# Heterogeneous detrapping process

Earlier, we have explored how power-law distributed detrapping times lead to 1/f noise. Moreover, we have also established that power-law distribution can be obtained from the superposition of exponential distributions (see this post). Now, what implications does the synthesis of these findings hold?

## Hooge's parameter

As discussed in [1], such construction of the detrapping process enables the derivation of a readily interpretable expression for the Hooge's parameter:

$$\alpha_H = \frac{\langle \tau_{\text{min}} \rangle}{\langle \theta \rangle}.$$

The provided expression suggests that Hooge's parameter depends on both the expected detrapping time from the shallowest trapping center, $$\langle \tau_\text{min} \rangle$$ and the material's purity, represented by the average free-flight time, $$\langle \theta \rangle$$. Hooge's parameter is important in characterizing the noise behavior in semiconductor devices, serving as a key metric in evaluating their reliability and performance.

## Spurious low-frequency cutoff

Moreover, such formulation predicts the presence of a low-frequency cutoff in finite experiments. This cutoff can be mitigated by averaging data over multiple experiments or observing a large number of charge carriers. The phenomenon arises from the fact that the low-frequency cutoff is related to the smallest actually observed detrapping rate during the experiment. Even if the model parameter $$\gamma_\text{min}$$ is set to zero, in practical scenarios, the effective $$\gamma_\text{min}$$ will never reach zero. Furthermore, the cutoff frequency will always be higher than the lowest observable frequency.

Our calculation predict that the cutoff frequency [1]:

$$f_c \approx \gamma_\text{min}^{(\text{eff})} \approx \frac{R + \gamma_\text{max} \langle\theta\rangle}{R T} .$$

In the above $$R$$ stands for the number of single charge carrier experiments. It can be also replaced by $$N$$ the number of charge carriers within a single experiment. Given that $$N \gg \gamma_\text{max} \langle\theta\rangle$$, no low-frequency cutoff will be noticeable, as then $$f_c < 2/T$$. Although, $$f_c > 1/T$$ will be true for any duration $$T$$.

## Interactive app

The interactive app below allows you to explore heterogeneous detrapping process. It plot for distinct plots: (top left) distribution of detrapping rates $$\gamma$$, (top right) temporal dependence of smallest $$\gamma$$ observed during the current run, (bottom left) distributions of detrapping and trapping times, and (bottom right) power spectral density of the signal. Observe that low-frequency cutoff is related to the smallest $$\gamma$$ observed (i.e., compare rightmost plots).

Observe that, while the theoretical curve predicts (black curve in top right figure) a continuous gradual decrease of the smallest $$\gamma$$ observed, but the experimental smallest $$\gamma$$ observed decreases in steps. This is because theoretical curve predicts expected value of smallest $$\gamma$$ observed.

Note: The app was includes an additional parameter $$N_{success}$$. It will be discussed in a forthcoming post (the link will be added after that post is published). When exploring the app in the context of this post simply keep the default $$N_{success} = 1$$.

## References

• A. Kononovicius, B. Kaulakys. 1/f noise in electrical conductors arising from the heterogeneous detrapping process of individual charge carriers. arXiv:2306.07009 [math.PR].