Multi-Dimensional Charge Diffusion–Collection Model in Semiconductor Devices Subjected to Single Ionizing Particles

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We propose a comprehensive analytical tool that can be integrated into more complex simulation codes for simulating diffusion-dominated charge collection processes induced by single events. The model links diffusion, recombination, and charge collection by an idealized sensitive node, providing a complete analytical expression for the collected charge as a function of time. Its general formalism in 1D, 2D, and 3D makes it useful for analyzing elementary charge-transport mechanisms in simplified microelectronic structures. A minimal Python implementation is also proposed to facilitate integration into simulation workflows. The model may be used as a first-order analytical approximation or scaling tool for single-event sensitivity studies.

Abstract

Analytical modeling of charge collection is essential for predicting single-event effects in integrated circuits subjected to radiation. This work proposes a unified model of charge collection by diffusion in semiconductors in one, two, and three dimensions, accounting for recombination effects. We derive exact expressions for the carrier density, diffusion current, and collected charge using the solution of the diffusion equation for a point source. We formulate the collected charge using Bessel functions, which allow for a more general and fully analytical description of the problem. The model emphasizes the role of geometry by explicitly accounting for the dimensionality of the problem. It also establishes that, for any dimension, the collected current peaks before the carrier density does. We also propose analytical expressions for the collection efficiency and the recombination factor, with simplified forms in the absence of recombination. A minimal Python implementation is provided to facilitate the practical application of the model. Finally, we outline how to use the proposed model to perform realistic simulations of single events and relate the results to the soft error rate of a given device.

1. Introduction

The passage of an ionizing particle through a semiconductor device deposits energy along its track and generates electron–hole pairs [1,2]. The resulting excess carriers, which spread through the semiconductor by diffusion, may recombine during transport and can be collected at sensitive nodes of the device, in particular under the influence of electric fields associated with biased electrodes or reverse-biased junctions [3]. The spatial and temporal evolution of this charge cloud and the carrier collection at sensitive nodes determine the electrical transient induced in the device. Accurate modeling of these coupled processes is therefore essential to predict the response of CMOS devices to ionizing radiation [4,5,6] and to estimate the associated soft error rate (SER), which is the probability that a single particle induces a transient malfunction without causing permanent damage [2,7]. In modern CMOS technologies, as device dimensions and operating voltages decrease, circuits become more sensitive to radiation-induced transient effects [8,9]. Thus, predictive physics-based models are required for reliability assessment and design optimization. This issue is relevant not only to conventional CMOS technologies but also to emerging device platforms intended for extreme-environment electronics. These platforms include irradiation-tolerant oxide semiconductors and oxide-based neuromorphic systems that have recently been investigated under proton irradiation [10,11]. These studies demonstrate the increasing interest in radiation-resistant materials and architectures and further encourage the development of generic charge diffusion–collection models that can capture the primary transport and recombination mechanisms across various semiconductor platforms.

Charge transport and collection in semiconductor devices can be investigated using advanced numerical tools, such as technology computer-aided design (TCAD) simulators [12,13,14] or multiphysics finite-element software [15]. These approaches are essential for realistic device-level simulations because they account for detailed geometries, doping profiles, electric fields, boundary conditions, and coupled physical mechanisms [5,6,16]. However, these approaches are often computationally expensive and can obscure the explicit dependence of charge collection on geometry, transport parameters, and recombination. In contrast, analytical approaches provide direct physical insight and are useful for compact SER modeling, rapid parametric studies, and validating more complex computational models.

Several analytical and semi-analytical descriptions of charge transport and collection have been proposed since the pioneering work of Kirkpatrick, Messenger, Edmonds, and others [17,18,19,20,21,22], followed by numerous developments dedicated to radiation effects in semiconductor devices [23,24,25,26,27,28,29,30,31,32,33,34,35]. However, a compact unified formulation that describes diffusion-driven charge collection in one, two, and three dimensions, while consistently including recombination and geometry-dependent collection factors, is still lacking.

The objective of this work is therefore to develop a general analytical framework for diffusion-driven charge transport and collection from a point-like initial charge distribution in a semiconductor medium. The model yields closed-form or semi-analytical expressions of the carrier density, diffusion current, and collected charge as functions of time, distance, dimensionality, and carrier lifetime. Geometry is incorporated through explicit dimension-dependent factors, while recombination is introduced through a finite lifetime. The collected charge is expressed in terms of special functions, which allows both transient and asymptotic regimes to be described within the same framework. This model is not intended to replace numerical device simulations, but rather to provide a complementary analytical framework. By reducing the problem to idealized diffusion–recombination–collection configurations, the proposed model gives explicit expressions of the collected charge as a function of time and highlights the role of the main physical parameters.

The main novelty of the proposed model lies in its unified treatment of dimensionality, recombination, and collection geometry. It provides a compact description that can be directly evaluated numerically and implemented in reliability codes, while preserving the physical dependencies needed to interpret charge-collection efficiency. The framework can therefore serve as a building block for single-event simulations and for SER estimation in semiconductor devices exposed to ionizing radiation.

The paper is organized as follows. In Section 2, we introduce the physical model and the governing diffusion equation. Section 3 derives the analytical expressions for carrier density and diffusion current in arbitrary dimensions. Section 4 provides an analysis of the characteristic times associated with charge and current maxima. In Section 5, we review the time-dependent collected charge and its formulation using special functions. Section 6 focuses on the integral collected charge and asymptotic behaviors. Section 7 provides a comprehensive overview of the collection efficiency and the impact of recombination. Section 8 offers a summary of the model and outlines its practical implementation. Finally, Section 9 discusses and illustrates how to use the proposed model to perform realistic single-event simulations and link them to the soft error rate of a given device.

2. Model Foundations

We first consider a point charge $Q=q{n}_0$ generated at $t=0$ at point M in a bulk semiconductor, and focus on the collection of this charge at point P ($\mathbf{r}$) by a collector [35]. Figure 1 shows a schematic representation of this problem in 1D, 2D, and 3D. The distance between M and P is denoted by x in 1D and $r=\parallel \mathbf{r}\parallel$ in 2D and 3D. The collector is reduced to a point in 1D; it corresponds to a small segment (length $\ell \ll 2\pi r$) in 2D and to a small surface ($A_S\ll 4\pi r^2$) in 3D. An excess carrier generated at the point source and transported at the level of the collector is considered to be collected upon its first passage through the collector.

In the pure diffusion regime and in the presence of carrier recombination, the transport of the excess carrier density $n_e$ (expressed in the physical unit of $m^{-N}$) in the semiconductor domain is governed by the following diffusion law [17,18,19]:

$${\displaystyle \frac{\partial n_e(r,t)}{\partial t}}+{\displaystyle \frac{n_e(r,t)}{\tau }}=D{\nabla }^2{n}_e(r,t)$$

(1)

where $\tau$ is the carrier lifetime and D is the diffusion coefficient.

In the following, we examine the exact solution of Equation (1) in terms of excess carrier density, diffusion current, and collected charge (at the level of the collector contact) in the 1D, 2D, and 3D cases and in the presence or absence of carrier recombination. When possible, equations are first established for the general case of N dimensions (N = 1, 2, or 3).

3. Carrier Density and Diffusion Current

The excess carrier density at time t and distance r, originating from the initial charge $n_e(r,0)=n_0{\delta }^N\left(r\right)$, is the solution of Equation (1) and is given by [18]:

$$n_e(N,r,t)={\displaystyle \frac{n_0}{{\left(4\pi Dt\right)}^{\frac{N}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$$

(2)

where N is the dimension of the problem (N = 1, 2, or 3) and ${\delta }^N\left(r\right)$ is the N-dimensional Dirac delta function.

For electrons, the diffusion current density with the conventional direction of the electrical current is

$${\mathbf{J}}_{\mathrm{diff}}(N,r,t)=qD\nabla n_e(N,r,t)$$

(3)

Introducing Equation (2) into Equation (3) gives

$${\mathbf{J}}_{\mathrm{diff}}(N,r,t)=-{\displaystyle \frac{qr}{2t}}{n}_e(N,r,t){\mathbf{e}}_r=-{\displaystyle \frac{q{n}_{0}r}{2t{\left(4\pi Dt\right)}^{N/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right){\mathbf{e}}_r$$

(4)

where ${\mathbf{e}}_r={\displaystyle \frac{\mathbf{r}}{\parallel \mathbf{r}\parallel }}$ is the unit radial vector. The sign − indicates that, in the conventional current direction, the current flows toward the origin, even though the electrons diffuse outward on average.

To find the total current flowing through the collector, we integrate the radial component of ${\mathbf{J}}_{\mathrm{diff}}$ over the whole collector surface inside the semiconductor, as explained and discussed in [35] for realistic collector structures integrated in front-end microelectronics. This effective collection surface in N dimensions is denoted $A_N$ and is equal to $A_1=1$, $A_2=\ell$ and $A_3=A_S$. Since the current density is constant on this surface,

$$I_{\mathrm{diff}}(N,r,t)=〈{\mathbf{J}}_{\mathrm{diff}}·{\mathbf{e}}_r〉A_N=J_{\mathrm{diff},r}(N,r,t)A_N$$

(5)

From Equation (4), $J_{\mathrm{diff}}(r,t)=-{\displaystyle \frac{qr}{2t}}{n}_e(r,t)$, we obtain

$$I_{\mathrm{diff}}(N,r,t)=-{\displaystyle \frac{qr}{2t}}{n}_e(r,t)A_N$$

(6)

and then obtain the following:

$$|I_{\mathrm{diff}}(N,r,t)|={\displaystyle \frac{q{n}_0{A}_{N}r}{2t{\left(4\pi Dt\right)}^{\frac{N}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$$

(7)

Introducing the surface area of a hypersphere of radius r in N dimensions:

$$S_N\left(r\right)={\displaystyle \frac{2{\pi }^{N/2}}{\mathrm{\Gamma }(N/2)}}{r}^{N-1}$$

(8)

where $\mathrm{\Gamma }\left(x\right)$ is the ordinary gamma function [36], we can rewrite Equation (7) as follows:

$$|I_{\mathrm{diff}}(N,r,t)|=q{n}_0\times {\displaystyle \frac{A_N}{S_N\left(r\right)}}\times {\displaystyle \frac{r{S}_N\left(r\right)}{2t{\left(4\pi Dt\right)}^{N/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$$

(9)

In Equation (9), the first term corresponds to the initially generated point charge at the origin at $t=0$, the second term ${\mathrm{\Omega }}_N=A_N/S_N\left(r\right)$ is a pure geometrical term discussed below, and the third term is a time- and space-dependent function. ${\mathrm{\Omega }}_N$ corresponds to the collector’s solid angle, in N dimensions, viewed from the source, divided by the solid angle for the entire space, also in N dimensions. For $N=1$, ${\mathrm{\Omega }}_1=1/2$, for $N=2$, ${\mathrm{\Omega }}_2=\ell /\left(2\pi r\right)$ and for $N=3$, ${\mathrm{\Omega }}_3=A_S/\left(4\pi r^2\right)$.

4. Maximum Times for Carrier Density and Current

The times at which $n_e(r,t)$ and $I_{diff}(r,t)$ reach a maximum can be easily calculated from the time derivative of Equations (2) and (9), respectively. After straightforward calculations, we obtain the following:

$${\left.{\displaystyle \frac{\partial n_e(N,r,t)}{\partial t}}\right|}_{t=t_{max}^{n,N}}=0\iff t_{max}^{n,N}={\displaystyle \frac{\tau }{4}}\left(\sqrt{N^2+{\displaystyle \frac{4r^2}{D\tau }}}-N\right)$$

(10)

$${\left.{\displaystyle \frac{\partial I_{diff}(N,r,t)}{\partial t}}\right|}_{t=t_{max}^{i,N}}=0\iff t_{max}^{i,N}={\displaystyle \frac{\tau }{2}}\left[\sqrt{{\left({\displaystyle \frac{N}{2}}+1\right)}^2+{\displaystyle \frac{r^2}{D\tau }}}-\left({\displaystyle \frac{N}{2}}+1\right)\right]$$

(11)

From Equations (10) and (11), for a given physical dimension N and carrier lifetime $\tau$, we can establish that $t_{max}^{i,N}<t_{max}^{n,N}$: the current systematically reaches its maximum slightly before the carrier density. This result is consistent with what was established in 3D without recombination [35]. This is because the current depends on the spatial derivative of the density, introducing an additional factor of 1/t, and altering the temporal balance between the power-law decay and the exponential decay. In other words, the current is proportional to the gradient of carrier density, which peaks before the density itself peaks.

In the absence of recombination, i.e., for large values of $\tau$, Expressions (10) and (11) reduce to much simpler forms. Indeed, using the Taylor expansions, $\sqrt{N^2+ϵ}\approx N+ϵ/2N$ for small $ϵ=4r^2/D\tau$ and $\sqrt{{(N/2+1)}^2+ϵ}\approx (N/2+1)+ϵ/\left(2(N/2+1)\right)$ for small $ϵ=r^2/D\tau$ leads to (see also

Table 1. Values of the times at which $n_e(r,t)$ and $I_{diff}(r,t)$ reach their maxima as a function of the problem dimension (case without recombination).

	1D	2D	3D
$t_{max}^{i,N}$	${\displaystyle \frac{r^2}{6D}}$	${\displaystyle \frac{r^2}{8D}}$	${\displaystyle \frac{r^2}{10D}}$
$t_{max}^{n,N}$	${\displaystyle \frac{r^2}{2D}}$	${\displaystyle \frac{r^2}{4D}}$	${\displaystyle \frac{r^2}{6D}}$

):

$$\underset{\tau \to \infty }{lim}{t}_{max}^{n,N}={\displaystyle \frac{r^2}{2ND}}\underset{\tau \to \infty }{lim}{t}_{max}^{i,N}={\displaystyle \frac{r^2}{2D(N+2)}}$$

(12)

For N = 3, Equation (12) gives the expressions previously reported in [35]. Figure 2 shows the carrier density $n_e$ (Equation (2)) and the diffusion current $I_{diff}$ (Equation (9)) as a function of time for several values of the carrier lifetime in dimension N = 1. The values of $t_{max}^n$ and $t_{max}^i$, given by Equations (10) and (11), respectively, are also indicated, illustrating the time lead of the current over the carrier density, as previously discussed.

In addition, Figure 3 shows the evolution of $t_{max}^n$ and $t_{max}^i$ with the carrier lifetime $\tau$, as a function of the dimension N. The time difference between the charge and current peaks is greatest when there is no recombination; it approaches zero as recombination increases, i.e., as the carrier lifetime decreases. This figure also highlights a result that is not evident when comparing Equations (10) and (11): $t_{max}^n$ for $N=3$ is equal to $t_{max}^i$ for $N=1$.

5. Time Variations of the Collected Charge

The collected charge at point P can be evaluated from the time integral of the diffusion current:

$$q_{col}(N,r,t)={\int }_0^t\left|I_{diff}(N,r,t’)\right|dt’={\displaystyle \frac{q{n}_0{A}_{N}r}{2{\left(4\pi D\right)}^{\frac{N}{2}}}}\times {\mathcal{I}}_N(r,t)$$

(13)

with

$${\mathcal{I}}_N(r,t)={\int }_0^t{t}^{\prime -(\frac{N}{2}+1)}exp\left(-{\displaystyle \frac{r^2}{4D{t}^{\prime }}}-{\displaystyle \frac{t^{\prime }}{\tau }}\right)d{t}^{\prime }$$

(14)

This integral has no simple analytical closed-form due to the presence of the second term $-t/\tau$ in the exponential function. However, it can be expressed exactly using ${\mathcal{K}}_{-N/2}$, a special form of the incomplete modified Bessel function of the second kind [37,38,39], in the following form:

$${\mathcal{I}}_N(r,t)=t^{-N/2}{\mathcal{K}}_{-N/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$$

(15)

with

$${\mathcal{K}}_{\nu }(x,y)={\int }_1^{\infty }{u}^{-\nu -1}exp\left(-xu-{\displaystyle \frac{y}{u}}\right)du,$$

(16)

where $x>0$, $y\ge 0$, and $\nu \in \mathbb{R}$. ${\mathcal{K}}_0(x,y)$ is also called the “leaky aquifer integral” [37,38,39].

The final expression of the collected charge in N dimensions is then:

$$q_{col}(N,r,t)={\displaystyle \frac{q{A}_N{n}_{0}r}{2{\left(4\pi Dt\right)}^{N/2}}}\times {\mathcal{K}}_{-N/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$$

(17)

For $\tau \to +\infty$, i.e., in the absence of recombination, the second argument of the function ${\mathcal{K}}_{-N/2}$ is zero, and the function reduces to the following:

$${\mathcal{K}}_{-N/2}(a,0)=a^{-N/2}\mathrm{\Gamma }\left({\displaystyle \frac{N}{2}},a\right)\Re \left(a\right)>0$$

(18)

where $a={\displaystyle \frac{r^2}{4Dt}}$ and $\mathrm{\Gamma }(s,x)$ is the upper incomplete gamma function [40,41]. This leads to the following expression of the collected charge without recombination:

$$q_{\mathrm{col}}(N,r,t)={\displaystyle \frac{q{A}_N{n}_0}{2{\pi }^{N/2}{r}^{N-1}}}\mathrm{\Gamma }\left({\displaystyle \frac{N}{2}},{\displaystyle \frac{r^2}{4Dt}}\right)$$

(19)

We examine and discuss, in the following, these different expressions for N = 1, 2, and 3 with and without carrier recombination.

5.1. One-Dimensional Case ($N=1$)

For $N=1$, $A_1=1$, and Equation (17) reduces to the following:

$$q_{\mathrm{col}}^{1\mathrm{D}}(r,t)={\displaystyle \frac{q{n}_{0}r}{4\sqrt{\pi Dt}}}{\mathcal{K}}_{-1/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$$

(20)

The function ${\mathcal{K}}_{-1/2}$ can be written in the following form (using Equation 7.4.33 from [42]):

$${\mathcal{K}}_{-1/2}(x,y)={\displaystyle \frac{\sqrt{\pi }}{2\sqrt{x}}}\left[e^{-2\sqrt{xy}}erfc(\sqrt{x}-\sqrt{y})+e^{2\sqrt{xy}}erfc(\sqrt{x}+\sqrt{y})\right]$$

(21)

where $erfc\left(z\right)$ is the complementary error function [43,44].

Equation (20) gives with $x={\displaystyle \frac{r^2}{4Dt}}$ and $y={\displaystyle \frac{t}{\tau }}$:

$$q_{\mathrm{col}}^{1\mathrm{D}}(r,t)={\displaystyle \frac{q{n}_0}{4}}\left[e^{-r/\sqrt{D\tau }}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}-\sqrt{{\displaystyle \frac{t}{\tau }}}\right)+e^{r/\sqrt{D\tau }}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}+\sqrt{{\displaystyle \frac{t}{\tau }}}\right)\right].$$

(22)

In the absence of carrier recombination, i.e., when $\tau \to \infty$, $\sqrt{{\displaystyle \frac{t}{\tau }}}\to 0$ and $e^{-r/\sqrt{D\tau }}\to 1$, we obtain the compact form:

$$q_{\mathrm{col}}^{1\mathrm{D}}(r,t)=q{n}_0\times {\displaystyle \frac{1}{2}}\times erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}\right).$$

(23)

Evidently, Equation (19) leads to the same result for $N=1$ without recombination:

$$q_{\mathrm{col}}^{1\mathrm{D}}(r,t)={\displaystyle \frac{q{A}_1{n}_0}{2\sqrt{\pi }}}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{r^2}{4Dt}}\right)={\displaystyle \frac{q{n}_0}{2}}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}\right)$$

(24)

since $\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}},x\right)=\sqrt{\pi }erfc\left(\sqrt{x}\right)$ [40].

5.2. Two-Dimensional Case ($N=2$)

For $N=2$ and $A_2=\ell$, Equation (17) can be rewritten as follows:

$$q_{\mathrm{col}}^{2\mathrm{D}}(r,t)={\displaystyle \frac{q\ell n_{0}r}{8\pi Dt}}{\mathcal{K}}_{-1}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$$

(25)

where ${\mathcal{K}}_{-1}$ cannot be reduced to a simple combination of erf or erfc functions. Nevertheless, in the absence of recombination, ${\mathcal{K}}_{-1}(a,0)=a^{-1}\mathrm{\Gamma }(1,a)=e^{-a}/a$ and a more compact form can be obtained:

$$q_{\mathrm{col}}^{2\mathrm{D}}(r,t)=q{n}_0\times {\displaystyle \frac{\ell }{2\pi r}}\times exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$$

(26)

5.3. Three-Dimensional Case ($N=3$)

Finally, for $N=3$ and $A_3=A_S$, Equation (17) gives the following:

$$q_{col}^{3D}(r,t)={\displaystyle \frac{q{A}_S{n}_{0}r}{2{\left(4\pi Dt\right)}^{3/2}}}\times {\mathcal{K}}_{-3/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$$

(27)

For $\tau \to +\infty$, i.e., without recombination, the second argument of the function ${\mathcal{K}}_{-3/2}$ is zero, and this quantity reduces to the following:

$${\mathcal{K}}_{-3/2}(a,0)=a^{-3/2}\mathrm{\Gamma }\left({\displaystyle \frac{3}{2}},a\right)\Re \left(a\right)>0$$

(28)

with $a={\displaystyle \frac{r^2}{4Dt}}$, which leads to the same expression of the collected charge without recombination as already established in Ref. [35]:

$$q_{col}^{3D}(r,t)={\displaystyle \frac{q{A}_S{n}_0}{2{\left(\pi \right)}^{\frac{3}{2}}{r}^2}}\times \mathrm{\Gamma }\left({\displaystyle \frac{3}{2}},{\displaystyle \frac{r^2}{4Dt}}\right)$$

(29)

Note that this incomplete gamma function takes the following special form [40]:

$$\mathrm{\Gamma }\left({\displaystyle \frac{3}{2}},a\right)={\displaystyle \frac{\sqrt{\pi }}{2}}erfc\left(\sqrt{a}\right)+\sqrt{a}{e}^{-a}$$

(30)

Equation (29) without recombination can be rewritten as follows:

$$q_{col}^{3D}(r,t)=q{n}_0\times {\displaystyle \frac{A_S}{4\pi r^2}}\times {\displaystyle \frac{2}{\sqrt{\pi }}}\left[{\displaystyle \frac{\sqrt{\pi }}{2}}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}\right)+{\displaystyle \frac{r}{2\sqrt{Dt}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)\right].$$

(31)

To conclude, note that this result can also be obtained directly from Equation (27), by using the expression of ${\mathcal{K}}_{-3/2}(x,y)$ (which can be derived from Equation (21)):

$$\begin{array}{cc}\hfill {\mathcal{K}}_{-3/2}(x,y)& ={\displaystyle \frac{\sqrt{\pi }}{4x^{3/2}}}[(1+2\sqrt{xy})e^{-2\sqrt{xy}}erfc(\sqrt{x}-\sqrt{y})\hfill \\ \hfill & +(1-2\sqrt{xy})e^{2\sqrt{xy}}erfc(\sqrt{x}+\sqrt{y})]\hfill \\ \hfill & +{\displaystyle \frac{1}{x}}{e}^{-x-y}.\hfill \end{array}$$

(32)

This gives an alternative expression to Equation (27) without the use of the special function ${\mathcal{K}}_{-3/2}$:

$$\begin{array}{cc}\hfill q_{col}^{3D}(r,t)& ={\displaystyle \frac{q{A}_S{n}_0}{8\pi r^2}}[\left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)e^{-r/\sqrt{D\tau }}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}-\sqrt{{\displaystyle \frac{t}{\tau }}}\right)\hfill \\ \hfill & +\left(1-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)e^{r/\sqrt{D\tau }}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}+\sqrt{{\displaystyle \frac{t}{\tau }}}\right)]\hfill \\ \hfill & +{\displaystyle \frac{q{A}_S{n}_0}{4{\pi }^{3/2}r\sqrt{Dt}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right).\hfill \end{array}$$

(33)

When $\tau \to \infty$, $\sqrt{{\displaystyle \frac{t}{\tau }}}\to 0$ and ${\displaystyle \frac{r}{\sqrt{D\tau }}}\to 0$, the simplification of Equation (33) logically gives the same result as Equation (31):

$$q_{col}^{3D}(r,t)={\displaystyle \frac{q{A}_S{n}_0}{4\pi r^2}}erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}\right)+{\displaystyle \frac{q{A}_S{n}_0}{4{\pi }^{3/2}r\sqrt{Dt}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right).$$

(34)

6. Integral Collected Charge

The integral collected charge at point P corresponds to the charge collected over the time interval from zero to infinity:

$$Q_{col}(N,r)={\int }_0^{\infty }\left|I_{diff}(N,r,t)\right|dt={\displaystyle \frac{q{n}_0{A}_{N}r}{2{\left(4\pi D\right)}^{\frac{N}{2}}}}\times {\mathcal{I}}_N(r,\infty )$$

(35)

where

$${\mathcal{I}}_N(r,\infty )={\int }_0^{\infty }{t}^{\prime -(\frac{N}{2}+1)}exp\left(-{\displaystyle \frac{r^2}{4D{t}^{\prime }}}-{\displaystyle \frac{t^{\prime }}{\tau }}\right)d{t}^{\prime }$$

(36)

This gives the following:

$${\mathcal{I}}_N(r,\infty )=2{\left({\displaystyle \frac{4D}{\tau r^2}}\right)}^{N/4}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(37)

where $K_{\nu }\left(x\right)$ is the modified Bessel function of the second kind (also called the Macdonald function) given by the integral form [37]:

$$K_{\nu }\left(x\right)={\displaystyle \frac{1}{2}}{\int }_0^{\infty }{t}^{\nu -1}exp\left(-{\displaystyle \frac{x}{2}}(t+{\displaystyle \frac{1}{t}})\right)dt$$

(38)

6.1. Uni-Dimensional Case (N = 1)

For $N=1$, we have the following [42,45]:

$$K_{1/2}\left(x\right)=\sqrt{{\displaystyle \frac{\pi }{2x}}}{e}^{-x}$$

(39)

which gives

$${\mathcal{I}}_1(r,\infty )={\displaystyle \frac{2\sqrt{\pi D}}{r}}exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(40)

that immediately gives

$$Q_{col}^{1D}\left(r\right)=q{n}_0\times {\displaystyle \frac{1}{2}}\times exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(41)

When $\tau \to \infty$, ${\displaystyle \frac{r}{\sqrt{D\tau }}}\to 0$, and the integral charge without recombination is equal to the following:

$$Q_{col}^{1D}\left(r\right)=q{n}_0\times {\displaystyle \frac{1}{2}}$$

(42)

It is evident that the same expressions as in Equations (41) and (42) can be obtained when considering the limits of Equations (22) and (23), respectively, as t approaches infinity.

6.2. Bi-Dimensional Case (N = 2)

Similarly, for N = 2:

$${\mathcal{I}}_2(r,\infty )={\displaystyle \frac{4}{r}}\sqrt{{\displaystyle \frac{D}{\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(43)

and the collected charge is therefore,

$$Q_{\mathrm{col}}^{2\mathrm{D}}\left(r\right)={\displaystyle \frac{q{n}_0\ell }{2\pi \sqrt{D\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(44)

When $\tau \to \infty$, ${\displaystyle \frac{r}{\sqrt{D\tau }}}\to 0$ and $K_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)\to {\displaystyle \frac{\sqrt{D\tau }}{r}}$, so in the absence of recombination:

$$Q_{\mathrm{col}}^{2\mathrm{D}}\left(r\right)=q{n}_0\times {\displaystyle \frac{\ell }{2\pi r}}$$

(45)

6.3. Three-Dimensional Case (N = 3)

Now for $N=3$, we have [42]:

$$K_{3/2}\left(x\right)=\sqrt{{\displaystyle \frac{\pi }{2x}}}{e}^{-x}\left(1+{\displaystyle \frac{1}{x}}\right)$$

(46)

Then,

$${\mathcal{I}}_3(r,\infty )={\displaystyle \frac{4\sqrt{\pi }{D}^{3/2}}{r^3}}\left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(47)

and therefore

$$Q_{\mathrm{col}}^{3\mathrm{D}}\left(r\right)=q{n}_0\times {\displaystyle \frac{A_S}{4\pi r^2}}\times \left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(48)

For $\tau \to \infty$, the case without recombination is immediate:

$$Q_{\mathrm{col}}^{3\mathrm{D}}\left(r\right)=q{n}_0\times {\displaystyle \frac{A_S}{4\pi r^2}}.$$

(49)

6.4. Alternative Formulation and Expression

A more general formulation of $Q_{col}\left(r\right)$ as a function of N can be directly obtained by inserting Equation (37) in Equation (35). After simplification, we obtain the following:

$$Q_{col}(N,r)={\displaystyle \frac{q{n}_0{A}_N}{{\pi }^{N/2}{\left(4D\tau \right)}^{N/4}}}{r}^{1-{\displaystyle \frac{N}{2}}}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(50)

Using the surface area of the hypersphere given in Equation (8), the integral collected charge can also be expressed as follows:

$$Q_{col}(N,r)=q{n}_0{\displaystyle \frac{A_N}{S_N\left(r\right)}}{\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(51)

From this equation, we can verify that

$$\underset{\tau \to \infty }{lim}{Q}_{col}(N,r)=q{n}_0\times {\displaystyle \frac{A_N}{S_N\left(r\right)}}$$

(52)

Figure 4 shows the time evolution of $q_{col}(r,t)$ given by Equation (17) for different carrier lifetimes and for $N=1$, $N=2$, and $N=3$, with a source–collector distance of r = 2 µm. As expected and for any dimension N, the collected charge increases over time and as the carrier lifetime increases, reaching a maximum when $t\to \infty$ (red dots). The absolute maximum of the collected charge is obtained in all dimensions for $t\to \infty$ when $\tau \to \infty$. These maximum values are respectively given by Equations (51) and (52), in the presence or absence of carrier recombination, respectively.

Finally, Figure 4 illustrates how the dimensionality of the diffusion problem influences the time dynamics of charge collection and the asymptotic collected fraction, with or without carrier recombination. On the one hand, the comparison between the curves should be interpreted as an indication of the role of dimensionality in the collection process, rather than as an absolute comparison between devices. Indeed, the cases $N=1$, $N=2$, and $N=3$ correspond to collection geometries defined in different spatial dimensions, with intrinsically different collector surfaces. As a result, the absolute value of the collected charge cannot be directly compared across dimensions without specifying an additional geometrical normalization. The curves mainly show that dimensionality modifies the rise time and the limiting collected fraction through the dimensional dependence of the diffusive flux toward the collector. On the other hand, the time dynamics of the collected charge result from the competition between diffusion and recombination. Initially, the collection is low because the carriers have not yet reached the collector. It then increases over a characteristic diffusion time $t_D\sim r^2/D$, and finally reaches a plateau. In the absence of recombination ($\tau \to \infty$), all of the initial charge is ultimately collected, independently of N, because recombination losses are absent. Conversely, for a finite carrier lifetime, the plateau is lower than one and depends strongly on the dimension: in higher-dimensional idealized geometries, a larger fraction of the carriers can contribute to the flux through the effective collector before recombining. Thus, within the assumptions of the model, the 3D formulation leads to faster and more efficient collection than the corresponding 2D and 1D idealized cases.

7. Collection Efficiency and Recombination Factor

The collection efficiency of the collector is defined as the ratio of the charge actually collected at point P when $t\to \infty$ to the charge initially generated by the particle at point M:

$${\eta }_{\mathrm{col}}(N,r)={\displaystyle \frac{Q_{col}(N,r)}{q{n}_0}}$$

(53)

From Equation (51), we immediately have

$${\eta }_{\mathrm{col}}(N,r)={\displaystyle \frac{A_N}{S_N\left(r\right)}}{\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(54)

that can be written as following:

$${\eta }_{\mathrm{col}}(N,r)={\displaystyle \frac{A_N}{S_N\left(r\right)}}\times f_{\mathrm{recomb}}\left(r\right)$$

(55)

with

$$f_{\mathrm{recomb}}(N,r)={\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$$

(56)

The recombination factor $f_{\mathrm{recomb}}$ can be interpreted as a survival factor for carriers undergoing recombination. $f_{\mathrm{recomb}}=1$ corresponds to negligible recombination ($r\ll L_D$), while $f_{\mathrm{recomb}}<1$ means that some carriers that could have reached P in the absence of recombination did not because they recombined along their diffusion path.

Considering developments and simplifications for N = 1, 2, and 3, we have the following:

$$f_{\mathrm{recomb}}(N,r)=\left\{\begin{array}{cc}exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right),\hfill & N=1,\hfill \\ {\displaystyle \frac{r}{\sqrt{D\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right),\hfill & N=2,\hfill \\ \left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right),\hfill & N=3.\hfill \end{array}\right.$$

(57)

Figure 5 shows the evolution of the recombination factor $f_{recomb}$ as a function of r for different values of the carrier lifetime $\tau$ in dimension $N=3$. This figure illustrates how the recombination factor decreases sharply with distance from the collector and how it depends on the lifetime $\tau$. For small values of r, almost all of the charge is collected; however, beyond a critical distance that corresponds to the diffusion length, $L_D$, most of the charge is lost through recombination. This distance increases with $\tau$ and follows $L_D\sim \sqrt{D\tau }$, reflecting the competition between diffusion and recombination.

8. Model Synthesis and Numerical Verification

This section summarizes all the equations of the proposed diffusion–collection model in three reference tables.

Table 2. Summary of the main general equations in N-dimensional diffusion–collection model with recombination. The names and input parameters of the corresponding Python functions described in Appendix A are also indicated.

Model Quantity	Dimension N	Python Function
Collector surface	$A_N$	A_N(N)
Excess carrier density $n_e(N,r,t)$	${\displaystyle \frac{n_0}{{\left(4\pi Dt\right)}^{\frac{N}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	n_e(N,r,t)
Max. time for carrier density $t_{max}^{n,N}$	${\displaystyle \frac{\tau }{4}}\left(\sqrt{N^2+{\displaystyle \frac{4r^2}{D\tau }}}-N\right)$	t_n_max(N,r)
Diffusion current $|I_{diff}(N,r,t)|$	${\displaystyle \frac{q{n}_0{A}_{N}r}{2t{\left(4\pi Dt\right)}^{\frac{N}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	I_diff(N,r,t)
Max. time for current $t_{max}^{i,N}$	${\displaystyle \frac{\tau }{2}}\left[\sqrt{{\left({\displaystyle \frac{N}{2}}+1\right)}^2+{\displaystyle \frac{r^2}{D\tau }}}-\left({\displaystyle \frac{N}{2}}+1\right)\right]$	t_i_max(N,r)
Collected charge $q_{col}(N,r,t)$	${\displaystyle \frac{q{A}_N{n}_{0}r}{2{\left(4\pi Dt\right)}^{N/2}}}\times {\mathcal{K}}_{-N/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$	q_col(N,r,t)
Integral collected charge $Q_{col}(N,r)$	$q{n}_0{\displaystyle \frac{A_N}{S_N\left(r\right)}}{\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	Q_col(N,r)
Collection efficiency ${\eta }_{col}^{diff}(N,r)$ for $t\to \infty$	${\displaystyle \frac{A_N}{S_N\left(r\right)}}{\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	eta(N,r)
Recombination factor $f_{recomb}(N,r)$	${\displaystyle \frac{2^{1-\frac{N}{2}}}{\mathrm{\Gamma }(N/2)}}{\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)}^{N/2}{K}_{N/2}\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	f_rec(N,r)

contains the most general equations, which can be used for any of the considered dimensions (N = 1, 2, or 3) and with recombination. It also lists the names and entry parameters of the corresponding functions given in Appendix A for implementing the model in Python (version 3.12.7). The general expressions for collected charge, integral collected charge, and collection efficiency use special functions, or Bessel integrals, that can be expressed using standard functions in certain cases.

Table 3. Synthesis of the main equations in 1D, 2D, and 3D for the diffusion–collection model with recombination.

Dimension N	1	2	3
Collector surface $A_N$	1	ℓ	$A_S$
Excess carrier density $n_e(N,r,t)$	${\displaystyle \frac{n_0}{2\sqrt{\pi Dt}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{n_0}{4\pi Dt}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{n_0}{{\left(4\pi Dt\right)}^{\frac{3}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$
Max. time for carrier density $t_{max}^n$	${\displaystyle \frac{\tau }{4}}\left(\sqrt{1+{\displaystyle \frac{4r^2}{D\tau }}}-1\right)$	${\displaystyle \frac{\tau }{2}}\left(\sqrt{1+{\displaystyle \frac{r^2}{D\tau }}}-1\right)$	${\displaystyle \frac{3\tau }{4}}\left(\sqrt{1+{\displaystyle \frac{4r^2}{9D\tau }}}-1\right)$
Diffusion current $|I_{diff}(N,r,t)|$	${\displaystyle \frac{q{n}_{0}r}{4\sqrt{\pi D}{t}^{3/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{q{n}_0\ell r}{8\pi D{t}^2}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{q{n}_0{A}_{S}r}{16{\pi }^{3/2}{D}^{3/2}{t}^{5/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}-{\displaystyle \frac{t}{\tau }}\right)$
Max. time for current $t_{max}^i$	${\displaystyle \frac{3\tau }{4}}\left[\sqrt{1+{\displaystyle \frac{4r^2}{9D\tau }}}-1\right]$	$\tau \left[\sqrt{1+{\displaystyle \frac{r^2}{4D\tau }}}-1\right]$	${\displaystyle \frac{5\tau }{4}}\left[\sqrt{1+{\displaystyle \frac{4r^2}{25D\tau }}}-1\right]$
Collected charge $q_{col}(N,r,t)$	${\displaystyle \frac{q{n}_{0}r}{4\sqrt{\pi Dt}}}{\mathcal{K}}_{-1/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{q\ell n_{0}r}{8\pi Dt}}{\mathcal{K}}_{-1}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$	${\displaystyle \frac{q{A}_S{n}_{0}r}{16{\left(\pi Dt\right)}^{3/2}}}\times {\mathcal{K}}_{-3/2}\left({\displaystyle \frac{r^2}{4Dt}},{\displaystyle \frac{t}{\tau }}\right)$
Integral collected charge $Q_{col}(N,r)$	${\displaystyle \frac{q{n}_0}{2}}\times exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	${\displaystyle \frac{q{n}_0\ell }{2\pi \sqrt{D\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	${\displaystyle \frac{q{n}_0{A}_S}{4\pi r^2}}\times \left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$
Collection efficiency ${\eta }_{col}^{diff}(N,r)$ for $t\to \infty$	${\displaystyle \frac{1}{2}}\times exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	${\displaystyle \frac{\ell }{2\pi \sqrt{D\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	${\displaystyle \frac{A_S}{4\pi r^2}}\times \left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$
Recombination factor $f_{recomb}(N,r)$	$exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	${\displaystyle \frac{r}{\sqrt{D\tau }}}{K}_1\left({\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$	$\left(1+{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)exp\left(-{\displaystyle \frac{r}{\sqrt{D\tau }}}\right)$

and

Table 4. Synthesis of the main equations in 1D, 2D, and 3D for the diffusion–collection model without recombination.

Dimension N	1	2	3
Collector surface $A_N$	1	ℓ	$A_S$
Excess carrier density $n_e(N,r,t)$	${\displaystyle \frac{n_0}{2\sqrt{\pi Dt}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$	${\displaystyle \frac{n_0}{4\pi Dt}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$	${\displaystyle \frac{n_0}{{\left(4\pi Dt\right)}^{\frac{3}{2}}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$
Max. time for carrier density $t_{max}^n$	${\displaystyle \frac{r^2}{2D}}$	${\displaystyle \frac{r^2}{4D}}$	${\displaystyle \frac{r^2}{6D}}$
Diffusion current $|I_{diff}(N,r,t)|$	${\displaystyle \frac{q{n}_{0}r}{4\sqrt{\pi D}{t}^{3/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$	${\displaystyle \frac{q{n}_0\ell r}{8\pi D{t}^2}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$	${\displaystyle \frac{q{n}_0{A}_{S}r}{16{\pi }^{3/2}{D}^{3/2}{t}^{5/2}}}exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$
Max. time for current $t_{max}^i$	${\displaystyle \frac{r^2}{6D}}$	${\displaystyle \frac{r^2}{8D}}$	${\displaystyle \frac{r^2}{10D}}$
Collected charge $q_{col}(N,r,t)$	${\displaystyle \frac{q{n}_0}{2}}\times erfc\left({\displaystyle \frac{r}{2\sqrt{Dt}}}\right)$	${\displaystyle \frac{q{n}_0\ell }{2\pi r}}\times exp\left(-{\displaystyle \frac{r^2}{4Dt}}\right)$	${\displaystyle \frac{q{A}_S{n}_0}{2{\left(\pi \right)}^{\frac{3}{2}}{r}^2}}\times \mathrm{\Gamma }\left({\displaystyle \frac{3}{2}},{\displaystyle \frac{r^2}{4Dt}}\right)$
Integral collected charge $Q_{col}(N,r)$	$q{n}_0\times {\displaystyle \frac{1}{2}}$	$q{n}_0\times {\displaystyle \frac{\ell }{2\pi r}}$	$q{n}_0\times {\displaystyle \frac{A_S}{4\pi r^2}}$
Collection efficiency ${\eta }_{col}^{diff}(N,r)$ for $t\to \infty$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{\ell }{2\pi r}}$	${\displaystyle \frac{A_S}{4\pi r^2}}$

present these equations in one, two, and three dimensions, with and without recombination, respectively. These tables provide simplified expressions for certain quantities.

All of the model’s equations are analytical and correspond to exact solutions derived from the initial diffusion equation (Equation (1)) and its solution in terms of excess charge density (Equation (2)). However, we also checked all the equations numerically, starting from the numerical solution of Equation (1) with the initial condition given in Section 3. We solved this equation using a finite-difference discretization in space and an implicit backward Euler scheme in time [46]. We approximated the diffusion term using centered finite differences on a uniform, one-dimensional grid. We used a geometrically spaced time grid to capture transient and long-time dynamics [46].

Figure 6 illustrates a typical comparison of analytical and numerical results for the time evolution of charge collected at a specific distance from a point charge, both with and without carrier recombination. As expected, these results highlight the validation of Equations (29), (33), (48), and (49) in three dimensions.

9. Discussion

In this section, we discuss and demonstrate how to use the above equations to perform realistic single-event simulations and link them to the soft error rate of a given device. We will also address the question of generalizing our model.

9.1. Realistic Single-Event Simulations and Link with the SER

When a charged particle enters a semiconductor material, it gradually loses its kinetic energy as it penetrates deeper into the material, until it comes to a stop or exits the area of interest within the component or circuit [2]. The diffusion–collection model involves the conversion of energy lost by the particle into electron–hole pairs. These pairs are then rearranged into a succession of point charge densities of electrons and holes, also referred to as “charge packets”, as illustrated in Figure 7 [6]. This process occurs immediately after the energy deposition and the creation of the pairs, at the initial time $t=0$. The model then operates under the assumption that the transport of the charge generated by the particle along its trajectory can be regarded as a superposition of independent transport processes for all point charge packets. In practice, only one type of carrier is considered for transport, i.e., electrons or holes, even though it is ambipolar in nature. This is because the collector collects either electrons or holes, depending on its type and/or the local bias state. Thus, the transport of each charge packet reduces to the problem addressed in this paper: the diffusion of a point charge in space and time with or without recombination.

Figure 7 illustrates this key point of the diffusion–collection approach: the transformation of continuous charge along the particle track into discrete charges for subsequent transport and collection using equations developed in the model. Note that in 1D, this transformation is reduced to a single point charge at the intersection of the semiconductor domain and the particle path. In two and three dimensions, the total charge $Q_{dep}$ generated by the particle in the domain of interest is divided into a series of point charges, $q_i$, located at distances $r_i$ from the collector:

$$Q_{dep}={\displaystyle \sum _{i=1}^n}{q}_i$$

(58)

Each point charge, $q_i$, independently diffuses and recombines in the semiconductor, contributing $I_{diff}^i$, $q_{col}^i$, and $Q_{col}^i$, respectively, to the collected current, the collected charge, and the integral collected charge at the collector level. Thus, the total collected current, charge, and integral charge due to the complete particle track can be obtained by summing these contributions:

$$I_{col}^{track}\left(t\right)={\displaystyle \sum _{i=1}^n}{I}_{diff}^i\left(t\right)q_{col}^{track}\left(t\right)={\displaystyle \sum _{i=1}^n}{q}_{col}^i\left(t\right)Q_{col}^{track}={\displaystyle \sum _{i=1}^n}{Q}_{col}^i$$

(59)

It is important to note at this point that the diffusion–collection model is based on a linear transport approximation. In this approximation, the carriers generated by the ionizing particle diffuse and recombine independently. This assumption corresponds to a linear, low-injection regime that is valid as long as the generated charge density is low enough that it does not perturb the local electric field or the transport and recombination parameters. Therefore, the model does not include collective effects, such as plasma effects, field screening, nonlinear recombination, or circuit-level feedback. These simplifications define the model range of validity.

Figure 8 provides a numerical example in 3D of the collected current and the charge induced by the passage of a heavy ion close to a sensitive node in a silicon circuit. In this example (see the inset in Figure 8), a 5 MeV Al-27 heavy ion is incident normally on the silicon surface at a distance $l_0$= 1.5 µm from the collector and comes to rest after 3.6 µm in the Si volume (as given by SRIM [47,48]). The particle LET is about 10 MeV cm² mg⁻¹ [47,48] and the ion generates a total charge $Q_{dep}=0.37$ pC along its track. For simplicity and clarity, this charge is divided into only five point charges that are regularly spaced along the particle path. Figure 8 shows the five individual contributions to the total collected quantities, $I_{col}^{track}$ and $q_{col}^{track}$, as well as the resulting time profiles of these global quantities, obtained by summation. In practice, values of n ranging from 20 to 100, as a function of track length, provide satisfactory numerical accuracy. In the case of multiple secondary particles, as encountered in neutron interactions with target material atoms, the generated charge is decomposed/discretized on different particle tracks. Then, the summation in Equation (59) is applied to the total number of charge packets associated with all secondary ionizing particles.

Computing the collected charges due to a set of incoming ionizing particles or nuclear reactions in a semiconductor domain that accounts for the geometry of a given device or circuit (possibly including multiple sensitive nodes or collectors) makes it possible to identify and count all events that induce a collected charge greater than the critical charge, $Q_{crit}$:

$$Q_{col}^{track}>Q_{crit}$$

(60)

where $Q_{crit}$ is the minimum charge required to cause a circuit to malfunction due to a particle strike [2]. This enables a direct estimation of the soft error rate of the considered device or circuit. This topic is beyond the scope of this study. However, the development of a code for estimating the soft error rate is essentially based on the equations presented in this paper. The other aspects of the code are limited to reading an event database, generating clusters of charge packets, performing various geometric calculations—including the distances between point charges and the circuit’s collector(s)—and applying criteria for detecting logic errors.

9.2. Generalization of the Model

The generalization of the proposed diffusion–collection model must be understood in relation to its underlying assumptions. The model is formulated for an idealized semiconductor medium with effective transport parameters, such as diffusion coefficient and carrier lifetime, and simplified collection geometries. Consequently, the model applies to situations in which charge transport is diffusion-dominated, recombination can be represented by an effective lifetime, and the collection region can be approximated as planar, cylindrical, or spherical. These situations include weak-field regions, undepleted substrates, epitaxial layers, and regions located far enough from high-field junctions that diffusion remains the main mechanism controlling the delayed component of the charge collection. The same mathematical framework can be used for different semiconductor materials, provided that the relevant material parameters are adapted. In particular, the diffusion coefficient, mobility, carrier lifetime, ionization energy, recombination properties, and effective collection length must be chosen consistently with the material under consideration. While the model is not restricted to silicon in its mathematical form, its quantitative predictions depend on the availability and accuracy of these material-dependent parameters. This is especially important for wide-bandgap or oxide semiconductors, as carrier mobility, trapping, defect-assisted recombination, and radiation-induced changes in material properties may differ significantly from conventional silicon technologies.

The applicability of the model to realistic semiconductor devices depends on the device architecture and biasing conditions. In modern CMOS devices, FinFETs, fully depleted structures, oxide semiconductor transistors, and neuromorphic devices, the electric field distribution, doping profile, interfaces, contacts, and boundary conditions may be highly nonuniform. In such cases, the present analytical model should not be viewed as a replacement for full device simulations. Instead, it offers a simplified description that captures the diffusion–recombination aspect of charge transport. This description can be used to interpret numerical simulations, estimate collection times, evaluate limiting cases, and provide compact input parameters for higher-level soft-error-rate models.

The main limitation of the current formulation is its inability to explicitly resolve complex, three-dimensional device geometries; nonuniform electric fields; field-enhanced drift; carrier trapping at interfaces; nonlinear recombination; and time-dependent material degradation under irradiation. When these effects dominate the transient response, TCAD or multi-physics numerical simulations are still necessary. Nevertheless, the analytical framework is useful as a reference model because it identifies the dependencies of collected charge on distance, dimensionality, diffusion coefficient, carrier lifetime, and collection geometry. The framework’s strength lies not in providing an exact description of every device configuration, but rather in its ability to offer a transparent baseline for diffusion-driven charge collection across various materials and device structures.

10. Conclusions

In conclusion, this paper presents a unified analytical framework for modeling charge diffusion and collection in semiconductor materials. This framework is valid in one, two, and three dimensions and includes carrier recombination. Starting from the diffusion equation for an initial point charge, we derived analytical expressions for the carrier density, diffusion current, and collected charge as functions of time and distance.

The proposed model emphasizes the role of geometry through dimension-dependent factors, providing a consistent description of the diffusion–collection process across different configurations. The use of special functions allows for an exact treatment of recombination effects while maintaining analytical compactness. The analysis also reveals important physical characteristics, such as the temporal shift between the maxima of the current and carrier density and the effect of recombination on collection efficiency.

Beyond its theoretical value, this framework is a practical building block for modeling and simulating radiation-induced transient effects and the soft error rate (SER) in semiconductor devices. Its compact formulation and straightforward implementation make it well-suited for integration into reliability-oriented simulation tools. Future extensions may include electric field effects and more complex device geometries.