Extrinsic semiconductor formula

Explore the extrinsic semiconductor formula, its role in n-type & p-type semiconductors, and an example calculation for silicon-based materials.

Understanding the Extrinsic Semiconductor Formula

Extrinsic semiconductors play a crucial role in modern electronic devices, enabling the flow of electrical current through controlled pathways. The extrinsic semiconductor formula, derived from the law of mass action, is essential for determining the properties of doped semiconductors. This article delves into the equation and its implications in the field of semiconductor technology.

Extrinsic Semiconductors: A Brief Overview

Extrinsic semiconductors are created by introducing impurities or dopants into an intrinsic semiconductor, typically composed of a pure element like silicon or germanium. Depending on the type of dopant used, extrinsic semiconductors can be classified as n-type or p-type:

• n-type: Formed by introducing donor atoms that provide extra electrons to the semiconductor lattice, resulting in a majority of negative charge carriers (electrons).
• p-type: Formed by introducing acceptor atoms that create holes in the lattice, resulting in a majority of positive charge carriers (holes).

The extrinsic semiconductor formula allows us to calculate the concentration of electrons and holes in a doped semiconductor, providing valuable information for designing and optimizing electronic devices.

The Extrinsic Semiconductor Formula

The extrinsic semiconductor formula is derived from the law of mass action, which states that the product of the electron and hole concentrations remains constant at a given temperature for an intrinsic semiconductor. Mathematically, the law of mass action can be expressed as:

n_ip_i = n_ep_e

Where:

1. n_i is the intrinsic electron concentration
2. p_i is the intrinsic hole concentration
3. n_e is the extrinsic electron concentration
4. p_e is the extrinsic hole concentration

For n-type and p-type extrinsic semiconductors, the extrinsic electron and hole concentrations can be calculated using the doping concentration (N_D for n-type, N_A for p-type) as follows:

n-type: n_e ≈ N_D, p_e = (n_i²)/N_D

p-type: p_e ≈ N_A, n_e = (n_i²)/N_A

These formulas are instrumental in determining the behavior of extrinsic semiconductors and designing efficient electronic devices.

Conclusion

The extrinsic semiconductor formula is a fundamental tool in understanding the properties of doped semiconductors. By calculating electron and hole concentrations, engineers can optimize electronic devices for improved performance and efficiency. As semiconductor technology continues to advance, the extrinsic semiconductor formula will remain an essential piece of knowledge for professionals in the field.

Example of Extrinsic Semiconductor Calculation

Let’s consider a silicon-based n-type extrinsic semiconductor with a doping concentration of 10¹⁶ donor atoms per cm³ at room temperature (300 K). To calculate the extrinsic electron and hole concentrations, we first need to determine the intrinsic electron concentration (n_i) for silicon at room temperature.

For silicon, the intrinsic carrier concentration (n_i) can be found using the following equation:

n_i = B_TT^3/2exp(-E_g/2k_BT)

Where:

• B_T is a temperature-dependent constant
• T is the temperature in Kelvin
• E_g is the energy bandgap of silicon (1.1 eV)
• k_B is Boltzmann’s constant (8.617 x 10^-5 eV/K)

For silicon at room temperature, B_T ≈ 5.23 x 10¹⁵ cm^-3K^-3/2, and T = 300 K. Using the above equation, we can calculate n_i ≈ 1.5 x 10¹⁰ cm^-3.

Now, we can use the extrinsic semiconductor formula for n-type semiconductors to calculate the extrinsic electron and hole concentrations:

n_e ≈ N_D = 10¹⁶ cm^-3

p_e = (n_i²)/N_D = (1.5 x 10¹⁰)²/(10¹⁶) ≈ 2.25 x 10⁴ cm^-3

Thus, the extrinsic electron concentration (n_e) is approximately 10¹⁶ cm^-3, and the extrinsic hole concentration (p_e) is approximately 2.25 x 10⁴ cm^-3 for this n-type silicon semiconductor at room temperature.