With the groundwork laid in the last two chapters, we can now simulate the motion of electrons in the presence of electric and magnetic fields. The right combination of E and B fields will trap the electrons near the target surface, creating high electron densities and consequently high sputter rates.
The electron motion is governed by the Lorentz force law:


(4 1) 
where F is the force on the electron and q is the electron charge. We know from NewtonÕs second law that force equals mass times acceleration. Using this, along with the fact that the acceleration is the second derivative of position, we can write equation (4 1) as


(4 2) 
where is the second derivative of position with time. This is a vector equation, and for us it will be more convenient to write it as three equations, one for each direction. If we donÕt remember the definition of a cross product, Mathematica can remind us:
The electron equations of motion are then


(4 3) 


(4 4) 


(4 5) 
where the first derivative of position is used to represent the velocity components. By solving these three coupled differential equations, we can determine the position of a single electron over time. The Mathematica function NDSolve is designed just for this type of problem. LetÕs try it out with a simplified B field, rather than a full magnet array. Consider two long parallel magnets, as shown in Figure 4 1.
Figure 4 1. Simplified magnet array.
We will set each magnet to be 1 cm x 1cm x 40 cm in size with a magnetization of Br = 1.4 T. Using our methodology from Chapter 2, we can write this as
Code 4 1. Definition of 2 magnet array.
With the B field functions from Chapter 2 (Error! Reference source not found.), we can plot the field. The upper surface of the target is imagined to be at z=0, indicated by the horizontal line in the figure. The voltage applied there will create our E field.
Code 4 2. StreamPlot is used to plot the B field vectors.
Figure 4 2. Field lines from two magnets.
From Equation Error! Reference source not found., the field varies linearly with distance from the target. In Mathematica we can write:
Code 4 3. Definition of linear E field.
The E field is zero in the x and y directions. In the Mathematica code for the z direction, we specify the field using the Piecewise function. This allows us to distinguish the sheath and nonsheath regions. Alternatively, an If statement can be used, but this can cause problems when using NDSolve.
We will assume the target is set to 300 V and the sheath is 1mm thick. Near the target, the electric field reaches 600,000 V/m, and then linearly drops to zero at the edge of the sheath.
Code 4 4. Plotting the electric field.
Figure 4 3. Linear electric field in sheath.
We can now solve our differential equations by calling the NDSolve function:
Code 4 5. Solving the electron equations of motion.
The first portion of the NDSolve expression lists our three equations of motion, for the x, y, and z directions. The next section has the initial conditions for the electron. In our case these are the starting position at the target surface (0, 0.007, 0) and the starting velocity (0, 0, 0). Next we list the variables we are solving for (x, y, z) and the range of times we are solving over. The last section has various solver settings. These are used to guide the solver to find a solution. In our case, boosting the accuracy and precision goals leads to a more accurate solution.
It takes a few minutes to solve this expression. Once it is done, Mathematica returns an interpolating function for each variable. We can plot these using a parametric plot to see the trajectory the electron takes over the target.
Code 4 6. Electron trajectory plotted using ParametricPlot
Figure 4 4. Electron trajectory.
The electron follows a zigzag path, drifting in the x direction. The side view of this trajectory can be seen in Figure 4 5, superimposed over the B field.
Figure 4 5. Side view of electron trajectory.
The electron leaves the target surface almost vertically at y = 0.007 m and then begins circling around a B field line until it reaches the target again. The E field then repels it and it retraces its path to where it started. In this way, the electron is trapped near the target surface. In addition to this backandforth motion in the yz plane, there is the ExB drift in x direction that takes the electron down the magnetron.
To ensure our solution is accurate, we should confirm that energy is conserved. At any given point the total energy of the electron consists of its kinetic energy and its potential energy. The latter comes about from its position in the electric field. Only when in the sheath will the electron have any potential energy.
The kinetic energy is given by


(4 6) 
The potential energy can be found by integrating the force on the electron as it moves from its current position to the sheath boundary, z = s.


(4 7) 
We can have Mathematica do this integration, using the Integrate function:
The results can be simplified with the Collect function:
The expression can be made simpler still by writing it in terms of E instead of V. This can be done by taking our expression for Ez, solving it for Vd, and substituting that expression into the potential energy expression above. An additional simplification using the Apart and Simplify functions gets us to the final form:
Thus the potential energy can be simply written as


(4 8) 
This, combined with the kinetic energy, can be used to plot the total energy, as shown in Code 4 7.
Code 4 7. Plotting the total electron energy.
In Figure 4 6, we can see that the total energy is conserved. This confirms that NDSolve is giving a reasonable result. We can also plot the potential and kinetic energy individually and see how the energy cycles back and forth between the two (Figure 4 7 and Figure 4 8). Each time the electron returns to the target surface, all of its energy is converted into potential energy, and its speed drops to zero.
Figure 4 6. Total energy of the electron is conserved
Figure 4 7. Kinetic energy of electron.
Figure 4 8. Potential energy of electron.
After an electron is emitted from the target surface, it quickly accelerates to high speeds. This can be seen in Figure 4 9. With 300 V applied to the target, the electron reaches speeds of over 8 x 10^{6} m/s in less than one nanosecond. While this is quite fast, it is still just a few percent of the speed of light. That is what allows us to ignore relativistic effects and use classical physics in solving for the electron trajectory.
Code 4 8. A ParametricPlot is used to plot electron position vs velocity.
Figure 4 9. Electron velocity after leaving target surface.
One can make some very interesting figures by plotting the phase space coordinates. For instance in Figure 4 10, the y component of velocity is plotted against the y position. In Figure 4 11, the same is done for the x and z components.
Figure 4 10. Y position versus y velocity.
Figure 4 11. Z and x positions versus z and x components of velocity.
A key aspect of magnetron sputtering is the fact that electrons follow a closed loop as they move over the target surface. This is due to their drift velocity. The first thing to point out about the drift velocity is that different electrons have different drift rates. Those in the center of the racetrack go much faster. Those further away from the center line of the racetrack drift more slowly. We can see that by launching some additional electrons.
In Figure 4 12, we launch from three different y positions, 0.007 m, 0.015 m, and 0.020 m, rerunning NDSolve for 0.1 ms for each case.
Code 4 9. ParametricPlot3D used on a list of solutions.
Figure 4 12. Launching electrons from three starting locations.
The inner trajectory is our original launch point. By moving the launch point out to 0.015 m, the trajectory is wider, and the drift velocity is much lower. Going further out to 0.020 m, the electron is only weakly bound by the magnetic field and doesnÕt seem to orbit the B field lines. This can be seen more clearly in a side view (Figure 4 13).
Figure 4 13. Side view of electron trajectories.
This electron drift down the racetrack is generally referred to as ExB drift. However, other mechanisms also contribute to the drift. In addition to ExB drift, there is gradient drift and curvature drift (Chen 1984). All three cause the electron to move in the x direction. We can estimate the magnitude of these mechanisms to get a feel for which dominates. The average drift velocity of the electron launched from y = 0.07 m is
On average, the electron is moving at 1.4 x 10^{6} m/s in the x direction. The ExB drift is given by
We can plot this for our electron and see how it varies with position.
Code 4 10. Position versus ExB drift velocity.
Figure 4 14. ExB drift velocity.
As expected, the ExB drift velocity can be quite high when the electron is in the sheath, reaching more than 6 x 10^{6} m/s. But outside of the sheath it goes to zero. The average drift velocity is calculated as follows:
Code 4 11. Calculating average ExB drift.
This is quite similar to the average drift velocity seen by our electron, which suggests ExB drift is the dominant mechanism. This is consistent with experimental measurements of Bradley, et al. (Bradley, Thompson and Gonzalvo 2001)
Even though the ExB drift drops to zero outside the sheath, our plot of trajectory (Figure 4 11) shows an apparent drift in the x direction all of the time. This suggests that the other drift mechanisms do play some role as well.
The other two sources of electron drift in a B field are gradient drift and curvature drift. The grad B drift is given by Chen (Chen 1984):
where is the electron velocity perpendicular to the field and is the Larmor radius given by
The curvature drift is given by
where is the electron velocity parallel to the the B field and r_{c} is the radius of curvature of the B field.
The Grad B drift can be calculated with this function:
Code 4 12. A function for the Grad B drift velocity.
Plotting this drift (Figure 4 15) we see that is it quite small compared to ExB drift. On average the curvature drift is only
202772
Figure 4 15. Velocity due to gradient drift.
For the curvature drift, we need to estimate the curvature of the B field. From Figure 4 2 we can see that for an electron starting at y = 0.007 m, the curvature is about 1.5 cm. A more precise calculation shows the average curvature of the B field along this trajectory to be 1.38 cm. The radius vector can then be written as
The curvature drift can then be calculated with this expression:
A plot of the curvature drift is shown in Figure 4 16.
Figure 4 16. Velocity due to curvature drift.
The peak curvature drift is about half the size of the ExB drift. More importantly, this peak drift occurs exactly when the ExB drift is zero. This can be seen by overlaying the two plots:
Figure 4 17. Curvature drift overlaid on ExB drift.
The average curvature drift is
From the average drift values we can say that for this case, the electron drift is due primarily to ExB, with small contributions from Grad B drift and curvature drift. We made a few simplifying assumptions and as a result the three mechanisms donÕt add up to the average drift velocity calculated at the start of this section. Still, these calculations help us understand the nature of the drift of electrons around the racetrack.
In practice it is difficult to maintain a constant magnetic field strength around the entire racetrack. In particular, the turnaround region typically has a weaker field. What is the effect of magnetic field gradients on the electron trajectory? Buyle et al. found that the height, width and velocity of the electron trajectory all change as the electron transitioned from a region of weak field to high field (Buyle, et al. 2004). We can do a similar calculation by modifying our magpack list slightly:
We have taken each long magnet and divided it in two, with a weak end and a strong end. Figure 4 18 shows the B field in the target plane half way between the magnets.
Figure 4 18. B field strength along length of magnets.
In order to find the electron trajectory, we run the NDSolve function like before. It takes a little longer to run because our magpack has twice the magnets in it.
Code 4 13. Solving electron equations of motion.
The trajectory is shown in Figure 4 19. In the weak field region, the electron has a higher drift velocity, so the pathline is more spread out. As the electron moves into the stronger field, its drift velocity slows and the pattern becomes tighter. This means that the electron spends less time in the weak field region. This leads to less time ionizing argon atoms and thus a lower sputter rate in the weak region.
Figure 4 19. Effect of B Field gradient on electron trajectory.
A side view of the trajectories can be seen in Figure 4 20. The particle initially follows an arc close to the surface. Much of the time the electron is in the 1 mm sheath. Once it transitions into the stronger field, the electron moves up to a higher B field line. This both broadens its arc and gets it out of the sheath for much of the time. Both of these have implications for sputtering. The broader arc should result in a wider erosion groove in the target, boosting target utilization. The higher trajectory means that more ions are formed above the sheath. As they are attracted to the target, the ions fall through the full sheath potential (300 V in this case), transmitting maximum energy to the target. When ionization occurs inside the sheath, the ions fall through only a portion of the sheath region and thus donÕt pick up the full 300 eV of energy.
Figure 4 20. The electron move up to a higher trajectory as B field strengthens.
We can easily try the reverse case, where the electron starts in a strong field and transitions to a weak field. The magpack then becomes
The trajectory plots are shown in Figure 4 21. In this case, the electron starts in a trajectory with a slow drift velocity and then transitions into a faster one. The arc also goes in the opposite direction. It starts high, and as the field weakens, the electron moves down to a lower B field line.
Figure 4 21. As the B field weakens, the drift velocity increases.
Figure 4 22. The weakening B field forces the electrons closer to the target surface.
Buyle (Buyle, et al. 2004) used these results to explain the cross corner effect—that is the observation of higher erosion rates just after the turnaround, at both ends of the target. They noted that as the electron comes out of the turnaround into a stronger field, it both slows down and drops to a lower orbit. Both effects increase the electron density there, leading to more ionization and higher erosion rates.
Using the magnetron from Section 2.2, letÕs model the electron as it makes its way around the magnetron. We define out magnet array like before:
Figure 4 23. The magnet array used for a full magnetron simulation.
The magnetic field lines in the middle portion of the array can be easily plotted:
Code 4 14. Using StreamPlot for B field vectors
Figure 4 24. Magnetic field vectors at x = 0. The horizontal line represents the target surface.
We can launch a particle from anywhere along the racetrack and then solve for its position in the usual way. LetÕs see how the motion evolves as the electron moves through the turnaround region.
Code 4 15. NDSolve is used to find the electron trajectory at the end of the magnet array.
As shown in Figure 4 25, the electron trajectory has three distinct motions. On a fine scale the electron is circling around the local B field line. On a bigger scale, it is following an arcshaped path from the target surface upward and then back down to the target. Lastly, it is following the ExB drift direction, which takes it around the racetrack.
This last motion can be seen in Figure 4 26 where the trajectory is superimposed on both the racetrack (dashed line) and the contours of the parallel B field. There are several interesting things to note in this figure. First, the electron position in the B field shifts dramatically as it reaches the end of the magnet array. At the start of its trajectory, the electron is primarily in the strongest part of the B field (indicated by lighter shading). In the turnaround region, not only is the field weaker overall, the electron drifts into an even weaker portion of the field. As we saw earlier, this weaker B field results in a faster ExB drift, leading to less time spent in the turnaround region and less ionization and sputtering. So just from this plot we would expect the target erosion to be less at the turnaround than in the straight section of the magnet array.
Figure 4 25. Trajectory of an electron starting at (0.10, 0.007).
Another interesting aspect of the trajectory is the relative positions of the electron trajectory and the dashed Bz = 0 line. The general rule of thumb is that the racetrack will be centered around this line. This is true when the field lines follow nice, symmetric arcs, as in Figure 4 2. However, the field lines for this magnet array are not symmetric, as shown in Figure 4 24. At the target surface, the Bz = 0 point is closer to the outer magnets, rather than centered between the inner and outer magnets. A slice through the turnaround region (not plotted) would show the opposite trend. From this we can conclude that the racetrack is only approximately located by the Bz = 0 line.
The electron motion around the field lines also changes as the electron moves through the turnaround. The spirals become larger and more closely spaced. This indicates that some of the electronÕs kinetic energy has been diverted from ExB drift into spiral motion. We can check this by finding the average velocity in the x direction at the start and the end of the trajectory:
The electron has slowed by nearly 40 percent in the x direction. This would suggest that the erosion rate could be higher on the outboard side of the turnaround compared to the inboard side. This cross corner effect, which is experimentally observed (Fan, Zhou and Gracio 2003), is generally what limits target lifetimes.
Figure 4 26. Electron trajectory superimposed on racetrack (dashed line) and contours of the B field parallel to the target.
We can look at a side view of the electron trajectory to get further insights into the effect of the turnaround. Figure 4 27 shows three planes along the electron path where we examine the trajectory. These are the initial profile, the profile halfway through the turnaround, and the final profile. The initial and final profiles are compared in Figure 4 28.
Figure 4 27. The rectangles indicate where we examine the trajectory in Figure 4 28 and Figure 4 29.
The initial profile shows a small amount of motion around the B field lines as noted above. The dashed line shows the profile at the end of the simulation. The orbits are much larger and the width has increased slightly. This would suggest that the erosion groove will be somewhat wider on the outbound side of the turnaround.
Figure 4 28. Electron path at the start and end of its trajectory.
A comparison between the initial profile and the midpoint profile is shown in Figure 4 29. Here the width of the arc has shrunk compared to the incoming profile, suggesting a narrower erosion groove in the turnaround.
Figure 4 29. Electron path at the start and midpoint in its trajectory.
From the above results, we can see that even these simple models of electron motion can provide useful insights into the behavior of the magnetron. They allow us to better understand the nature of electron drift as well as the effect of changing field strength on electron motion. However, to go further, we need to include the effect of electronargon collisions. That is the subject to which we now turn.