Semiclassical Transport Theory of Charge Carriers, Part II: Macroscopic Approaches

Semiclassical Transport Theory of Charge Carriers, Part II: Macroscopic Approaches

DOI: 10.4018/978-1-5225-2312-3.ch003
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1. Overview And Chapter Objectives

During the last two decades, a considerable interest has been devoted for the study of high-field transport because of the scaled-down dimensions of devices. The scaled-down dimensions imply the existence of high electric fields. With high electric field, the velocity-field characteristics become non-linear and the velocity eventually saturates at high applied fields. The nonlinear response of the carrier velocity to a high electric field has been extensively explored, both theoretically and experimentally. As we have pointed out earlier, the high-field transport of charge carriers can be described within the framework of Boltzmann transport equation (BTE). However, for large perturbations due to high fields, the collision term, in the BTE, cannot be linearized using a simple microscopic relaxation time. Also, the carrier effective mass concept does not hold valid as long as the inter-valley and interband transitions are concerned, which are common at high fields. Therefore, several other methods have been explored to solve the BTE, by the first order Chapmanp-Enskog (C-E) expansion or the spherical harmonic expansion (SHE) or the Monte Carlo (MC) method or the hydrodynamic (moment) method.

Upon completion of this Chapter, the readers and students will be able to:

  • Be acquainted with the treatment of non-linear transport with hydrodynamic moments of the BTE,

  • Understand the hydrodynamic model (HDM) of semiconductors and its ability to address the non-local transport effects

  • Understand the energy-dependent transport parameters and their underlying physics and where and how they are used in TCAD simulation programs.


2. Hydrodynamic Model (Hdm)

The hydrodynamic moment method was proposed by Grad (1949) as an alternative approach to solve the Boltzmann transport equation (BTE) in the study of aero-dynamic flow. The hydrodynamic model (HDM), which is based on the moment method, was suggested by Bløtekjaer (1970) to study the hot electron transport in semiconductor devices. Unfortunately, Blotekjaer’s formulation, upon which most hydrodynamic models are based, made use of the heated Maxwellian distribution to calculate the collision terms. This assumption is not valid at high electric fields (in hot carrier nonlinear regime). Since then, several contributions have been introduced to improve the hot carrier transport models in semiconductors. For instance, Shur and Eastman (1979) could describe the velocity overshoot of hot carriers by simple conservation equation.

The so-called energy-transport model (ETM), which is a reduced version of the HDM with no convection terms, was introduced by Cook and Frey (1982). The ETM is actually based on the early work of Stratton (1962,1972), about transport of hot and cold carriers. The Stratton model and variant ETM’s usually make use of a priori distribution functions to calculate the collision terms and transport parameters.

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