` T. E. Sheridan and J. Goree`

` Department of Physics and Astronomy`

` The University of Iowa, Iowa City, IA 52242-1410`

We have investigated the turbulent transport of electrons across the magnetic
field in a sputtering magnetron. Density fluctuations in the range 5 kHz to 1
MHz were recorded; the rms level was less than 5.2 percent. Measured electron
confinement times range from 0.65 to 0.85 usec as the discharge current density
is varied from 25 to 150 A/m^{2}`. Assuming that the
turbulence consists of ion acoustic modes, we find that the electron
confinement time scales as the -0.15 power of the mean squared electric field
fluctuation, much less than the -1 power scaling predicted by turbulent
transport models. From this we conclude that low frequency turbulence is not
responsible for electron transport across the magnetic field.`

__I. Introduction__

` Magnetrons are electrical discharge devices used widely for sputter
deposition of thin films. ^{1,2} Several configurations are common,
including planar, cylindrical, and sputter-gun designs. All of them have a dc
magnetic field which helps to confine electrons in an E` x

` The magnetic field of a magnetron is intended to provide a degree of
electron confinement so that ionization can be sustained with a reduced neutral
gas pressure. This confinement is not perfect, and indeed it must not be, so
that electrons escape at a rate balancing the loss of ions to the cathode. In
order to escape the plasma, electrons which were trapped generally must cross
magnetic field lines; therefore, in order to understand the performance of the
magnetron it is necessary to model the cross-field electron transport. `

` Rossnagel and Kaufman claimed that data from their magnetron
experiments agreed with the Bohm diffusion law for cross-field
diffusion.`^{3,4 }`Being an empirical scaling law, Bohm
diffusion is not a physical model in itself. There are at least three classes
of mechanisms which can result in Bohm-like electron transport, one of which is
plasma turbulence. ^{5} Rossnagel and Kaufman speculated that this was
responsible for the transport they observed. Previously, low-frequency plasma
turbulence was observed in the range 50 - 500 kHz by Thornton.^{6} No
experiments have been reported, however, to determine whether there is in fact
any correlation between the turbulence and the transport in the
magnetron.`

` Turbulent electric fields result in random `d` E `x

` In this paper we report an experiment to test the hypothesis that
low-frequency turbulence accounts for the cross-field electron transport. We
used a dc-biased planar magnetron and a Langmuir probe similar to those used by
Rossnagel and Kaufman,`^{8 }`as described in Section II.
Density fluctuations in the same frequency range as reported by
Thornton ^{6} were observed. By characterizing them in the time and
frequency domains we obtain the rms density fluctuation
<|`d

`II. Apparatus`

` The magnetron used for this experiment is planar, with a cylindrically
symmetric magnetic field. The magnetic field is created by a cylindrical plug
magnet surrounded by a ring of 30 bar magnets (Fig. 1). All the magnets are
made of cast Alnico 5. The maximum value of the radial magnetic field, 220 G,
occurs at the surface of the cathode at a distance of approximately 1.2 cm from
the center. The outer radius of the discharge, determined both from density
measurements and the edge of the etch track, is 2.5 cm. The cathode is copper
and the gas is argon. `

` The voltage is provided by a linear power supply operating in the current
regulated mode. With a neutral pressure of 1 Pa, the discharge voltage
typically ranges from -357 V to -463 V, which corresponds to a range of Idis of
50 mA to 300 mA. A switching power supply was also tested, but its voltage
ripple (5% at 50 kHz) created density fluctuations much larger than the ambient
level observed using a linear supply; therefore, use of the switching supply
was deemed inappropriate for turbulence experiments.`

` Because of the cylindrical symmetry in the magnetic field (neglecting the
spaces between the discrete magnets in the outer ring), the magnetic field can
be described solely by the azimuthal component of the magnetic vector potential
A`[[theta]]`(r,z), where r represents radial position,
`[[theta]]` is azimuthal angle, and z is axial position. This is a
useful parameter since neither A`[[theta]]` nor the electrostatic
potential, `[[phi]]`, depend on `[[theta]]`, i.e., it is an
ignorable coordinate. Thus the `[[theta]]` component of the canonical
momentum, P`[[theta]]` = mr ^{2}`[[theta]]

^{
, }

`where q is the particle's charge and m is the mass. To determine the
volume of the magnetic trap we have plotted in Fig. 2 the first term for
P`[[theta]]` = 0. When the electrostatic potential resulting from the
sheath is also taken into consideration, the shape of the contours agree
qualitatively with the measured plasma density profile.`

` The Langmuir probe is a tungsten wire 0.25 mm in diameter and 3.0 mm long.
It is wire-wrapped and inserted in an alumina tube with a diameter of 1.6 mm.
Only the wire wrap touches the inside of the alumina tube in order to prevent
shorting once the probe tip and the alumina tube are sputter coated with
copper. ^{8} The probe is oriented parallel to the surface of the
cathode on a ray that intersects the symmetry axis of the cathode. It can be
moved in both the r and z directions, allowing two-dimensional mapping of
plasma parameters.`

__III. Data Acquisition and Analysis__

` The purpose of the experiment is to determine how the confinement time
`[[tau]]` scales with <|`d`E| ^{2}>. The value of
`[[tau]]

` The confinement time can be written as ^{5}`

` [[tau]]`` = N / (dN/dt), `

`where N is the total number of electrons in the magnetic trap. The loss
rate dN/dt is given by Idis/e, where e is the electron charge. Then
`[[tau]]` can be expressed as`

` [[tau]] ``= e N / Idis . (1)`

`To measure N we use the Langmuir probe to determine the local density of
electrons at a number of different points in the trap. First, Vp is extracted
from the probe characteristic; this is the voltage at which the derivative of
electron current, I, with respect to the retarding potential is a
maximum. ^{10} Data points for I < 0.75 I(Vp) are then fit to a
model for the characteristic that is the sum of a linear term and an
exponential term. The electron temperature, Te, is determined from the growth
rate of the exponential and the density is found from the values of I(Vp), Te,
and the probe area. The density measurements are then integrated over the
volume of the trap to give N.`

` Electric field fluctuations are computed from probe current fluctuations.
We set the probe bias to the plasma potential and measure both the dc current
drawn by the probe, I, and the fluctuations in the current, `d`I. Since
I is proportional to n and the average electron velocity, then `d`n/n ~
`d`I/I. The time history of the current fluctuations is taken by
monitoring the voltage across a 46.9 [[Omega]] resistor with a capacitively
coupled 8-bit transient recorder sampling 8192 points at an interval of 0.2
usec between points. This digitizing rate gives a Nyquist frequency of 2.5
MHz. A 1300 pF capacitor is put in parallel with the resistor to act as a low
pass current filter with a bandwidth of 2.6 MHz, thereby preventing aliasing.
Using this procedure, the time history of `d`n/n was recorded.`

` The record of the density fluctuations can then be related to `d`E.
Assuming that the electrons are in thermal equilibrium, the electron density
must obey` `ne(`[[phi]]`) = noexp( e`[[phi]]`/kTe),
(2)`

`where k is Boltzmann's constant. Provided that `d`n/n << 1,
which we found to be true, an expansion of Eq. (2) indicates that the potential
fluctuations, `d[[phi]], ` are related to the density fluctuations
by`

` d``n/n ~ (e/kTe) `d[[phi]] .` `

`The electric field is given by E =
-`

d`Ekf = - ik `d[[phi]]`kf, (3)`

`where i is [[radical]]-1, k is the wave number, and f is the frequency. To
proceed further we have to make an assumption about the dispersion relation of
the turbulent modes.`

` Since the electrons are magnetized and experience a dc electric field,
they drift at the E `x

` 2[[pi]]f = csk (4)`

` when f << fpi (3.3 MHz at n = 10 ^{16}
m^{-3}).`

` When the dispersion relation presented in Eq. (4) is used in Eq. (3) and
the resulting expression is squared, we find that`

` ``
^{.} (5)`

`We can then use the fast Fourier transform and the discrete form of
Parseval's theorem ^{11} to evaluate <|`d

`IV. Results`

` All data were taken with a neutral pressure of 1.1 Pa of argon. Six runs
were done, each with a different value of Idis, ranging from 54 to 303 mA
(corresponding to average current densities of approximately 25 A/m ^{2}
to 150 A/m^{2}). `

` For each run we determined the number of electrons in the trap, N, and
then `[[tau]]` using Eq. (1). These values are listed in Table I. To
compute N, we measured the electron density at 17 points in the magnetic trap
ranging over r = 0.6 cm to 2.1 cm and z = 0.43 cm to 1.28 cm, and then
integrated over volume. In performing the integral the values of density
measured at z = 0.43 cm were assumed to extend back to the surface of the
cathode. We found that `[[tau]]` ranges from 0.65 usec at the lowest
discharge current to only 0.85 usec at the highest, and in fact becomes nearly
constant at the higher discharge currents.`

` In order to determine <|`d`E| ^{2}> we recorded
`d

` Figure 3 shows a short section of the density fluctuations recorded at
position 1 with a discharge current of 0.127 A, and Fig. 4 shows the averaged
power spectrum for the entire record. The low frequency peak in the spectrum
is close to the Ar ion cyclotron frequency, which is 5.0 kHz. Above 200 kHz
the spectrum begins to roll off and is comparable to the background noise at 1
MHz.`

` Here <|`d`E| ^{2}> was determined using Eq. (5) and
integrating the spectrum of |`d

` Values for the rms electric field,
(<|`d`E| ^{2}>)^{1/2}, range from 87 V/m at the
lowest discharge current to only 29 V/m at the highest. These values are
somewhat less than the perpendicular component of the dc electric field, which
is ~300 V/m. The rms density fluctuations were small, generally less than 4
percent (see Table I). `

` The scaling of `[[tau]]` with <|`d`E| ^{2}> is
shown in Fig. 5 for positions 1 and 2. For both positions the line of best fit
(on a log-log plot) has a slope of -0.15. That is to say, `[[tau]]
[[proportional]]

` The scaling of `[[tau]]` with <|`d`E| ^{2}>
does not agree with the prediction based on a random walk argument that
`[[tau]] [[proportional]]

`Acknowledgements`

` This work was supported by the Iowa Department of Economic Development and
an IBM Faculty Development Award. J. Goree is supported in part by the Norand
Applied Academics Program.`

`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, in Thin Film Processes, edited by J. L.
Vossen and W. Kern (Academic Press, New York, 1978), p. 131.

^{3 }S. M. Rossnagel and H. R. Kaufman, J. Vac. Sci. Technol.
__A5__, 88 (1988).

^{4 }S. M. Rossnagel and H. R. Kaufman, J. Vac. Sci. Technol.
__A6__, 223 (1988).

^{5 }Francis F. Chen, Introduction to Plasma Physics and Controlled
Fusion, Second Edition (Plenum Press, New York, 1984), p. 190.

^{6 }John A. Thornton, J. Vac. Sci. Technol. __15__, 171
(1978).

^{7} A. A. Vedenov, Theory of Turbulent Plasma, (London Iliffe
Books Ltd., New York, 1968), p. 96.

^{8 }S. M. Rossnagel and H. R. Kaufman, J. Vac. Sci. Technol.
__A4__, 1822 (1986).

^{9 }George Schmidt, Physics of High Temperature Plasmas, Second
Edition, (Academic Press, New York, 1979), p. 222.

^{10 }T. E. Sheridan, Rev. Sci. Instrum. (to be published).

^{11} William H. Press, Brian P. Flannery, Saul A. Teukolsky, and
William T. Vetterling, Numerical Recipes, (Cambridge University Press,
Cambridge, 1986), Chap. 12.

`Table I. `` Plasma parameters as a function of the discharge
current, including `[[tau]]`, N, and parameters measured at positions 1
and 2.`

`_____________________________________________________________________`

_____________________________________________________________________

I`dis N `[[tau]]` position Vp Te ne `d`n/n
<|`d`E| ^{2}>`

` Amps `x `10 ^{11} usec Volts eV `x

^{}____________________________________________________________________

`0.0536 2.17 0.648 1 -2.38 4.49 1.49 5.12 7630`

2 -1.09 3.76 0.98 4.04 5660

0.0856 4.12 0.759 1 -1.24 3.77 2.69 3.73 3700

2 -0.38 3.14 1.94 2.82 2470

0.1268 6.52 0.826 1 -0.89 3.55 4.16 2.69 2160

2 -0.01 2.81 3.26 2.09 1270

0.1678 8.77 0.836 1 -0.73 3.40 5.60 2.46 1760

2 0.03 2.72 4.16 2.04 1120

0.2267 12.1 0.856 1 -0.74 3.29 7.69 2.07 1350

2 0.04 2.59 6.01 1.90 1050

0.3034 16.2 0.854 1 -0.70 3.14 1.05 2.13 1170

2 -0.16 2.54 8.56 1.99 820

`_____________________________________________________________________`

_____________________________________________________________________

Figure Captions

Fig. 1. Schematic drawing of the magnetron.

Fig. 2. Contour plot of the effective potential seen by an electron with
P[[theta]]` = 0. Probe positions 1 and 2 are indicated.`

`Fig. 3. Time history of density fluctuations taken at position 1 for a
discharge current of 0.127 mA. Points were taken every 200 nsec, and the first
251 points out of 8192 recorded are shown.`

`Fig. 4. Spectrum of |`d`n/n| ^{2}. This spectrum was
computed by averaging together 15 overlapping spectra each containing 1024
points.`

`Fig. 5. Scaling of `[[tau]]` with
<|`d`E| ^{2}>. Note the log-log axes. The broken line is
the prediction of turbulence theory. In both cases the line of best fit has a
slope of -0.15, much less than theoretically predicted.`

`
`