**1. Introduction**

Natural radiation at ground level, including both terrestrial cosmic rays and telluric radioac‐ tivity (alpha decay from radioactive ultra-traces in materials), is considered today as a major reliability issue for integrated circuits (IC), since they are increasingly sensitive to this radia‐ tion as long as CMOS technologies scale down [1]. The basic mechanism taking place when an ionizing particle crosses a circuit is the generation in the semiconductor region of a highdensitychargetrack;thegeneratedchargecanbecollectedatcircuitlevelthroughbiasedcontacts andespeciallyreverse-biaseddrainjunctionsandcreateparasitic transient currentsatthe circuit nodes. In the case of a memory cell, the collected charge may be sufficient to induce the cell upset; this phenomenon is called single event upset (SEU). When combinational logic is concerned, the charge induced by the particle may lead to a single event transient (SET) [2].

Researchers have always used modeling and simulation approaches to better understand the physical processes and to predict the consequences on the circuit operation of such single events. The study of these phenomena, and more precisely, the computation of the electrical response of devices and circuits submitted to ionizing radiation imply the accurate modeling of various physical mechanisms at different scales, as charge generation, transport and collection of charges within the circuit. The solutions developed to solve this kind of problem include full numerical methods (such as TCAD) and approximate analytical models used in circuit simulators [3]. Full numerical methods, such as TCAD, exactly solve this coupled problem (radiation + electrostatics + transport) and offer a very accurate solution. But the huge computation time and computer resources needed to solve this high complexity problem restrict its application to small simulation domains, limited to, at most, several devices (very small circuits). In order to overstep these limitations, several approximated solutions have been developed in the literature: these approaches generally calculate the current transients induced by single events (SETs) in devices and circuits by decoupling the radiation-induced charge transport from the electrostatic problem. The so-called diffusion-collection model is one of the most interesting and physically accurate approaches developed for this purpose [4– 6]. This model makes the assumption that the main carrier driving mechanisms is a pure spherical ambipolar diffusion in the semiconductor region. Moreover, the model can be easily implemented since an ionizing particle track can be divided into a series of discrete punctual charges to compute (or to analytically derive) single or multiple node charge collection. Although interesting improvements have been made [6, 7] and in spite of its capability to be parallelized [8], the diffusion collection model does not correctly take into account the electric field effects in the vicinity of the collecting contact, which is a serious intrinsic limitation. To overcome this drawback, the diffusion coefficient D (i.e. the mobility μ) and the carrier collection velocity v are generally fine-tuned on results obtained from numerical simulation (TCAD) in order to fit the transients for a series of typical radiation events. But, with this procedure the value of D (or μ) does not quantify the real diffusion process (the diffusion will be overestimated or underestimated), since, it artificially reproduces the combination of both the diffusion process and the collection by a reverse-biased junction. Another limitation of this approach is its inability to take into account a non-constant electric field or the nodes bias changes during the collection process.

This chapter presents a new computational method for accurately modeling single-event transient current and charge collection at circuit level. This approach is called random-walk drift-diffusion (RWDD) [9, 10] and consists in a fast Monte Carlo particle method based on a random-walk process [11] that includes both the diffusion and the drift of carriers in an electric field that is nonconstant in both space and time. This method has been successfully used in previous works, as shown in Refs. [12, 13]. In the present work, we considerably improve this approach by a series of developments that make it capable of fast and intensive simulation of full circuits with an accuracy degree similar to that of the TCAD simulation. The considered improvements are briefly described below, but they will be extensively detailed in the following sections. Firstly, the method models the charge carrier transport and collection process using exclusively physical parameters (for example for the carrier mobility or the minority carrier lifetime) and physical constants. In particular, transport and collection processes are modeled by the same mathematical equations, without any fitting parameter. Secondly, the new method implements a model fully derived in C++ environment, which authorizes the use of powerful capabilities such as advanced structures (classes and containers) to model the different problems to solve (geometry, particle track, charge transport, collection, recombination and extraction). Finally, the third important improvement concerns the intrinsic ability of the model to be parallelized on graphic unit processors (GPUs) for charge carrier transport and dynamically coupled with SPICE to take into account the impact of charge collection on biased nodes not considered as stand-alone contacts but embedded in a circuit and connected to other nodes.

**1. Introduction**

116 Modeling and Simulation in Engineering Sciences

changes during the collection process.

Natural radiation at ground level, including both terrestrial cosmic rays and telluric radioac‐ tivity (alpha decay from radioactive ultra-traces in materials), is considered today as a major reliability issue for integrated circuits (IC), since they are increasingly sensitive to this radia‐ tion as long as CMOS technologies scale down [1]. The basic mechanism taking place when an ionizing particle crosses a circuit is the generation in the semiconductor region of a highdensitychargetrack;thegeneratedchargecanbecollectedatcircuitlevelthroughbiasedcontacts andespeciallyreverse-biaseddrainjunctionsandcreateparasitic transient currentsatthe circuit nodes. In the case of a memory cell, the collected charge may be sufficient to induce the cell upset; this phenomenon is called single event upset (SEU). When combinational logic is concerned, the charge induced by the particle may lead to a single event transient (SET) [2]. Researchers have always used modeling and simulation approaches to better understand the physical processes and to predict the consequences on the circuit operation of such single events. The study of these phenomena, and more precisely, the computation of the electrical response of devices and circuits submitted to ionizing radiation imply the accurate modeling of various physical mechanisms at different scales, as charge generation, transport and collection of charges within the circuit. The solutions developed to solve this kind of problem include full numerical methods (such as TCAD) and approximate analytical models used in circuit simulators [3]. Full numerical methods, such as TCAD, exactly solve this coupled problem (radiation + electrostatics + transport) and offer a very accurate solution. But the huge computation time and computer resources needed to solve this high complexity problem restrict its application to small simulation domains, limited to, at most, several devices (very small circuits). In order to overstep these limitations, several approximated solutions have been developed in the literature: these approaches generally calculate the current transients induced by single events (SETs) in devices and circuits by decoupling the radiation-induced charge transport from the electrostatic problem. The so-called diffusion-collection model is one of the most interesting and physically accurate approaches developed for this purpose [4– 6]. This model makes the assumption that the main carrier driving mechanisms is a pure spherical ambipolar diffusion in the semiconductor region. Moreover, the model can be easily implemented since an ionizing particle track can be divided into a series of discrete punctual charges to compute (or to analytically derive) single or multiple node charge collection. Although interesting improvements have been made [6, 7] and in spite of its capability to be parallelized [8], the diffusion collection model does not correctly take into account the electric field effects in the vicinity of the collecting contact, which is a serious intrinsic limitation. To overcome this drawback, the diffusion coefficient D (i.e. the mobility μ) and the carrier collection velocity v are generally fine-tuned on results obtained from numerical simulation (TCAD) in order to fit the transients for a series of typical radiation events. But, with this procedure the value of D (or μ) does not quantify the real diffusion process (the diffusion will be overestimated or underestimated), since, it artificially reproduces the combination of both the diffusion process and the collection by a reverse-biased junction. Another limitation of this approach is its inability to take into account a non-constant electric field or the nodes bias

This chapter describes in-depth the RWDD approach and illustrates all of these points in the three following sections. Section 2 details the physical insights of the model. The practical implementation of the method using an object-oriented programming language and its parallelization on GPUs is presented in Section 3. Besides, the capability of the approach to treat multiple-node charge collection is presented. Section 4 details the coupling of the model either with an internal routine or with SPICE program for circuit solving. Finally, Section 5 illustrates this proposed approach at device and circuit-level, considering four different test vehicles in 65 nm technologies: a stand-alone transistor, a CMOS inverter, a SRAM cell and a flip-flop circuit. RWDD results are compared with data obtained from a full three-dimensional (3D) numerical approach (TCAD simulations) at transistor level. The importance of the circuit feedback on the charge-collection process is also demonstrated for devices connected to others.
