UMN K100 Run67 - Impact Ionization Model Notes

Matt Fritts, 29 April 2018

Observation: at a given bias voltage, the rate at which the charge collection efficiency (CCE) grows just after biasing is strongly dependent on the overall event rate, i.e. on the ionization rate, \( R_{ion} \) (the rate at which charge carriers are produced by ionizing events). Assume that the CCE growth rate is proportional to the impact ionization target population growth rate, \( \dot{N}_{iit} \). An increase in ionization rate by a factor of about 1.14 (with Ba versus without) results in an increase in CCE growth rate by a factor of about 2.4 (averaged over 6V and 8V data, see Peak Parameters note).

Suppose that the targets are generated at a rate proportional to the rate at which charge carriers are created, \( R_{ccc} \). If \( R_{ccc} = R_{ion} \) then it would not explain why a 1.14 increase in \( R_{ion} \) produces a much greater increase in \( \dot{N}_{iit} \). But impact ionization amplifies the charge carrier production rate beyond the ionization rate. Suppose

1. \( \dot{N}_{iit} = kR_{ccc} \)
2. \( R_{ccc} = R_{ion}(1+pN_{iit}) \)
3. \( N_{iit} = 0 \) at \( t = 0 \)

Then

4. \( N_{iit} = \frac{1}{p} (e^{pkR_{ion}t} – 1) \)
5. \( \dot{N}_{iit} = kR_{ion}e^{pkR_{ion}t} \)

In the above, \(k\) is essentially a probability that a drifting charge carrier will produce an impact ionization target. \(p\) is a probability that a charge carrier will impact with a target and produce another charge carrier.

Let’s now use the observation that a 1.14 increase in ionization rate produces a 2.4 increase in the initial growth rate of targets. Thus

6. \(2.4 = 1.14e^{pk\Delta R_{ion}t_{meas}} \)

Here \( \Delta R_{ion} \) is the difference between the high and low ionization rates, and \( t_{meas} \) is the approximate amount of time over which the CCE growth rate was measured. The lower ionization rate is on the order of 30 Hz times 300 keV divided by 3 eV, or 3e6 Hz. \( \Delta R_{ion} \) is about 0.14 times this, 4e5 Hz. \( t_{meas} \) is around 5 minutes. So \( \Delta R_{ion}t_{meas} \) is around 1e8. This gives

7. \( pk \approx \frac{0.8}{\Delta R_{ion}t_{meas}} \approx 10^{-8} \)
8. \( \sqrt{pk} = 10^{-4} \)

suppose \( p=k \), and concentration \(N\) of impurities that can be activated into impact ionization targets.

9. \( p = \sigma N L \)

given the Van der Waals radius of Ge let’s suppose \( \sigma < 10^{-15} cm^2 \)

10. \( N = \frac{p}{\sigma L} > 10^{-4}/(10^{-15} cm^2)/(3 cm) = 3\times 10^{10} cm^{-3} \)

This is a plausible order-of-magnitude for an impurity concentration.

Some model difficulties:

• When a carrier produces an impact ionization target, is it trapped? If so wouldn't this effectively quench the rate of charge carrier creation? Maybe this isn't a difficulty - maybe it's just an adjustment on \(k\)
• When a carrier impacts a target and an impact ionization occurs, does this "deactivate" the target? If so then we need to add a negative term to the impact target growth rate.

plans for extending this model

• check whether the exponential growth in this model can fit the roughly linear CCE growth rate seen in data
• it's observed that growth rate depends (exponentially) on bias. This suggests \(p\) or \(k\) or both increase with bias. At low enough bias, \(pk\) is small enough that no impact ionization is seen.
• also need to explain why CCE turns around after a few minutes then falls again. My current thinking is that as space charge develops over time the effective bias is reduced and therefore \(pk\) falls and impact ionization happens less and less. (If activated targets don't deactivate when impact ionization occurs, then it must be \(p\) that falls as effective bias falls.)
• since the CCE growth only occurs at positive bias, could be that only holes are producing the impact ionization. But in this model the targets could be activated by either carrier, or both of them.
• could this model work with the notion that a target is produced by a hole getting trapped on a particular impurity type, and that impact ionization occurs when a hole hits the trapped hole, releasing it? If so then it would have to be the case that the impact ionization deactivates the target, which might break the model described above.
• why does the CCE growth decrease with radius? Generally expected that impurity concentration increases with bias. Could be that the decrease in effective bias after a space charge develops is stronger at higher radius.
• do we expect a localized effect in target density for the column below a source position? The rate of Am events is small compared to the overall rate, but it may be a large component of the carriers drifting in that column (particularly for holes which move straight in the z-direction).