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We derived exact analytical expressions for the variance, the third central moment, and the skewness of the multiplication gain distribution in uniform avalanche structures. The model assumes Poissonianity and locality of the ionization process and is valid for arbitrary values of the electron and hole ionization coefficients,

## 1 Introduction

The stochastic nature of the impact ionization mechanism due to energetic charge carriers in avalanche semiconductor devices [1] causes the overall multiplication gain to be distributed according to a non-trivial probability density function, with important implications on the behavior of the noise. The noise mechanisms and their impacts on the overall device properties have been the subject of a remarkable series of studies since the 1960s (e.g., [2] among the firsts). This was partly due to the versatility of this class of devices, making them suitable for a vast range of applications, and partly due to the complexity of the description of the avalanche multiplication phenomenon *per se*.

For instance, [3] McIntyre derived an expression for the noise spectral density due to the self-multiplication of the leakage current and possibly of the photo-generated and thermal-generated currents in uniform (i.e., one-dimensional) avalanche diodes as a function of the electron and hole impact ionization coefficients

where

The knowledge of the variance of the multiplication gain can be of particular interest in spectral resolution applications [12] as it directly contributes to the uncertainty in energy measurements pertaining to single pulses. It has to be noted, however, that the statistical moments of

To overcome this limitation, in this work, we present an analytical derivation of the third central moment and the skewness of the multiplication gain distribution for arbitrary

Apart from a purely theoretical interest, knowledge of the skewness of the multiplication gain can contribute to better modeling of avalanche structures and better understanding and interpretation of their behavior [12, 19]. For example, it is worthwhile to mention the increasing interest in avalanche devices in fields beyond optical photon science, such as the case of low-gain avalanche diodes (LGADs) with energy-resolving capabilities in high-energy physics with tracking and timing applications [20] and soft X-ray synchrotron applications [21]. In this context, we provide a simple model for the evaluation of the impact of the skewness of the multiplication gain on spectral measurements of ionizing radiation.

## 2 Materials and methods

The investigated avalanche structure consists of an

### 2.1 Mean multiplication gain

To facilitate reading, we provide a brief summary of the derivation of the mean multiplication gain as a function of

After differentiating, we obtain

whose solutions are

By substituting Equation 7 into Equation 4, for

and analogously, substituting Equation 6 into Equation 4 for

Thus,

Two particular cases are given as follows:

*Case*

If the electron and hole ionization coefficients are equal,

*Case*

If the electron and hole ionization coefficients are related by a fixed ratio

### 2.2 Variance of the multiplication gain

Let us indicate the variance of the multiplication gain by

The total variance

After differentiating, we obtain

which is a first-order ordinary differential equation of the first order with variable coefficients in the form:

where

The general solution is

where the function

The initial value

The components of Equations 16, 17 can be elaborated as follows: for instance, from Equation 6, it holds that

which obviously implies that

Then, using Equation 4, for

which is needed to solve the following equality:

Furthermore, using Equations 5, 18, we can write that

The initial value

which, using Equation 20, can be simplified into

By substituting Equation 26 into Equation 16 and simplifying, we finally obtain

Now, this formulation is consistent with the excess noise factor computed by [3] in Equation 13 (minus the term

*Case*

Like the mean

*Case* ^{1}

The integral

Therefore,

Finally, using Equation 12, we can express the variance resulting from the injection of holes only

### 2.3 Skewness of the multiplication gain

The skewness (or third standardized moment) of the multiplication gain is defined as

where ^{2} The composition rule for the third central moment of a random sum is reported in Appendix 3. In particular, for an electron traversing a displacement element

The total third central moment

which, after differentiation, yields

Like in the case of Equation 14, the general solution of Equation 36 is given in the form

where the function

The initial value

The components of Equations 37, 38 can be elaborated further. For instance, we observe that

where in the last passage, we used Equation 14. Then, using Equations 18, 21, we notice that the denominator of Equation 38 is reduced to

By substituting Equation 43 into Equation 37 and simplifying, we finally obtain

*Case*

Like the mean

*Case*

The term

which, by substituting into Equation 44, gives

Finally, using Equation 12, we can express the third central moment resulting from the injection of holes only

Similarly, for electrons only

### 2.4 Impact on spectral measurements

When the initial number of electron–hole pairs starting the avalanche is also a statistical quantity, like upon the interaction of the semiconductor device with ionizing radiation such as an X-ray photon or a charged particle, the total number of generated pairs after multiplication follows the composition rules of a random sum. Let

According to Equation B1 in Appendix 1, the mean value of

where in the last passage, we used the common relationship that the mean number of initial electron–hole pairs equals the ratio between the energy deposited in the semiconductor by the impinging particle

The variance of

where in the last passage, we used the common relationship that the variance of the initial number of electron–hole pairs equals its mean value scaled by an approximately constant material-specific factor

The third central moment of

The determination of the value of

where

and the parameter

which are nominally accurate in the asymptotic regime when

which allow for the computation of ^{3}

## 3 Results

Figure 1 shows the behavior of the third central moment

**Figure 1**. Third central moment of the multiplication gain distribution as a function of the mean value for several values of the ratio

Figure 2 shows the skewness

**Figure 2**. Skewness of the multiplication gain distribution, computed using Equation 34, as a function of the mean value and for several values of the ratio

Figure 3 shows, as an example, the skewness

**Figure 3**. Skewness of the distribution of the total signal

## 4 Conclusion

We derived exact analytical expressions for the variance, the third central moment, and the skewness of the multiplication gain in uniform avalanche structures as a function of the starting position of the electron–hole pair generating the avalanche. The expressions were obtained by solving integral equations based on the property of additivity of the central statistical moments of the sum of random variables and moment composition rules of the random sum. The assumptions include Poissonianity and locality of the ionization process. The model is valid for arbitrary values of electron and hole ionization coefficients

Expressions are then provided for the particular case where the ionization coefficients are related by a constant ratio

The impact of the first three central moments of the multiplication gain distribution on spectral measurements of ionizing radiation was also evaluated through the use of the composition rules for a random sum. In this framework, we adopted the COM–Poisson distribution, a generalization of the Poisson distribution which takes into account the under-dispersion effect parameterized by the Fano factor as a description of the distribution of the initial number of photo-generated electron–hole pairs.

## Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author contributions

PZ: writing–original draft and writing–review and editing.

## Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

## Conflict of interest

Author PZ was employed by DECTRIS Ltd.

## Publisher’s note

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## Footnotes

^{1}Please refer to [16] for a critical review of this approximation to the purpose of the excess noise factor in the noise spectral density of current fluctuations.

^{2}The property of additivity of the central moments of the sum independent stochastic variable holds up to the third degree, so the method presented in the text cannot be extended to higher moments.

^{3}For example, in silicon, assuming

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## Appendix

Let

### 1 Mean value of a random sum

The mean value of the random sum

### 2 Variance of a random sum

The variance of a random sum

### 3 Third central moment of a random sum

The third central moment of a random sum

Proof

The third central moment

which, in our case, means

By applying the theorem of total moment on a random sum, we can write

By inverting Equation B6, we know that for any

and from basic statistics, we also know that

so that Equation B8 becomes

and therefore, Equation B7 can be written as

By substituting Equation B14 into Equation B6 and obtaining the mean value of a random sum from Equation B1 and the variance of a random sum from Equation B2, we get

Keywords: avalanche diode, LGAD, multiplication gain variance, multiplication gain skewness, energy resolution

Citation: Zambon P (2024) Variance and skewness of the multiplication gain distribution in uniform avalanche diodes. *Front. Phys.* 12:1452279. doi: 10.3389/fphy.2024.1452279

Received: 20 June 2024; Accepted: 05 August 2024;

Published: 21 August 2024.

Edited by:

Lodovico Ratti, University of Pavia, ItalyReviewed by:

Philipp Windischhofer, The University of Chicago, United StatesEdoardo Milotti, University of Trieste, Italy

Copyright © 2024 Zambon. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: P. Zambon, pietro.zambon@dectris.com