Module 6 part A4

Unipolar Transport of Electron Holes in Semiconductors

Unbipolar transport refers to the movement of electron holes in semiconductors as charge carriers. When an external electric field is applied to a semiconductor, it exerts forces on the charge carriers: positive charges move in the direction of the electric field, while negative charges (electrons) move in the opposite direction. As a result, positive charges accumulate on one side of the semiconductor, and negative charges on the other side, creating an internal electric field that opposes the external field.

Movement of Charge Carriers

The emergence of the internal electric field causes electrons and holes to interact more closely. Specifically, the holes will drift toward electrons, and the electrons will drift toward holes, resulting in a phenomenon known as bipolar transport where both charge carriers move together.

Equations and Internal Electric Field Calculation

The internal electric field can be calculated using the Poisson equation, which considers the charge of the carriers and the difference in the concentration of excess carriers over the permittivity of the semiconductor. When excess carriers are generated, such as through light excitation or the application of an electric field, electrons and holes are typically produced in pairs.

Due to the principle of quasi charge neutrality, the concentrations of excess electrons ( ( ext{delta } n )) and excess holes ( ( ext{delta } p )) remain relatively close. However, even slight variations in these concentrations can lead to a sufficiently pronounced internal electric field that influences the movement of carriers, causing them to drift together.

Generation and Recombination Rates

As both charge carriers move together, it is critical that their generation rates ( ( g_n ) for electrons and ( g_p ) for holes) and recombination rates remain equal. This maintenance of symmetry is essential to uphold quasi neutrality within the semiconductor.

Transport Equations for Co-moving Carriers

To derive transport equations for the co-moving electrons and holes, we refer back to the time-dependent diffusion equations previously discussed. These equations, which account for excess carrier changes over time, harmonize the processes of diffusion, generation, and recombination. By substituting the respective concentrations and mobility values for both types of carriers into the diffusion equations and simplifying, we ultimately arrive at expressions for the effective diffusion coefficients and mobilities for electron holes.

Type-Specific Behavior

In semiconductors where there is a predominance of either electron or hole carriers (i.e., p-type or n-type materials), we see pronounced behaviors. For p-type materials, the electron concentration is significantly less than that of holes, and conversely for n-type materials. This characteristic division allows distinct formulations for analyzing the transport equations for minority carriers in each type of semiconductor.

Steady State Conditions

Various conditions can simplify the analysis of unipolar transport equations. In steady-state conditions, the concentrations of carriers do not change over time ( ( rac{d ext{delta } n}{dt} = 0 )). Other scenarios, such as a uniform distribution of excess carriers, the absence of external electric fields, zero generation rates, or infinite carrier lifetimes, may lead to further simplifications.

Application of Conditions to Solutions

These established conditions can help us effectively apply and solve unipolar transport equations for different semiconductors. In upcoming supplementary notes, we will explore practical examples demonstrating how to leverage these conditions, through specific scenarios in unipolar transport systems and their equations.