Monte Carlo Simulation of Ionization in a Magnetron Plasma
J.E. Miranda, M.J. Goeckner, J. Goree, and T.E. Sheridan
Department of Physics and Astronomy
The University of Iowa
Iowa City, Iowa 52242-1410
A Monte Carlo simulation of electrons emitted from the cathode of a planar magnetron is tested against experiments that were reported by Wendt, Lieberman and Meuth [J. Vac. Sci. Technol. A6, 1827 (1988)] and by Gu and Lieberman [J. Vac. Sci. Technol. A6, 2960 (1988)]. Comparing their measurements of the radial profile of current and the axial profile of optical emission to the ionization profiles predicted by the model, we find good agreement for a typical magnetic field strength of 456 G. We also find that at 456 G the product of the average number of ionizations, [Ni, and the inverse secondary electron emission coefficient, [[gamma]], is ~1. This indicates that secondary emission contributes significantly to the ionization that sustains the discharge. At 171 G, however, , revealing that cathode emission is inadequate to sustain a discharge at a low magnetic field.
Magnetrons are glow discharge sputtering devices used for thin film deposition and etching.1,2 They employ electric and magnetic fields that are configured to trap electrons and enhance ionization near a cathode surface. Ions produced in the plasma are accelerated by the electric sheath toward the cathode. They strike the cathode surface, producing the desired sputtering as well as the emission of secondary electrons that help sustain the discharge.
The electrons in the plasma can be categorized according to their point of origin: (1) the cathode surface, (2) the sheath, and (3) the main plasma region. Electrons in the third category are called the bulk because they comprise the largest part of the electron population. The total electron loss rate, including all three categories, is balanced in equilibrium by the ion loss rate, as described by Sheridan, Goeckner, and Goree.3
The same authors described a model of the transport that is applicable only to the energetic electrons, i.e., to those electrons with enough energy to ionize neutral atoms.4 This includes all the electrons in the first two categories plus the tail of the bulk distribution. The model is useful for predicting where the ionization takes place. Electrons are assumed to interact only with prescribed static electric and magnetic fields and undergo collisions with neutrals. These collisions reduce an electron's energy and scatter its velocity vector, eventually leading to its escape from the trap region. While the electrons also undergo Coulomb collisions with ions, these are negligible in comparison to collisions with neutrals for the range of pressures and electron densities found in sputtering magnetron discharges.4 Ions are assumed to be accelerated by the electric field directly to the cathode from the ionization sites, their trajectories not influenced by the magnetic field.
In a fully self-consistent model, the electric fields would evolve in order to balance the loss rates of ions and electrons, exactly as they do in the actual plasma. For practicality, the electric field is prescribed in our model and does not evolve in time; therefore, an accurate expression for the electric field relies on the use of empirical data. In particular, the electric field model must be supplied with laboratory measurements of the bulk electron temperature and density, Te and ne. (We did not develop the model to predict Te or ne.)
This physical model for energetic electrons has been implemented numerically in a Monte Carlo code, which has been documented in Ref. 4. It takes into account ionization, excitation, and elastic collisions by making use of differential cross sections for argon. As an initial test of this code, the authors compared its prediction of the radial ionization profile to the etch track profile measured in a planar magnetron, and found good agreement. The device that was simulated was the Iowa magnetron,4 which has a cylindrically symmetric magnetic field formed by a set of permanent magnets. One magnet is on the axis and the other is a concentric outer ring of the opposite polarity. The code was used to simulate only the electrons emitted from the cathode, ignoring the electrons born in the sheath and in the main plasma region.
We report in this paper a test of the model that is more rigorous than the test that was described when the code was first reported.4 The experiments we evaluate were carried out at the University of California at Berkeley by Wendt, Lieberman and Meuth5 and by Gu and Lieberman.6 Our presentation begins in Sec. II with a synopsis of the experimental and simulation methods. We then show results in Sec. III for the ionization efficiency predicted by the code and a comparison of the radial and axial profiles from experiment and simulation.
A. Berkeley experiments
Wendt et al.5 and Gu and Lieberman6 performed their experiments with a cylindrically symmetric planar magnetron that has a variable strength magnetic field produced by an electromagnet. The coil windings are encircled with an iron core that has a circular slit to form the magnetron's magnetic field. This magnetron differs significantly from the Iowa magnetron in its magnetic field geometry, yet the experiments that have been reported for these two devices are readily compared because they had in common the use of argon gas and a copper cathode.
Wendt et al.5 reported measurements of radial ion current profiles, which were made using probes embedded in the cathode. Since this current is produced by ions that fall directly to the cathode from the site of an ionization event, it can be compared to the radial profile of ionization predicted by our Monte Carlo simulation. This test of the model has an advantage over the previously reported comparison to the etch track profile;4 the radial current profile is a better measurement because it is recorded in real time for a single set of discharge parameters.
Gu and Lieberman6 reported measurements of the axial profile of broadband optical emission. The emission is produced principally by electron-impact excitation of neutral atoms to a highly excited state. As a proxy for these excitation events, we use the ionization collisions for argon. This is an appropriate approximation since ionizing collisions and most excitation transitions have cross sections with nearly the same dependence on electron energy.7,8
The locations of ionizations coincides with the glow emitted by neutral argon atoms. However, the sputtered metal atoms also emit light after they pass through the discharge, and this can overwhelm the argon neutral emission, especially at high discharge currents.9 For sputtered copper atoms, the brightest spectral line is at 511 nm, resulting from the metastable state 2D5/2, which has a lifetime of ~1 usec.10,11 We have visually observed this green glow in the Iowa magnetron everywhere downstream from the discharge. Consequently, copper emission is not appropriate for comparison to the simulation of the locations of ionizing collisions. Gu and Lieberman did not filter out the copper glow. Therefore, we must explain from the outset that while their axial profile data is the best available, it is not as well suited for comparing to our simulation as the radial profiles of Wendt et al.
Using the Monte Carlo code, we simulated the electrons emitted from the cathode of the Berkeley magnetron and produced spatial profiles of the ionization. We used a one-dimensional electric potential; it is composed of a linear presheath that is connected to a sheath potential predicted by a numerical solution of Poisson's equation.12 The electron orbits and collisions were handled the same way as in Ref. 4, including the approximations that all excitation collisions result in an energy loss of 11.6 eV, that two-step ionization processes13,14 can be ignored, and that the kinetic energy of the electron ejected in an ionization event is negligible. When the incident electron energy is several hundred eV, the kinetic energy of the ejected electron is typically between 0 and 10 eV,15,16 and by neglecting this we underestimate the energy loss in an ionizing collision, [[Delta]]Ki, by about 30 percent.
The magnetic field for the Berkeley magnetron, as shown in Fig. 1, was computed by Wendt using the POISSON finite-element code.5 By varying the current in the electromagnet coil, the Berkeley researchers were able to adjust the magnetic field strength up or down while maintaining its shape. We performed simulations for two field strengths: |B|max = 456 G, which is comparable to the field in the Iowa magnetron, and |B|max = 171 G, which is rather weak.17
We have chosen the parameters of the simulation to conform as closely as possible to those of the experiments, as listed in Table I. Argon at a pressure of 0.67 Pa was used. The electric field model in the code must be supplied with values of the discharge voltage Vdis, the electron temperature Te, and the Debye length [[lambda]]D. The Berkeley experimenters reported values for Vdis and the sheath thickness d for each of their runs, but they did not indicate Te or [[lambda]]D. We have chosen Te = 5 eV, based on the measurements of Rossnagel and Kaufman,18 and we computed [[lambda]]D by using d in Child's Law.12 These estimates for Te and [[lambda]]D are not used for any purpose other than the electric field model. Given the assumptions that the discharge is radially uniform and that Child's Law is applicable, these estimates are probably accurate to within 50 percent. We determined that this accuracy is satisfactory by testing several values of [[lambda]]D within this range in the simulations described below and finding no statistically significant differences in the results.
The equation of motion was integrated in three dimensions using a fourth-order Runge-Kutta routine. A fixed time step [[Delta]]t was selected to be small enough to provide for good energy conservation and to be small compared to the mean collision time and the gyroperiod. By examining a few representative electron orbits, we found that many of the electrons emitted from the cathode are trapped in a ring-shaped region a few millimeters above the cathode. This was also observed in the simulation reported previously for the Iowa magnetron.4 The orbits in the two magnetrons differ, however, in the way that electrons escape from the trap region. In the Berkeley device, we found that they escape toward the sides of the cylindrical vessel, not toward the anode plate at the end of the vessel opposite the cathode as they do in the Iowa magnetron.
An orbit was terminated when the electron escaped from the work space (5.22 cm axial by 14 cm radial), or its total energy fell below the ionization potential qVi = 15.8 eV of argon, or a time limit of 10 usec elapsed, whichever came first. A new orbit was then begun on the cathode surface at a random location chosen in a manner consistent with the history of previous ionizations.4 This procedure was repeated for an ensemble of 75 or more electrons.
We used the simulation results to compute the average number of ionizations per electron emitted by the cathode, [Ni. This value can be compared to the maximum possible number of ionizations,
Nmax [[equivalence]] Int (-qVdis / [[Delta]]Ki ) , (1)
where the Int operator indicates truncation to an integer, and [[Delta]]Ki represents the average energy lost in an ionizing collision. The ratio [Ni / Nmax thus serves as a figure of merit for a magnetron at a given gas pressure.19
We will show in Sec. III that when a magnetron device has extremely effective electron confinement, [Ni / Nmax will be as high as 0.8 for PAr ~ 1 Pa, while a device that provides virtually no confinement at all will have [Ni / Nmax on the order of 0.1. The figure of merit can never reach a value of 1.0 due to the energy lost in excitation collisions.
The simulation also predicted the density of ionizations, [[rho]]i(r,z), on the r-z plane, where r is the radial coordinate and z is the axial coordinate measured from the cathode surface. Using [[rho]]i(r,z), we computed the radial ionization profile
and the axial ionization profile
Gu and Lieberman's optical collimator viewed the discharge tangentially to the cathode, thereby integrating the emission over a pencil-shaped volume of diameter 0.28 mm.6 Our definition of nz(z) in Eq. 3 is also a volume integral. While we could calculate the integral for the same small volume, we choose instead to improve our statistics by counting the number of ionizations inside the entire slab parallel to the cathode between z - [[Delta]]z/2 and z + [[Delta]]z/2, where the spatial resolution is defined to be [[Delta]]z = 0.5 mm. This discrepancy in the shapes of the volumes in the experiment and simulation is equivalent to a different radial weighting in the integral, and it is probably insignificant due to the large radius of the plasma ring.
A. Ionization Efficiency
We found that the average number of ionizations per fast electron, [Ni, was 17.53 for the Berkeley magnetron with a 456 G field and Vdis = 362 V. This yields a figure of merit [Ni / Nmax = 0.80, indicating that the confinement of fast electrons is very effective.
For the discharge with the 171 G field and Vdis = 520 V, we found that [Ni is only 3.72, which yields a figure of merit of 0.12. This reveals that the fast electrons are ineffectively confined when the field is weak. For comparison, [Ni was reported to be 14.26 for Vdis = 400 V in the Iowa magnetron,4 corresponding to a figure of merit of 0.56.
Since the simulation took into account only the electrons born on the cathode surface, one can determine whether secondary emission is adequate to sustain the discharge by considering the product [Ni [[gamma]] , where [[gamma]] is the secondary electron emission coefficient. Recalling that for each ion that strikes the cathode, the average number of secondary electrons emitted is [[gamma]], and assuming that all the ions are collected by the cathode, we see that a discharge can be sustained by cathode emission if [Ni [[gamma]] is comparable to unity. Because our simulation overestimates [Ni by about 30 percent, 19 the computed value of [Ni [[gamma]] must actually be somewhat larger than one in order for cathode emission to sustain the discharge. For 400 eV argon on copper, the secondary emission coefficient19 is [[gamma]] ~ 0.09. For the low magnetic field (171 G) discharge, , indicating that the electrons we have simulated cannot account for the ionization needed to sustain the plasma.
Considering that the experimenters were able to maintain a discharge at |B|max = 171 G, we conclude that there is some physics missing from our model that is important at low magnetic fields. Most likely, this indicates that the electrons born in the sheath and in the plasma, which we have not included in the simulation, dominate the ionization for discharges with weak magnetic fields. We would expect some of the electrons born in the sheath to have trapped orbits that periodically re-enter the sheath, resulting in a sheath avalanche; however, we have not performed any simulations to determine the effectiveness of this avalanche. In any case, such a weak magnetic field would not likely be used in a practical sputter deposition process.
On the other hand, in the higher field (456 G) and also in the Iowa magnetron, we find that [Ni [[gamma]] >= 1, revealing that cathode emission contributes significantly to the ionization and may indeed be able to sustain the discharge without assistance from electrons born in the sheath or in the main plasma.
The energy cost of creating an ion-electron pair can be estimated from the results above. This cost is CE = qVdis / [Ni [[gamma]], the ratio of the energy required to accelerate one ion across the cathode sheath to the number of ionizations resulting from the impact of one ion on the cathode. This cost depends on the secondary emission coefficient [[gamma]] of the material used in the cathode target. For the Berkeley magnetron at 456 G and assuming [[gamma]] = 0.09, we find that CE = 362 eV / 1.6 = 226 eV. The sputtering magnetron can be readily compared to other types of plasma sources based on the value of CE. While the ionization efficiency [Ni / Nmax is quite high for a magnetron, the energy cost is considerable. The principal energy loss is the deposition of ion kinetic energy into the cathode target.
B. Radial and Axial Profiles
As explained in Section IIA, the results reported by Wendt et al.5 are the best experimental data available for comparing to our simulation. Their radial ion current profile for |B|max = 456 G is reproduced in Fig. 2, where it is overlaid on the radial ionization profile, nr(r), from our simulation. The experiment and the simulation show excellent agreement: the radial profiles are peaked at the same location, and the widths of the peak are nearly the same.
The axial optical emission profile measured by Gu and Lieberman6 for |B|max = 456 G are compared to nz(z) from our simulation in Fig. 3. Both the experiment and the simulation exhibit a dark space adjacent at the cathode surface and a peak in the electron trap region. The simulation shows good agreement with the experiment for the first few millimeters.
However, farther from the cathode the glow observed in the experiment exceeds the simulation prediction. We have determined from visual and spectroscopic observations of the Iowa magnetron that in this region the copper glow dominates the optical emission. Accordingly, we dismiss the discrepancy between experiment and simulation found at z > 4 mm. Because the experimental axial profile measurements were made with a broadband optical detector, one cannot expect agreement as good as exhibited by the radial profiles described above.
We have tested a Monte Carlo simulation of electrons emitted from the cathode of a planar magnetron. To make a more rigorous test of this code than previously reported,4 we compared its predictions to the experimental results reported by Wendt, Lieberman and Meuth5 and by Gu and Lieberman6 for two different strengths of the magnetic field.
We found that the level of ionization due to secondary emission predicted by the code is adequate to sustain the discharge for the case with the typical field strength, |B|max = 456 G, but not for the weak field of 171 G. This suggests that the electrons born in the plasma are important for maintaining the ionization at low magnetic fields. We also found that the energy cost of producing an ion-electron pair in a sputtering magnetron is on the order of 200 eV, depending on |B|max and [[gamma]].
The radial ionization profile predicted by the simulation shows excellent agreement with experiment for the 456 G field. This confirms the earlier finding4 that the simulation is useful for predicting the profile of the ion flux to the cathode. The axial profiles also show agreement as well as can be expected given the experimenters' use of a broadband optical detector.
The authors are indebted to Amy Wendt for supplying us with her magnetic field data. This work was supported by the State of Iowa Department of Economic Development.
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 T.E. Sheridan, M.J. Goeckner, and J. Goree, U. of Iowa Report No. 89-19, and J. Vac. Sci. Technol. (submitted 1989).
4 T.E. Sheridan, M.J. Goeckner and J. Goree, U. of Iowa Report No. 89-14, and J. Vac. Sci. Technol. A8 (to be published 1990).
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 Lan Gu and M.A. Lieberman, J. Vac. Sci. Technol. A6, 2960 (1988).
7 It would be desirable to compare the simulated ionization density to the optical emission of a specific spectral line, preferably one that results from electron-impact excitation that has a cross section with an energy dependence that nearly matches that of ionizing collisions. Most spectral lines of neutral argon have such an energy dependence, but not all of them (cf. Ref. 8). Because the experimenters of Ref. 6 employed a broadband optical detector that did not isolate a specific spectral line, this limits the accuracy of comparing their experiment to our simulation, although the copper atom emission discussed in the text is probably a more severe problem.
8 A. Chutjian, and D.C. Cartwright, Phs. Rev. A 23, 2178 (1981).
9 S.M. Rossnagel and K.L. Saenger, J. Vac. Sci. Technol. A7, 968, (1989).
10 A.K. Anders, E.C. Harvey, and R.C. Tobin, Appl. Phys. Lett. 49, 923 (1986).
11 Charles H. Corliss and William K. Bozman, Experimental Transition Probabilities for Spectral Lines of Seventy Elements, National Bureau of Standards monograph No. 53 (1962) p. 80.
12 T.E. Sheridan and J. Goree, U. of Iowa Report No. 89-13, and I.E.E.E. Trans. Plasma Sci. (to be published).
13 C.M. Ferreira, J. Loureiro, and A. Ricard, J. Appl. Phys. 57, 82 (1985).
14 S. E. Savas, Appl. Phys. Lett. 48, 1042 (1986).
15 J. Bretagne, G. Callede, M. Legentil, and V. Puech, J. Phys. D: Appl. Phys 14, 1225 (1981).
16 R.J. Carman, J. Phys. D: Appl. Phys 22, 55 (1989).
17 The values for |B| cited by the Berkeley researchers5,6 are computed for the location on the cathode surface where it is largest, as shown with the arrow in Fig. 1. At this location, |B| is 1.65 times as large as at the point we usually cite, i.e., at the point on the cathode where the field is tangential. In this paper we adopt their convention.
18 S.M. Rossnagel and H.R. Kaufman, J. Vac. Sci. Technol. A4, 1822 (1986).
19 We actually overestimate [Ni by about 25 percent. This results from neglecting the kinetic energy of the electron born in an ionizing collision, as discussed above, by approximating that [[Delta]]Ki = - qVi = - 15.8 eV when we decrement the energy of the fast electron in the simulation. To compensate for this, we also use [[Delta]]Ki = - qVi in Eq. 1. The figure of merit [Ni / Nmax is thus not affected by the approximation.
20 Sanborn C. Brown, Basic Data of Plasma Physics, 2nd ed. revised (MIT Press, Cambridge Mass., 1966), p. 225. The value we cite for [[gamma]] is probably accurate to within 50 percent.
Parameters for planar magnetron experiments and simulations. The magnetic fields are listed for two points on the cathode: where |B| is maximum17 and where B is tangential to the cathode. The results from Ref. 4 for the Iowa magnetron are listed in the last column for comparison. The average number of ionizations, [Ni, is only for electrons emitted by the cathode.19 The secondary emission coefficient in the last line is assumed to be [[gamma]] = 0.09.
Berkeley5,6 Berkeley5,6 Iowa4
high |B|max low |B|max
Parameters reported for experiment
|B|max (G) 456 171 __
|Btan| (G) 277 104 245
PAr (Pa) 0.67 0.67 1.0
Vdis (V) 362 520 400
Idis (A) 0.5 0.5 0.15
Parameters assumed for simulation
Te (eV) 5 5 4
[[lambda]]D (mm) 0.0525 0.0802 0.1
[[Delta]]t (psec) 5 5 50
No. of electrons 75 600 600
[Ni 17.53 +/- 0.62 3.72 +/- 0.17 14.26 +/- 0.44
[Ni / Nmax 0.80 0.12 0.56
[Ni [[gamma]] 1.6 0.3 1.3
Fig. 1 Magnetic field of the Berkeley planar magnetron, computed by Wendt.5 The coordinates used in this paper, r and z, are shown in the axes. The cathode surface is located at z = 0 and 0 < r < 11.4 cm. The heavy arrow indicates the point on the cathode surface where |B| is maximum; values for |B|max are cited for this point.17
Fig. 2 Radial profiles for |B|max = 456 G. The experimental data of Wendt et al. shown here is the ion current collected by probes imbedded in the cathode.5 The error bars for the simulation results are based on the counting statistics for the number of ionizations in each radial bin.
Fig. 3 Axial profiles for |B|max = 456 G. The experimental curves of Gu and Lieberman shown here represent the combined optical emission of argon and copper; the small peak near the cathode surface is an artifact of their measurement technique.6 Only the first few millimeters are suitable for comparing experiment to simulation because of the copper glow. The parameters entering the simulation are the same as those for Fig. 2; they are listed in Table I.