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/m2. 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.
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 B drift loop in order to sustain a plasma. This field is low enough, typically 150 G, that only the electrons and not the ions are magnetized. The plasma is exposed to a cathode target where a plasma sheath serves the dual purposes of providing a repulsive electric field for electron confinement and of accelerating ions from the plasma into the cathode. These ions bombard the cathode, thus sputtering material from the target.
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 dE x B drifts across the magnetic field lines.5 Particles execute a random walk where the displacement is the dE x B drift velocity multiplied by the characteristic time scale of the fluctuating dE field. This gives rise to a so-called anomalous diffusion coefficient, D^, that is proportional to the mean square electric field, <|dE|2>.7 For cases where plasma transport is dominated by diffusion, the particle confinement time, [[tau]], is proportional to D^-1; therefore, turbulent transport is characterized by the scaling [[tau]] [[proportional]] <|dE|2>-1.
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 Thornton6 were observed. By characterizing them in the time and frequency domains we obtain the rms density fluctuation <|dn/n|2>1/2 as well as <|dE|2>. We also find [[tau]] and the scaling of [[tau]] on <|dE|2>, which is the basis for our conclusion that low frequency turbulence is not responsible for electron transport.
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]] = mr2[[theta]]. + qA[[theta]]r, is a constant of the motion. Because of this, charged particle motion can then be reduced to that of movement in a two-dimensional effective potential,9
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 <|dE|2>. The value of [[tau]] can be determined indirectly from the number of electrons in the magnetic trap and the rate at which they escape, and <|dE|2> is computed by recording the density fluctuation and assuming a specific dispersion relation for the turbulent waves.
The confinement time can be written as5
[[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, dI. Since I is proportional to n and the average electron velocity, then dn/n ~ dI/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 dn/n was recorded.
The record of the density fluctuations can then be related to dE. 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 dn/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
dn/n ~ (e/kTe) d[[phi]] .
The electric field is given by E = -[[gradient]][[phi]]. (This assumes that the modes are electrostatic, which is true at the low frequencies that are measured.) Considering only one-dimension, the Fourier components of E are related to those of [[phi]] by
dEkf = - 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 B velocity through the unmagnetized ions. For typical values of the radial magnetic field and the axial electric field (150 G and 300 V/m), the drift velocity, vd, is 2 x 104 m/s. The ion acoustic velocity, cs, is 3.1 x 103 m/s for Te = 4 eV in an argon plasma. When vd > cs the ion acoustic mode is unstable;9 therefore, it is reasonable to assume that the turbulent waves are ion acoustic waves. When the ions are unmagnetized, these modes obey the dispersion relation (2[[pi]]f)2 = (csk)2 (1 - f2/fpi2), where fpi is the ion plasma frequency. This reduces to
2[[pi]]f = csk (4)
when f << fpi (3.3 MHz at n = 1016 m-3).
When the dispersion relation presented in Eq. (4) is used in Eq. (3) and the resulting expression is squared, we find that
We can then use the fast Fourier transform and the discrete form of Parseval's theorem11 to evaluate <|dE|2> from the time history of dn/n. Note that the f2 weighting in Eq. (5) means that the highest frequency spectral components of the density fluctuations contribute the most to <|dE|2> .
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/m2 to 150 A/m2).
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 <|dE|2> we recorded dn/n at two positions in the plasma (see Fig. 2). Position 1 is at r = 1.6 cm, z = 0.68 cm and position 2 is at r = 2.1, z = 0.68 cm. The magnetic field measured with a Hall probe at position 1 is almost completely radial (Bz = 7.6 G and Br = 132 G) and at position 2 is at about 34deg. to the probe (Bz = -74 G and Br = 110 G).
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 <|dE|2> was determined using Eq. (5) and integrating the spectrum of |dE|2 up to 1 MHz. The integration was cut off above 1 MHz for two reasons. First, the signal-to-noise ratio is small above 1 MHz. Since Eq. (5) requires that the |dn/n|2 spectrum be weighted by f2 when <|dE|2> is computed, a poor signal-to-noise ratio at high frequencies introduces significant errors in <|dE|2> unless the range of integration is limited. Second, a 1 MHz integration bandwidth allows us to use the simplified ion acoustic dispersion relation, Eq. (4), which is accurate for f << fpi. The spike in the |dn/n|2 spectrum at low frequency makes a negligible contribution to <|dE|2> because of the f2 weighting.
Values for the rms electric field, (<|dE|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 <|dE|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]] <|dE|2>-0.15. The fit was determined independently for each position.
The scaling of [[tau]] with <|dE|2> does not agree with the prediction based on a random walk argument that [[tau]] [[proportional]] <|dE|2>-1 for turbulent transport; therefore, we conclude that low frequency (f < fpi) fluctuations are not responsible for cross field electron transport in the magnetron.
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.
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.
Idis N [[tau]] position Vp Te ne dn/n <|dE|2>
Amps x 1011 usec Volts eV x 1016 m-3 percent (V/m)2
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
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 |dn/n|2. This spectrum was computed by averaging together 15 overlapping spectra each containing 1024 points.
Fig. 5. Scaling of [[tau]] with <|dE|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.