Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

Chapter 4 The HCD Model Based on BTE Solution

As discussed in previous sections, carrier transport is the key to modeling HCD. The better the carrier transport analysis, the more precise the HCD model will be. Many approaches to solve the carier transport and obtain the carrier energy distribution function have been proposed. Some of these methods are more rigorous, while others are simplistic and less accurate. Ever since the shift of paradigm in hot-carrier degradation modeling from field driven approaches [150] to the energy driven concept [9, 11], the evaluation of the carrier distribution function has become a vital component. This has also been suggested in recent HCD modeling results where the entire Si-H bond breakage process, which is assumed to be the dominant contributor to HCD in devices, is described by the carrier energy distribution function [7, 8]. Ideally, within the semiclassical regime, the DF is obtained by solving the Boltzmann transport equation. Although both the Monte-Carlo approach and the spherical harmonics expansion method provide a solution to the BTE, the latter technique has a number of advantages: it allows to better resolve the high-energy tails of the DF (which are of crucial importance in the context of HCD), simulate long-channel devices under high operating/stress voltages in a reasonable time, and implement electron-electron scattering in a more straightforward manner and with less computational burden. Apart from these approaches, simplified techniques of carrier transport treatment such as the drift-diffusion and energy transport scheme are highly popular [33, 30]. Inspite of its limitations, the drift-diffusion method has been effective for the description of carrier transport even in modern devices [172].

In the following, a simple and efficient model for the carrier distribution function is derived. This approach to approximately calculate the DF is based on the drift-diffusion scheme where an analytical expression is used to mimic the carrier energy distribution function. To verify the obtained DF, a solution of the Boltzmann transport equation is used as a reference. The analytical method for calculating the DF is coupled with a physics-based model for hot-carrier degradation. The analytical approach for the DF is consistent with the HCD model as it contains terms for both hot and cold carriers which lead to the single- and multiple-carrier processes. We use two versions of the HCD model: one is based on more rigorous deterministic approach and is used for reference or verification, while the other one employs the DD-based concept. In the following chapters, both versions of the model are compared against experimental data for several devices and a conclusion on the validity of the DD-based version of the HCD model is drawn suggesting its efficiency for predictive HCD simulations under certain conditions.

4.1 Carrier Transport

The operation of semiconductor devices is based on the reaction of electrons and holes to the local and externally applied electrical forces. These forces are due to the potentials either existing or created inside the semiconducting material due to: (i) applied bias, (ii) depletion regions formed by immobile ionized impurities at thermal equilibrium (built-in potential), and (iii) other charge carriers or impurities (scattering potentials). Since a typical semiconductor device includes a large concentration of charge carriers and artificially introduced impurities or dopants, this many-particle system is difficult to model. Various models for carrier transport have been proposed with different levels of refinements and complexity that are applicable based on the device dimensions, usage and computational budget. Although it is attractive to choose the most fundamental concepts, they typically have high computational demands and for most engineering applications a more approximate but faster and computationally cheaper solution is acceptable.

This section provides an overview of the most important carrier transport models developed for semiconductor devices and discusses their efficiency and limitations. The carrier transport models can be categorized based on the device dimensions and characteristic lengths – de Broglie wavelength, mean free path, and phase relaxation length [173]. For an electron in silicon at room temperature, the de Broglie wavelength is approximately 8 nm, while the mean free path and the phase relaxation length are between 5 to 10 nm. If the device dimensions are much larger than the characteristic lengths, carrier transport is governed largely by scattering and the transport is called diffusive. Semi-classical models are quite accurate in this regime to describe the motion of charge carriers. However, in nanoscale devices, smaller than the characteristic lengths, quantum effects come into play and scattering becomes less important. Thus quantum transport models are required for an accurate description of the electrostatics in the device.

In this section, the semi-classical transport models are discussed which are most relevant for HCD modeling. The Boltzmann transport equation is the backbone of semi-classical transport. The different approaches to the BTE solution starting from the Monte-Carlo method, the spherical harmonics expansion, and the techniques based on the BTE moments, i.e. the drift-diffusion and energy transport schemes, see Figure 4.2 are described in order to understand the applicability of each of these methods.