Only carriers with high energies contribute to the avalanche process. Like all highenergy mechanisms, impactionization is a nonlocal process (compare Section 4.3.1). This leads to problems in classical driftdiffusion simulation environments, because only local quantities are available, while no exact information on the distribution function can be obtained. Also hydrodynamic simulations deliver only the mean energy. Thus, the information on highenergy tails of the carrier distribution function is not available (see Section 4.3.2). As a result, driftdiffusion and/or hydrodynamic schemes require modeling approaches which are only based on local quantities.
In driftdiffusion simulations, the only local quantity that allows conclusions on the carrier temperature and therefore on the impactionization rate is the electric field. Many authors who investigated the ionization coefficients, both, experimentally and theoretically, suggested an exponential relation to the electric field as [11]
An early approach deriving the value of the parameter from the BTE was performed by Baraff [195]. He showed that the different values 1 and 2 used for are the limiting cases for relatively low and high fields. However, there is no closed solution of his approach, but approximations have been presented in various publications, for example, by Crowell et al. [196],
Another localfield model was presented by Lackner [198], who derived an expression which leads to an extension of Chynoweth's law,
(5.15) 
(5.16) 
Slotboom et al. [199] have observed lower impact ionization rates for currents near the surface. As a consequence, models describing the transition between surface and bulk impact ionization have been developed [200]. However, Monte Carlo simulations have shown that there are no or only minor differences between surface and bulk impact ionization rates [201]. This means that there is no physical evidence of different rates near the surface and that those models are based on artifacts resulting from the approximate ionization rates based on the electric field.
In many applications, localfield based models deliver good results. However, from the physical point of view, impactionization is not field dependent. This is especially important in areas of rapidly changing electric fields and in small devices. This weakness can be observed in Fig. 5.4 for a 200nm and a 50nm structure. This figure compares generation rates calculated using different transport schemes. In this example, the electrons are accelerated from left to right. As soon as the electric field is risen, the localfield model predicts the carrier generation rate. However, physically more correct are the Monte Carlo reference simulations. They demonstrate that the carrier energy can follow changes in the electric field with a certain delay. At characteristic lengths shown in that figure, the validity of the localfield approach becomes questionable.
As long as device expansions are well above nm, the advantages of localfield models are the good integrability in driftdiffusion schemes which are the workhorse in device simulation tools. Most TCAD simulators therefore commonly include one or more of the local electric field models. A typical application where driftdiffusion models deliver good results are highvoltage LDMOS transistors. Fig. 5.5 shows some results based on (5.13) and also clearly shows the importance of this physical effect. A more comprehensive example incorporating breakdown and snapback simulations is presented in Section 5.3.
Local modeling approaches do not provide information about where carriers come from and if they have already gained high energy. But since electrons and holes have to be accelerated to gain at least the threshold energy before impactionization can occur, this information is relevant. The area where there is a high electric field but impactionization has not started yet is often called darkspace [202]. This darkspace has to be considered also in experimental extractions of the ionization coefficients. Measuring of the coefficients cannot be performed directly. Therefore, they are commonly extracted out of the multiplication factor using the ionization integral (5.10). While considering the darkspace, the boundaries of the ionization integral are shifted so that only regions where impactionization takes place are included. With this approximation Okuto and Crowell [202] could explain anomalous energies which were found previously for the parameters and in (5.14). They have obtained more realistic energies employing a pseudolocal approximation which is also valid for high fields. In their work, a model using apparent and real ionization coefficients was developed. This technique delivered reasonable results for the given onedimensional examples.
An interesting approach which includes local changes of the electric field is the method presented by Slotboom et al. [179] (compare (4.23)). The required postprocessing which is necessary after a conventional driftdiffusion simulation, makes this method elaborative to implement, especially if selfconsistent solutions are required.
The lucky electron concept [192,203] introduces a threshold energy level The carriers need to surmount this potential barrier in order to trigger impactionization. Hu et al. [203] describes that this energy can be reached if a carrier travels a sufficiently long distance without collisions. In this work by Hu, a compact model is formulated, which relates the substrate current as a consequence of impactionization with the drain current and the maximum electric field in the device. The drain current acts as a source function (the supply of carriers) and the peak field is used together with the (hotcarrier) mean free path to describe the ionization probability. This leads to
A method to utilize the lucky electron model in device simulation and to introduce the nonlocality has been proposed by Meinerzhagen [205]. In this work, the electric field line is followed starting from a point until the electrostatic potential difference of has been reached at a point The length along the field line between and is The generation rate in the point yields
(5.18) 
Another approach for modeling impactionization employs the carrier temperature, i.e. the local mean carrier energy, as the key parameter. The formalism for the rate calculation is very similar to the localfield model. For the transformation of the commonly used local field based models, the electric field can be replaced by the homogenous stationary energy balance equation, compare (4.19), to form [206]. Thus, Chynoweth's law can be transformed from (5.13) to a carrier temperature dependent model. An often used estimation combining all coefficients reads [207]


It seems natural to use the carrier energy in place of the electric field to model the impactionization rate and it is commonly used in the energytransport or hydrodynamic simulation frameworks. Hence, it is possible to approximately consider nonlocal issues like the darkspace phenomenon. However, the carrier temperature alone cannot reflect the existence and strength of highenergy tails, i.e. the amount of high energetic carriers available. Completely different shapes of the distribution function can lead to the same average energy [15]. Additionally, highenergy tails can also exist if the average energies are low (compare Fig. 4.2 and 4.12). Therefore, the carrier temperature is commonly overestimated in small devices, which also leads to an overestimation of the impactionization rate [208,186]. These effects, which are especially relevant for aggressively downscaled devices, can only be considered by incorporating the full distribution function.
Instead of using the electric field, the carrier temperature, hotcarrier subpopulations, or some nonlocal approximations, the most rigorous modeling approach is to directly incorporate the carrier distribution function This allows one to calculate the total generation rate using [187]
The integration boundaries in (5.20) show that only carriers above a certain energy threshold influence the generation rate and highlight that high energies are of vital importance. Approximations of the distribution function based on splitting hot and cold carrier fractions (as shown in (4.32) for the six moments model [134]) provide results in good agreement with the more precise but also very timeconsuming fullband Monte Carlo method (see Fig. 5.4).
In the work of Rauch and La Rosa [211,212], a compact modeling approach for the total substrate current is presented. Although this compact model cannot be used for TCAD, it highlights the importance of the shape of the distribution function. Here, the substrate current of an nMOSFET is formulated as
As shown in Fig. 5.6(a) the maximum of the integrand depends on the slope of the carrier energy distribution and results according to (5.25) in
(5.26) 
(5.27) 
In downscaled devices the carrier energy distribution exhibits a knee near the maximum energy available from the steep potential drop along the pinch off region which is approximated as [211]. Due to the constantly rising scattering rate and the abrupt decrease of the carrier distribution function near the knee point the maximum of the integrand in (5.23) at coincides with the knee near (see Fig. 5.6(b))
In this case the dominant energy is determined by the knee of the distribution function, which by itself depends on the bias conditions and therefore on the available energy [215]. Further influences of voltage changes seam not to shift this peak level. Hence, the final rate is proposed to be
(5.29) 

This particular idea of this compact model is to combine the two regimes in one model. It herby highlights the necessity of new modeling paradigms when proceeding to small, downscaled devices. The validity of both regimes has been shown in [211].