Momentum-Based Transport Models

The MC model is very CPU intensive and simpler but more CPU efficient models based on balance equations have been developed. The most widely used momentum-based models are the drift-diffusion (DD) and the hydrodynamic (HD) models [7.1–7.13]. Both models a

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Momentum-Based Transport Models

The MC model is very CPU intensive and simpler but more CPU efficient models based on balance equations have been developed. The most widely used momentum-based models are the drift-diffusion (DD) and the hydrodynamic (HD) models [7.1-7.13). Both models are derived by applying different degrees of approximation to balance equations of the type (2.39). The DD model is the simplest momentum-based model and consists of balance equations for the particle and current densities. Thus, only the first two moments of the distribution function are calculated instead of the full distribution function and a large fraction of the information contained in the distribution function is lost. On the other hand, the dimensionality of the problem is reduced by the integration of the k-space and the CPU efficiency is improved by orders of magnitude. In the case of the HD model the first four moments are considered including the particle density, current density, particle gas temperature, and the energy current density. This already enables the simulation of nonlocal effects, like the velocity overshoot, which have a strong impact on the device behavior of modern deep sub-micron devices. In the following, the derivation of the HD model for large-signal (Sec. 7.1), small-signal (Sec. 7.2), and noise simulations (Sec. 7.3) is discussed in detail. Since the DD model is a straight forward approximation of the HD model, its derivation is described in less detail (Sec. 7.4). For the DD model on the other hand the proof of the Nyquist theorem is given. Transport and noise parameters for both models are discussed in the last section.

7.1 The Hydrodynamic Model The generalized HD model of Refs. [7.12,7.13) extended to the case of a positiondependent band structure is used in this work, because it is applicable in the case of a nonparabolic band structure. First the HD model for holes is derived and then the results are given for the electron model. In the following the notation for the hole density is p(r, t) and for the electron C. Jungemann et al., Hierarchical Device Simulation © Springer-Verlag Wien 2003

7.1 The Hydrodynamic Model

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density n(r, t). Hole quantities are labeled by the subscript p and electron quantities by n. The HD model is based on balance equations of the type (2.39) for the four microscopic quantities X = 1, v, c, and CV. The corresponding macroscopic densities are for holes p

P (l)k ,

jp

p (v)k ,

(7.1) (7.2)

wp

p (c)k ,

(7.3)

p (cv)k .

(7.4)

sp

With Eq. (2.39) and X = 1 the hole continuity equation is obtained without approximations (7.5)

where GIl is the generation rate due to II. The balance equation for the current density (2.41) is derived with the macroscopic relaxation time approximation (2.42) for X = v

In order to simplify this equation rather stringent approximations are used. The nearly negligible forces due to the position-dependent band structure are calculated with the assumption that the velocity can be separated into position- and wave vector dependent

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