**2. Random-walk drift-diffusion (RWDD) model**

This section introduces the basis of the RWDD model and the modeling methodology used to describe the main simulation steps, i.e. the charge deposition induced by the passage of an ionizing particle at silicon level, the radiation-induced charge transport within the structure and the computation of the collection current on the collecting node(s).

#### **2.1. Charge deposition**

In the RWDD approach, an ionizing particle track crossing a circuit at silicon-level is modeled as a series of charge packets (**Figure 1**) spread along a straight segment whose length equals the ionizing particle range (R) in target material. The linear density of the charge packet along the particle track takes into account the non-constant linear energy transfer (LET) of the particle. SRIM tables are used to compute both LET and range using appropriate numerical functions [14]. The accuracy of this charge discretization is ensured by the degree of "granu‐ larity" [11] that can be fine-tuned by selecting the packets size, in practice from 1 to 100 elementary charges.

**Figure 1.** Cartoon illustrating the charge transport and collection processes after a 5 MeV alpha-particle strike in a re‐ verse-biased junction (its geometrical and electrical parameters correspond to the 65 nm node). The biased contact col‐ lects charges that diffuse in silicon and are accelerated by the electric field developed in the space charge region. (Reprinted with permission from Glorieux et al. [12], © 2014, IEEE).

#### **2.2. Charge transport modeling**

The transport of each charge packet starts immediately after the particle crosses the device (**Figure 1**). For modeling the charge transport, the RWDD approach implements the popular drift-diffusion model [15], where the electron (**Jn**) and hole (**Jp**) current densities are computed as the sum of two components: the drift current component (first term) that is driven by the electric field and the diffusion component (second term) that models the current induced by the gradient of carrier concentration. **Jn** and **Jp** are given by:

$$\mathbf{J}\_{\mathbf{n}} = \mathbf{q}\mathbf{n}\mu\_{\mathbf{n}}\mathbf{E} + \mathbf{q}\mathbf{D}\_{\mathbf{n}}\mathbf{grad}\left(\mathbf{n}\right) \tag{1}$$

$$\mathbf{J}\_{\mathbf{p}} = \mathbf{q}\mathbf{r}\mu\_{\mathbf{p}}\mathbf{E} - \mathbf{q}\mathbf{D}\_{\mathbf{p}}\mathbf{grad}(\mathbf{p})\tag{2}$$

In these equations, n and p are, respectively, the electron and hole densities; μn and μp are, respectively, the electron and hole mobilities; E is the electric field; and Dn and Dp are, respectively, the diffusion coefficients that may be calculated from the carrier mobility using the Einstein's equation:

$$\mathbf{D}\_{\rm n,p} = \frac{\mathbf{k}\_{\rm B} \mathbf{T}\_{\rm n}}{\mathbf{q}} \boldsymbol{\mu}\_{\rm n,p} \tag{3}$$

In Eq. (3), kB is the Boltzmann constant and T is the carrier temperature; as the carrier gas in the drift-diffusion approximation is assumed to be in thermal equilibrium, T is equal to the lattice temperature.

The next calculation step in conventional full numerical methods (TCAD) is to inject the current densities given by Eqs. (1) and (2) into the conservation laws for electrons and holes also called continuity equations; they are next self-consistently solved with Poisson's equation. For this purpose, the simulation domain is meshed, and all previous equations are discretized on this mesh grid and then solved. Contrary to this procedure, in the RWDD model, no meshing is need since charge packets have continuous coordinates. Also in RWDD, a random-walk algorithm [11] is used to model the diffusion process, and the drift-induced current is directly calculated using the electrical field present in the considered region. For a charge packet situated at the position (x, y, z) at time t, its new position at time t+δt is given by (x+δx, y+δy and z+δz), where dx, dy and dz are calculated as follows:

$$\begin{cases} \mathbf{dx} = \mathbf{N}\_1 \times \sqrt{\mathbf{D} \mathbf{d} \mathbf{t}} + \mathbf{E}\_\times \times \mu \mathbf{d} \mathbf{t} \\ \mathbf{dy} = \mathbf{N}\_2 \times \sqrt{\mathbf{D} \mathbf{d} \mathbf{t}} + \mathbf{E}\_y \times \mu \mathbf{d} \mathbf{t} \\ \mathbf{dz} = \mathbf{N}\_3 \times \sqrt{\mathbf{D} \mathbf{d} \mathbf{t}} + \mathbf{E}\_z \times \mu \mathbf{d} \mathbf{t} \end{cases} \tag{4}$$

In Eq. (4), N1, N2 and N3 are three independent normal random numbers, D is the diffusion coefficient, μ is the carrier mobility and E(Ex, Ey, Ez) is the electric field vector at the corre‐ sponding position and time.

#### **2.3. Collection current computation**

**2.1. Charge deposition**

118 Modeling and Simulation in Engineering Sciences

elementary charges.

In the RWDD approach, an ionizing particle track crossing a circuit at silicon-level is modeled as a series of charge packets (**Figure 1**) spread along a straight segment whose length equals the ionizing particle range (R) in target material. The linear density of the charge packet along the particle track takes into account the non-constant linear energy transfer (LET) of the particle. SRIM tables are used to compute both LET and range using appropriate numerical functions [14]. The accuracy of this charge discretization is ensured by the degree of "granu‐ larity" [11] that can be fine-tuned by selecting the packets size, in practice from 1 to 100

**Figure 1.** Cartoon illustrating the charge transport and collection processes after a 5 MeV alpha-particle strike in a re‐ verse-biased junction (its geometrical and electrical parameters correspond to the 65 nm node). The biased contact col‐ lects charges that diffuse in silicon and are accelerated by the electric field developed in the space charge region.

The transport of each charge packet starts immediately after the particle crosses the device (**Figure 1**). For modeling the charge transport, the RWDD approach implements the popular drift-diffusion model [15], where the electron (**Jn**) and hole (**Jp**) current densities are computed as the sum of two components: the drift current component (first term) that is driven by the electric field and the diffusion component (second term) that models the current induced by

In these equations, n and p are, respectively, the electron and hole densities; μn and μp are, respectively, the electron and hole mobilities; E is the electric field; and Dn and Dp are,

**J E grad <sup>n</sup>** = + qnμ qD n n n ( ) (1)

**J E grad <sup>p</sup>** = - qnμ qD p p p ( ) (2)

(Reprinted with permission from Glorieux et al. [12], © 2014, IEEE).

the gradient of carrier concentration. **Jn** and **Jp** are given by:

**2.2. Charge transport modeling**

At each time step of the simulation, the radiation-induced collection current is computed from the transport dynamic of minority charge carriers described in the previous Section 2.2. For the estimation of this current, two main procedures may be employed: the first technique is to use the semi-conductor transport equations and the second one is to employ the Schockely-Ramo's theorem. Simulation tools used in microelectronics generally consider the first option; the transport equations are implemented in TCAD simulator that numerically solve them selfconsistently with Poisson's equation. This approach considers the free-charge carrier distri‐ butions as continuous functions in time and space coordinates. The second option is generally used in instrumentation or high-energy physics for the calculation of detector responses to radiation events. In this second approach, the Ramo's theorem is used to treat each carrier considered individually and all the interesting effects due to particular carriers are summed [16]. In our work, we implemented the first formalism (transport equations) by applying the continuity equation at the collecting (drain) contact. The transient current at the collecting node is directly computed from the number of carriers Δn that reach this contact during the time step Δt, i.e.

$$\mathbf{I} = \mathbf{q} \times \frac{\Delta \mathbf{n}}{\Delta \mathbf{t}} \tag{5}$$

In this expression, the displacement current is neglected; that is a reasonable approximation in this case [17]. This collection current is then injected in the electrical simulation to model the circuit response.
