UMN K100 Run67 - Physical Impact Ionization Model and Rates

Matt Fritts, 2 July 2018 (updated 16 November 2018)

Note: some rates quoted in this note are in error.

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This note is an attempt at a model to explain the time-dependent behavior seen in Run 67. Specifically the observations we attemp to explain are

This note does not consider other observed behavior, such as the dampening of the effect in the ring channels (this note considers the center channel only), and the exponential falloff in charge collection that kicks in after several minutes of bias.

We assume that the dominant interactions between charge carriers and impurities are those identified in Note 3: electrons trapping on neutral donors, and holes impacting with overcharged donors to release an electron and neutralize the donor.

  1. \( e^- + D^0 \rightarrow D^- \)
  2. \( h^+ + D^- \rightarrow h^+ + e^- + D^0 \)

As pointed out in Note 3, this explains why only hole events produce the observed impact ionization. The qualitative explanation for the growth of the charge collection efficiency is that drifting electrons produce a growing population of charged donors which serve as impact ionization targets for holes. Now we'll try to see if this model can match quantitative observations.

We add the process of ionization from ionizing radiation. Rather than having one ionization producing an electron/hole pair, let's separate it into two processes, one producing holes and one producing electrons. The reason is because we are interested in carriers that drift through the detector bulk, but with the Am source there is an asymmetry in the number of carriers produced of each type that drift through the detector bulk.

  1. \( I_+ \rightarrow h^+ \)
  2. \( I_- \rightarrow e^- \)

Now write the rates of electron production, hole production, and the production of charged donor states in terms of the rates of the processes above. (We assume the population of neutral donor states is constant).

  1. \( R_h = R_+ \)
  2. \( R_e = R_- - R_{eD} + R_{hD} \)
  3. \( \dot{N}_{D-} = R_{eD} - R_{hD} \)

The rates of interactions with donor states can be written as follows:

  1. \( R_{eD} = kR_e \)
  2. \( R_{hD} = pN_{D-}R_h \)

where \(k\) is the probability that an electron is captured by a neutral donor and \(p\) is the probability per target that a hole impact ionizes a negatively charged donor.

Combining these yields

  1. \( \dot{N}_{D-} = \frac{k}{1+k}R_- - \frac{k}{1+k}R_+N_{D-} \)

If we assume that the population of negative donors is initially zero, we find

  1. \( N_{D-} = \frac{k}{p}\frac{R_-}{R_+}(1 - e^{-\frac{k}{1+k}R_+t}) \)
  2. \( \dot{N}_{D-} = \frac{k}{1+k}R_-e^{-\frac{k}{1+k}R_+t} \)

We can pause here to compare the equations above to some observed behavior. We expect the enhanced charge collection efficiency to be proportional to the population of overcharged donors. Equation 11 says this should rise and exponentially approach some saturation level. We observe a linear rise before (apparently) other effects quench the effect. It could be we are only seeing the early approximately-linear portion.

Equation 12 says that the growth rate of the charge collection efficiency should be proportional to the rate at which electrons are produced by ionizing events. Form Note 1 we can estimate that at any bias the CCE growth rate is about 2.4 times greater when the Ba source is present than when it is not. From energy spectra we estimate that the overall ionization rate with the Ba source present is about 1.14 times greater than when it is not. Initially it seems surprising that such a small increase in ionization rate has such a large effect in CCE growth rate. However in this model the position of ionizing events in the detector is relevant - we are not interested in the production of charge carriers that are immediately collected, as is the case for electrons produced by 60keV events from the Am source, which are near-suface events. We are interested in carriers that drift significant distances and thus can contribute to the trapping or impact processes in the bulk of the detector.

Suppose that the ionization rate from the four Am sources is 9 times the background rate, and that the rate from the Ba source is 1.4 times the background rate. (These are plausible values.) Then the total ionization rate would be 1.14 times greater with the Ba source present, as calculated from the spectra. But the Am sources do not contribute to \(R_-\), because this rate does not include immediately-collected electrons. So \(R_-\) would be 2.4 times greater with the Ba source present. This part of the model works.

Continuing the model, we can write an expression for the gain produced by impact ionization:

  1. \( G = 1+pN_{D-} \)
  2. \( \dot{G} = p\dot{N}_{D-} \)

Recall \(p\) is the probability per target that a hole will produce impact ionization. For holes from Am events in positive bias, they drift the entire detector thickness. So \(p\) would be the interaction cross section divided by the cross-sectional area of the detector (\(A\)=78.5 cm2):

  1. \( p = \frac{\sigma_{hD}}{A} \)
  2. \( \dot{G} = \frac{\sigma_{hD}}{A}\dot{N}_{D-} \)

Taking Nick's empirical formula from Note 1 for the Am-only case, we can convert from keV/min to gain/s (for a 60 keV base energy) yielding

  1. \( \dot{G} = ge^{V/V_c} \)

with \(g=3.06\times10^{-5}\) Hz, V the detector bias, and \(V_c = 1.64\) V.

If we assume \(k\) (the probability that an electron captures on a neutral donor) to be its maximum value of 1, and consider growth rate at time 0, we have from Equation 12

  1. \( \dot{N}_{D-} = \frac{1}{2}R_- \)

so combining Equations 16, 17, and 18,

  1. \( \sigma_{hD} = \frac{2Ag}{R_-}e^{V/V_c} \)

In the Am-only case the total event rate is around 30 Hz, and the mean energy deposition is around 300 keV. So if one ionization is produced for every 3 eV deposited, the total ionization rate is about \(3\times10^6\) Hz. \(R_-\) will be less than this; let's just estimate \(1\times10^6\) Hz. If we plug in these constants over the range in which we observe impact ionization we get the following cross sections:

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These cross section values are probably too high to be plausible. The model and the observations suggest that the impact ionization cross section increases with field, in contrast to the plots from Phipps shown in Note 3. So this model doesn't seem quite right, though it does a better job than previous ones of matching observed behavior.

The model in this note is approximate in that it does not fully take into account the locations at which charge carriers are produced. But a more detailed model with the same features would probably produce the same order-of-magnitude result.