Pressure dependence of ionization efficiency in sputtering magnetrons
T. E. Sheridan, M. J. Goeckner, and J. Goree
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242
Using a Monte Carlo simulation, we show how electron confinement allows sputtering magnetrons to operate at lower neutral pressures than similar unmagnetized devices. We find that at high and low pressures the ionization efficiency in a magnetron becomes constant, and it varies by only 40% between these two regimes. In contrast, the efficiency varies linearly with pressure in an unmagnetized discharge.
Sputtering magnetrons are plasma devices used for thin film deposition and sputter etching. In these devices, crossed electric and magnetic fields confine electrons in complicated orbits near a cathode target.1,2 These trapped electrons ionize the neutral background gas, creating ions which are accelerated to the cathode. Upon striking the cathode, these ions sputter material and cause the emission of secondary electrons, which are accelerated back into the trap where they create more ions, thus sustaining the discharge. Collisions with neutrals eventually scatter electrons out of the trap.2 Since the principal virtue of the magnetron is its ability to operate at a low neutral pressure, it is worthwhile to study electron transport mechanisms in the magnetron and their dependence on pressure.
In this letter we use a Monte Carlo simulation2,3 of the transport of electrons emitted from the cathode to explore the ionization efficiency and its dependence on neutral pressure.4 The device we simulate is our cylindrically symmetric planar magnetron.5 The Monte Carlo code was described in detail in Ref. 2, where we simulated a 1 Pa argon discharge. This code has also been used to simulate a magnetron1 which has an adjustable magnetic field.3
We now briefly review this code. Single electrons are started at rest from the cathode and are followed as they move about within a cylindrical simulation box which extends 4 cm above the cathode and to a radius of 4 cm from the symmetry axis. Orbits are computed using prescribed, time-independent magnetic and electric fields. An electron's orbit is terminated when its total energy becomes too small to ionize an argon atom, or when it moves out of the simulation boundaries. The next electron is then started at a radius chosen in a manner consistent with radial distribution of previous ionization events, assuming that ions fall to the cathode without radial deflection.
The magnetic field B was calculated2 based on the actual device, which has a central plug magnet surrounded by a ring of bar magnets.5 The field is 245 G and purely tangential to the copper cathode surface at a radius of 1.7 cm, where the etch track is deepest. We denote this radius a. Even though we perform our simulation for only this magnetron size, our results can be applied to larger or smaller devices by using a dimensionless neutral pressure a / mfp (where the mean free path is mfp = kB T/[[sigma]]P, [[sigma]]is the cross section for momentum transfer, and T is room temperature).
We assume a one-dimensional electric field perpendicular to the cathode surface.6 The variation of this field with distance from the cathode was found from a numerical solution of the planar sheath equation,7 assuming typical experimental parameters of 4 eV for the electron temperature, 0.1 mm for the Debye length, and -400 Vdc for the cathode bias. The resulting sheath width is about 3 mm. A 1.0 V / cm linear presheath is added to this sheath solution. The same electric field is used in the simulation here regardless of the pressure or magnetic field.
Collisions with argon neutrals are accounted for by using the total cross sections to evaluate the probability of elastic, excitation, and ionizing collisions at each time step. If a collision has occurred, the differential cross sections are used to scatter the velocity direction. The electron's energy is decreased by each collision. For ionizing collisions, we use the energy loss data reported by Carman for M-shell ejection,8 which is an improvement upon the fixed energy loss assumed in our earlier2,3 simulations.
For this letter we have used the simulation to determine the average ionization efficiency [[eta]], which is defined as follows. First, we denote the number of ionizations that a single electron performs as Ni. The average of Ni over an ensemble of simulated electrons is denoted by [Ni. Here [Ni is found directly from the simulation; in general it will increase with cathode bias and pressure. The maximum possible number of ionizations, [[Nu]]i max , is defined as the number of ionizations performed by a well-confined electron in the absence of excitation and elastic collisions;9 thus, [[Nu]]i max increases with cathode bias. We can now define the ionization efficiency [[eta]] as
[[eta]] [[equivalence]] [Ni / [[Nu]]i max . (1)
Using this definition, [[eta]] is the efficiency of ionization by cathode emission at a given pressure and cathode bias. The efficiency will always lie in the range 0 < [[eta]] < 1. If [[eta]] ~ 1, electrons are efficiently confined, while if [[eta]] << 1, the confinement is ineffective and electrons escape easily from the device. In practice, [[eta]] can never reach a value of 1 because many electrons lose energy in excitation collisions. As a result, the highest possible [[eta]] is about 0.9.
We ran the simulation for a wide range of neutral pressures10 from 0.2 to 100 Pa. Under these conditions, the mean free path (for electron momentum transfer in argon) ranges from 7.5 cm to 0.015 cm, assuming a cross section11 of 1 x 10-16 cm2. Note that over this pressure range the mean free path ranges over values significantly smaller than the system size (a / mfp>> 1) to those greater than the system size (a / mfp<< 1). For good statistics we used at least 300 electrons in each ensemble, and for good energy conservation we selected a time step between 5 and 20 ps.
The principal results of this letter are shown in Fig. 1 (a), where we plot the ionization efficiency [[eta]] against neutral pressure P. The unmagnetized and magnetron discharges differ most significantly at low pressures. Without the magnetic field B, we find that [[eta]] increases nearly linearly with P until it saturates at P = 50 Pa. On the other hand, with B, the ionization efficiency is almost independent of P. The two curves merge at about 50 Pa, where a / mfp >> 1. The weak dependence of [[eta]] on P demonstrates the effectiveness of the magnetron configuration in trapping electrons. This effectiveness is the reason magnetrons can be used for sputtering at low neutral pressures.
In Fig. 1(b) we examine in greater detail the pressure dependence of [[eta]] for nonzero magnetic field. Note that the average number of ionizations per electron varies by only 40% as the neutral pressure is scanned over 3 decades from 0.2 Pa to 100 Pa. Two limiting regimes, at low and high pressures, are evident in Fig. 1(b). At low pressures (P < 1 Pa), [[eta]] is constant because the mean free path is much longer than the system size (a / mfp<< 1). At high pressures (P > 50 Pa), [[eta]] saturates since the mean free path is much shorter than the system size. In this regime, collisions are frequent and electron transport becomes diffusive, leaving no chance that an electron will escape while it still has enough energy to ionize a neutral atom.
Even though the simulation predicts that [[eta]] becomes constant at low pressures, real magnetron discharges extinguish at very low pressures. This paradox is resolved by noting that our prescribed electric field omits self-consistent effects,12 which play a particularly significant role at low pressures. The electric field in a discharge self-consistently adjusts so that electrons and ions are expelled at the same rate. At low pressures, energetic electrons require more time to perform a given number of ionizations. The ion-transit time, however, is relatively invariable in a magnetron, meaning that at a sufficiently low pressure, the electrons will be forced out of their confined orbits too quickly, and the discharge will go out.
To demonstrate electron loss at low pressures, in figure 2 we show histograms of the number of ionizations Ni performed by each electron for a magnetron at P = 0.2 Pa and P = 100 Pa. Both histograms have a peak at 15 ionizations, representing electrons that do not escape from the simulation before expending most of their energy. This peak has a finite width owing to energy lost in excitation collisions, which account for about one quarter of all inelastic collisions. The histogram for the low pressure displays a flat tail that extends down to zero. This tail reveals that some electrons escape carrying energy that could have caused additional ionizations. It does not appear in the histogram for 100 Pa, where the ionization efficiency is saturated.
In summary, we have used a Monte Carlo simulation of our magnetron device to determine the average number of ionizations per electron. We find that the ionization efficiency varies by only 40% as the neutral pressure is scanned from 0.2 to 100 Pa. This weak dependence owes to the electron confinement; even at low pressures only a few electrons are lost before consuming their energy by ionizing neutral atoms. Without the magnetic field, the ionization efficiency has a strong linear dependence on pressure.
This work was supported by the Iowa Department of Economic Development.
11 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).
22 T. E. Sheridan, M. J. Goeckner and J. Goree, "Model of energetic electron transport in magnetron discharges," J. Vac. Sci. Technol. A8, 30 (1990).
33 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. A8, 1627 (1990).
44 S. M. Rossnagel and H. R. Kaufman, "Current-voltage relations in magnetrons," J. Vac. Sci. Technol. A6, 223-229 (1988), and Ref. 3 offer some evidence that bulk electrons also contribute to ionization, but we do not consider them in this letter.
55 T. E. Sheridan, and J. Goree, "Low frequency turbulent transport in magnetron plasmas," J. Vac. Sci. Technol. A7, 1014 (1989).
66 M. J. Goeckner, J. Goree, and T. E. Sheridan, "Monte Carlo simulation of ions in a sputtering magnetron," submitted to IEEE Trans. Plasma Sci., July 1990, reports simulations using an electric field that varies with both radius and height above the cathode.
7 T. E. Sheridan and J. Goree, "Analytic expression for the electric potential in the plasma sheath," IEEE Trans. Plasma Sci. 17, 884 (1989).
78 R. J. Carman, "A simulation of electron motion in the cathode sheath region of a glow discharge in argon," J. Phys. D: Appl. Phys. 22, 55 (1989).
9 The formal definition of Ni max is different from the one we used in earlier papers1,6 because here we have improved upon the simulation by not assuming a fixed energy loss in ionizing collisions.
10 S. M. Rossnagel, "Gas density reduction effects in magnetrons," J. Vac. Sci. Technol. A6, 19 (1988), reported that a sputtering wind heats and rarefies the neutrals in a sputtering discharge; we neglect this effect.
11 Makato Hayashi , Nagoya Institute of Technology Report No. IPPJ-AM-19, Research Information Center, IPP/Nagoya University, Nagoya Japan, 1981, errata 1982.
812 T. E. Sheridan, M. J. Goeckner, and J. Goree, "Electron and ion transport in magnetron plasmas," J. Vac. Sci. Technol. A8, 1623 (1990).
FIG. 1. (a) Pressure dependence of ionization efficiency [[eta]] [[equivalence]] [Ni / Ni max. Without the magnetic field B, [[eta]] varies almost linearly with pressure P. With B, [[eta]] is nearly independent of pressure, allowing the magnetron to operate at lower neutral pressures than unmagnetized devices. (b) The data with B is re-plotted (on a linear scale with a suppressed zero and one-standard-deviation error bars), revealing that [[eta]] is constant at high and low pressure. The dimensionless pressure in the bottom scale is the ratio of the magnetron radius a to the mean free path mfp.
FIG. 2. Histogram of ionizing collisions. At 0.2 Pa, (a) the distribution is peaked at 15 ionizations, but has a tail extending down to zero, showing that some electrons escaped while they still had the energy to create ions. At 100 Pa, (b) the distribution has no tail, due to the short mean free path.