Measurements performed in our magnetron earlier^{6} indicate that n
[[proportional]] N, no matter at which point the density is measured.
Therefore, the charged particle confinement time must be proportional to the
plasma density divided by the discharge current:

[[tau]] [[proportional]] n / Idis . (3)

Our magnetron, which is described in Ref. 6, has a planar copper cathode and a cylindrically symmetric magnetic field. The magnetic field is tangential to the cathode surface (z = 0) at r = 1.7 cm, and has a value of 245 G there. Here r is the distance from the symmetry axis of the magnetron, and z is the height above the cathode. In this experiment the cathode bias was provided by a linear dc power supply. Pressure measurements, which we corrected for gas type, were made using a Granville-Phillips Convectron gauge tube that was located well away from the magnetron.

A cylindrical Langmuir probe was used to measure the electron
density,^{6} which should be equal to the plasma density n. The
diameter of the tungsten probe tip, which was parallel to the cathode surface,
is 0.254 mm, and its length is 3.5 mm. Plasma parameters were extracted
from the probe characteristic using the method described in Ref. 6.

In these experiments we varied the discharge current and the ion mass while measuring the electron density at a single point in the trap region. This point was located at r = 1.7 cm (over the deepest part of the etch track) and at z = 1.03 cm. The reduction in the discharge current caused by inserting the probe was less than 5 percent. Measurements for each gas (He, Ne, Ar, Kr, or Xe) were performed at a constant discharge voltage Vdis (see Table I). The discharge current was varied by changing the neutral gas pressure P from 0.04 Pa to 16 Pa, with most of the data falling between 0.2 Pa and 10 Pa (see Fig. 1). Higher discharge voltages were needed with Kr and Xe to get a good discharge.

Current-pressure curves for the five gases used are shown in Fig. 1. It has
been reported that the relationship between discharge current and pressure is
characterized by at least three distinct regimes.^{8} In the normal
"A" operating regime, Idis varies approximately linearly with P, and the
characteristic current-voltage scaling for dc magnetrons (Idis [[proportional]]
Vdis^{x}, where x ~ 7) is observed. Most of the data presented here is
from this normal operating regime, in which it has been shown that ions travel
collisionlessly through the sheath.^{8}

In Fig. 2, which is a plot of the electron density versus discharge current,
we see that the electron density increases with ion mass for a given discharge
current. We must conclude from Fig. 2 that *charged particle confinement
times are longer for heavier ions*. To gain a further understanding of the
physics involved we next consider a simple model.

III. DISCUSSION

As we have demonstrated experimentally, at a given discharge current the
plasma density increases with ion mass. We have argued in general terms that
this happens because heavier ions are accelerated more slowly in a given
electric field. We now explore this idea in greater depth.

We examine a one-dimensional model where we assume that we are
given the source and potential profiles. A complete model, which we do not
present here, would include some way of calculating the source and potential
self-consistently. We will show that if we assume that the source and
potential profiles are similar in different discharges that [[tau]] should
scale as the square root of the ion mass. Comparison with the experimental
data presented in the previous section yields good agreement with this
predicted scaling at higher discharge currents and for heavier gases.

**A. Model with nonuniform source and potential**

The motion of ions must be approximately one-dimensional, as we explain here. In the presheath, the magnetic field must be nearly constant along any magnetic field line, and the electric field must therefore be nearly perpendicular to the magnetic field. Further, the plasma density on a magnetic field line must be fairly uniform, since the field line is approximately an equipotential. The motion of ions along a single electric field line thus should be modeled well using only one spatial coordinate.

We also assume that both the electric potential profile [[phi]](z) and the source profile g(z) along our chosen electric field line are known. As before, g is the number of ions created per unit time and volume. The cathode is located at z = 0 and the plasma fills the volume for z > 0. The potential is taken to have a single maximum located at z[[minute]]. For z > z[[minute]] ions are collected at the anode. The ion density at a given point can now be calculated.

The ion density for any z < z[[minute]] will be given by the following
integral^{9}

^{, (4)}

if ions are born with zero initial kinetic energy and their trajectories are
not affected by the magnetic field or by collisions.^{3} Because of
quasineutrality, n = ni = ne, this is the density that would be measured by a
probe placed at z. From Eq. (4), we conclude that the plasma density depends
on the ion mass and on the source and potential profiles between z and
z[[minute]].

The current density at the cathode, J, is given by

^{,} (5)

if we ignore the small current of electrons created at the cathode via secondary emission. Combining Eqs. (4) and (5) we find

^{. (6)}

^{Thus, the ratio n}(z) / Idis, which we have shown is proportional to
[[tau]], is a function of Mi, g, and [[phi]]. Note that Eq. (6) is independent
of all constant factors, including the ionization cross section, that are
implicit in g; therefore, one can compare values of [[tau]] measured for
different discharges with various gases.

If we assume that the profile of g, and the profile and magnitude of [[phi]], are the same for different Mi, then Eq. (6) predicts the following scaling:

[[tau]] [[proportional]] Mi^{1/2} . (7)

Why would we expect g and [[phi]] to obey such assumptions? We have shown
previously^{3} that the location of ionizations events is determined by
the motion of single electrons. This motion is dominated by the shape of the
magnetic field as long as the sheath width is much narrower than the magnetic
trap height. Since the magnetic geometry is fixed, we have good reason to
expect that the profile of the source will not depend strongly on the type of
gas used. Others have shown that that under some circumstances the magnetic
field can create a "magnetic presheath."^{10,11}

Thus, it does not seem unreasonable to expect that the profile, and possibly the magnitude, of [[phi]] in the presheath should be determined by the magnetic field and will not depend on the type of gas used. A comparison to our experimental results will indicate the extent to which this is true.

**B. Comparison of experiment with model**

We make a quantitative comparison between the scaling predicted in Eq. (7) and the experimental data in the following way. Since the data points for each gas in Fig. 2 fall on nearly straight lines, a power law of the form

n = *a* Idis^{b} (8)

can be used to approximate the relationship between the plasma density and
discharge current. The parameters *a* and *b*, which we found using
a non-linear least squares fit, are shown in Table I.

Values of the exponent *b* range from 1.24 to 2.18. The departure of
*b* from the predicted value of one [Eq. (7)] indicates the extent to
which the assumptions about g and [[phi]] used to obtain Eq. (7) are met. The
highest values of *b* are found for the heaviest gases, Kr and Xe, for
which we found it necessary to use higher discharge voltages. Note that for
all gases *b* > 1. This indicates that [[tau]] increases with Idis.
At higher discharge currents the sheath width decreases. It seems likely that
this decrease in sheath width leads to a decrease in the electric field in the
presheath, and a corresponding increase in [[tau]].

To determine the experimental scaling of [[tau]] with ion mass, we plot [[tau]] in units of n / Idis against Mi for several values of Idis in Fig. 3. Here n / Idis is determined using Eq. (8) and the fit coefficients found in Table I. Recall that our model predicts that the plasma density at a given discharge current (i.e., the confinement time) should scale like the square root of the ion mass provided that g and [[phi]] do not change. We find good qualitative agreement with the scaling predicted by our model, particularly for Idis = 1 A. This agreement indicates that the source and electric potential structure are similar for the different discharges in this case.

At lower discharge currents, and for He, Fig. 3 demonstrates a nonignorable
deviation from the predicted Mi^{1/2} scaling [Eq. (7)]. Since the
plasma density in these discharges is lower than for the heavier gases and
higher currents, the sheath width will be larger. The failure to more closely
follow the predicted scaling probably indicates that the sheath width has
become a significant fraction of the magnetic trap width, significantly
altering the profiles of both [[phi]] and g in these cases.

**IV. CONCLUSIONS**

We have experimentally demonstrated the importance of ions in regulating
electron transport in sputtering magnetron discharges. Currents and electron
densities for separate discharges using He, Ne, Ar, Kr, or Xe gases are
reported. The electron density increases with increasing ion mass for the same
discharge current, indicating that the confinement time of charged particles
increases with ion mass. Assuming that ions travel collisionlessly in a
one-dimensional electric potential and are not affected by the magnetic field,
we predict that the confinement time should be proportional to the square root
of the ion mass. To facilitate comparison between the experiment and this
prediction, we fit the data for each gas with a function of the form n = *a
*Idis^{b}. Good agreement with the predicted
Mi^{1/2} scaling was found for higher plasma densities.

Since we have shown that electron transport and ion transport are inextricably linked, we can predict that they must both be taken into account in any future model which purports to describe the global properties of the magnetron discharge.

**ACKNOWLEDGMENT**

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

**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 }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).

^{3 }T. E. Sheridan, M. J. Goeckner and J. Goree, "Model of energetic
electron transport in magnetron discharges," U. of Iowa Report No. 89-14, and
J. Vac. Sci. Technol. **A** (to be published).

^{4 }Francis F. Chen, __Introduction to Plasma Physics and Controlled
Fusion__, (Plenum Press, New York, 1974), pp. 139-140 and pp. 152-153.

^{5 }John A. Thornton, "Magnetron sputtering: basic physics and
application to cylindrical magnetrons," J. Vac. Sci. Technol. **15**,
171-177 (1978).

^{6 }T. E. Sheridan and J. Goree, "Low-frequency turbulent transport in
magnetron plasmas," J. Vac. Sci. Technol. **A7**, 1014-1018 (1989).

^{7 }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. **A**. (to appear May/June 1990).

^{8 }J. Goree and T. E. Sheridan, "Regimes of sputtering magnetron
operation," U. of Iowa Report No. 89-16, submitted to J. Vac. Sci. Technol.
**A**, 1989.

^{9 }Lewi Tonks and Irving Langmuir, "A General Theory of the
Plasma of an Arc," Phys. Rev. **34**, 876-922 (1929).

^{10 }R. Chodura, "Plasma-wall transition in a oblique magnetic field,"
Phys. Fluids **25**, 1628-1633 (1982).

^{11 }H. A. Hassan and R. K. Seals, Jr., "Collisionless Sheath near an
Electrode in the Presence of a Magnetic Field," Phys. Fluids **14**, 344-348
(1971).

Table I. Ion mass, Mi, discharge voltage, Vdis, and fit coefficients, *a*
and *b*, tabulated for each of the five noble gases used. In the
experiment, the plasma density n and the discharge current were varied by
changing the neutral gas pressure, as shown in Fig. 1. Measurements of n were
fit with a power law of the form n = *a *Idis^{b}. The departure
of *b* from the predicted value of unity [Eq. (7)] indicates the degree of
validity of the assumptions about the profiles of g and [[phi]] that were made
to obtain Eq. (7).

^{_______________________________________________________}

^{_______________________________________________________}

^{} gas Mi Vdis *a* *b*

(amu) (V) (10^{16} m^{-3} A^{-b})

^{_______________________________________________________}

^{}helium 4.003 -400 11.9 1.43

neon 20.18 -400 44.8 1.24

argon 39.95 -400 156. 1.63

krypton 83.80 -500 586. 1.84

xenon 131.3 -600 2490 2.18

^{_______________________________________________________}

^{_______________________________________________________}

^{}

FIGURE CAPTIONS

FIG. 1. Discharge current Idis as a function of neutral pressure P for He, Ne, Ar, Kr, and Xe discharges. A line showing the scaling Idis [[proportional]] P, which is the signature of the normal operation regime, is drawn for comparison. Note the data for each gas exhibit this scaling.

FIG. 2. Plasma density n measured at r = 1.7 cm and z = 1.03 cm, plotted against the discharge current Idis for five noble gases. Note that as the atomic mass of the gas increases, the plasma density is always larger for the same discharge current, clearly indicating that the charged particle confinement time increases with increasing ion mass.

FIG. 3. Confinement time [[tau]], in units of n / Idis [Eq. (3)], plotted
against the ion mass Mi for discharge currents of Idis = 0.1, 0.3, 1 A. The
scaling [[tau]] [[proportional]] Mi^{1/2} [Eq. (7)], found by assuming
similar source and potential profiles in the different discharges, is shown for
comparison. Good agreement is found at higher plasma densities, where we
expect the sheath width to be much narrower that the magnetic trap height.