Semiclassical Transport Theory of Charge Carriers, Part I: Microscopic Approaches

1. Overview And Chapter Objectives

The semiclassical approach is suitable for semiconductor devices whose feature dimensions are much greater than the mean free path between collisions (L >> λ_e). In silicon devices, where the electron mean free path λ_e ~100Ǻ, this approach can be safely used for devices having feature size L>0.1 μm. It helps us to calculate the main transport parameters (such as the charge carrier mobility, conductivity, etc.) and so to determine the electrical, thermal, magnetic and optical characteristics of a given semiconductor device.

Figure 1.

Semiclassical approaches for charge carrier transport

In classical statistical mechanics, the state of a particle is completely defined at a certain time t as a point in a space of six coordinates, called the phase-space (sometimes called the μ-space). The six coordinates of the phase-space are the position and momentum coordinates (x, y, z, p_x, p_y, p_z) as shown in Figure 2. Physically, it is not possible to do so in large systems, where the number of particles is in the order of 10²² particle per cm. Then one has to use the concept of probability or the distribution function of the system under certain conditions, like the condition of thermal equilibrium¹. Therefore, we usually speak about the so-called distribution function, which expresses the average number of particles at a certain position in the phase-space. The distribution function is often written in either of the following forms:f(x,p, t) or f(x,v, t) or f(x,k, t)where:

Evidently the momentum p and velocity v of a classical particle are interrelated by: p= mv. Also, the momentum of a semiclassical particle is related to the wave vector k by the De Broglie relation: p= h/λ = ħk. Note that, we may indifferently use v, k or p, as second argument of the carrier distribution function, in this book.

Figure 2.

Schematic illustration of a fiction distribution functions in the phase-space (x=x,y,z, and p=p_x,p_y,p_z) at a certain moment t = t_o

The semiclassical transport theory is based on the ensemble concept and the Boltzmann transport equation (BTE), which describes the evolution of the distribution function f(x, k, t) of a gas of particles under nonequilibrium conditions. The ensemble concept shows that a macroscopic observation is consistent with a very large number of microscopic configurations (particles), which are represented by points in the phase space. The ensemble therefore is basically represented by a normalized density distribution function f(x, k, t) in the phase-space.