Semiconductor Theory

Elements can generally be classified into three categories based on the energy gap between the valency band and the conduction band. This gap determines the amount of energy required to move an electron from the valency band to the conduction band. An important note is that electrons cannot be found within the gap. In an insulator, the gap is so large that this type of element is unable to conduct. In a conductor, the gap would be nonexistent, meaning that it is highly conductive. A semiconductor has a bandgap size that falls between insulators and conductors (Shenoy).

Intrinsic semiconductors are composed of one pure element, while extrinsic semiconductors have been doped with an element with either one more or one less valence electron than the main element. A p-type semiconductor is a semiconductor doped with an element of one less valence electron, which increases the number of positive holes. An n-type semiconductor is a semiconductor doped with an element of one more valence electron, which increases the number of free electrons. If a p-type and n-type touch each other, a p-n junction is formed. Positive holes from the p-side and electrons from the n-side diffuse towards the junction. When they contact each other, they cancel each other, creating a depletion region. The depletion region reveals the dopant ions, creating an electric field and a voltage difference across the depletion region called the barrier voltage (Electronics I Chapter 3.3.2 42).

There are two equations for the electric field, one for the p-side and one for the n-side. These equations are the following: E(x)=(-(eN_A)/(ε_0 ε_r ))(x+w_p ) for -w_p<x<0 and E(x)=((eN_D)/(ε_0 ε_r ))(x-w_n ) for 0<x<w_n, where E(x) is the electric field, e is the elementary charge, w_p and w_n are the widths of the depletion region for the p-side and n-side respectively, N_A and N_D are the concentration of doping elements for the p-side and n-side respectively, and ε_0 and ε_r are the permittivity of free space and relative permittivity of the semiconductor respectively. There are also two voltage equations for the p- and n-side, which are as follows: V(x)=((eN_A)/(2ε_0 ε_r )) (x+w_p )^2 for -w_p<x<0 and V(x)=V_0-((eN_D)/(2ε_0 ε_r )) (x-w_n )^2 for 0<x<w_n. V_0 from the last equation is the built-in potential and is given by this equation: V_0=(e/(2ε_0 ε_r ))(w_0^2)((N_A N_D)/(N_A+N_D )) where w_0 is the width of the total depletion region or the sum of the p-side and n-side depletion regions (Lecture 10).

Citations

Electronics I Chapter 3 Lecture

Lecture 10: PN Junctions in Equilibrium

Shenoy, M. (Writer). (n.d.). Conductors, Insulators, and Semiconductors [Video file]. Retrieved June 08, 2020, from https://www.khanacademy.org/science/in-in-class-12th-physics-india/in-in-semiconductors/in-in-band-theory-of- solids/v/conductors-insulators-and-semiconductors-class-12-india-physics-khan-academy