Intrinsic Carrier Concentration Calculator
Free calculate intrinsic carrier concentration (n_i) for silicon at any temperature. Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Typical: 300 K (27°C) for room temperature
Results
Enter temperature to calculate
What is Intrinsic Carrier Concentration?
Intrinsic carrier concentration (n_i) is the number of free electrons and holes per unit volume in a pure, undoped semiconductor at thermal equilibrium. It represents the natural concentration of charge carriers due to thermal excitation.
In silicon, n_i is highly temperature-dependent. At room temperature (300 K), n_i ≈ 1.45 × 10¹⁰ cm⁻³. As temperature increases, more electron-hole pairs are generated, dramatically increasing n_i.
This parameter is fundamental in semiconductor device physics, determining the conductivity of intrinsic material and influencing the behavior of doped semiconductors, especially at high temperatures.
Intrinsic Carrier Concentration Formula
Where:
- • n_i = Intrinsic carrier concentration (cm⁻³)
- • B = Material constant ≈ 5.29 × 10¹⁹ cm⁻³·K^(-3/2) for Si
- • T = Absolute temperature (K)
- • E_g = Bandgap energy (eV), temperature-dependent
- • k_B = Boltzmann constant = 8.617 × 10⁻⁵ eV/K
Temperature-dependent bandgap:
E_g(T) = 1.166 - (4.73×10⁻⁴ × T²)/(T + 636) eV
How to Calculate
-
1
Convert temperature to Kelvin
If given in °C, add 273.15. If in °F, convert via K = (°F + 459.67) × 5/9.
-
2
Calculate temperature-dependent bandgap
Use E_g(T) = 1.166 - (4.73×10⁻⁴ × T²)/(T + 636) eV for silicon.
-
3
Apply the formula
Calculate n_i = B × T^(3/2) × exp(-E_g/(2k_B T)) with B = 5.29×10¹⁹.
Practical Examples
Example 1: Room Temperature (300 K)
Calculate n_i for silicon at 300 K (27°C).
Solution:
E_g(300) = 1.166 - (4.73×10⁻⁴ × 300²)/(300 + 636) ≈ 1.124 eV
n_i = 5.29×10¹⁹ × 300^(3/2) × exp(-1.124/(2×8.617×10⁻⁵×300))
n_i ≈ 1.45 × 10¹⁰ cm⁻³
Example 2: High Temperature (400 K)
Calculate n_i at 400 K (127°C).
Solution:
E_g(400) ≈ 1.102 eV
n_i ≈ 1.38 × 10¹² cm⁻³ (much higher!)
Note: n_i increases exponentially with temperature
Applications
Device Modeling
Understanding carrier concentrations in intrinsic regions, calculating junction properties, and modeling device behavior.
Temperature Effects
Analyzing how device characteristics change with temperature, especially leakage currents and breakdown voltages.
Material Science
Characterizing semiconductor materials, understanding intrinsic conductivity, and material quality assessment.
Education
Teaching semiconductor physics, understanding thermal generation, and learning about intrinsic semiconductors.
Frequently Asked Questions
Why is n_i temperature-dependent?
Thermal energy creates electron-hole pairs by breaking covalent bonds. Higher temperature means more thermal energy, exponentially increasing the number of generated carriers.
What is the typical value at room temperature?
For silicon at 300 K, n_i ≈ 1.45 × 10¹⁰ cm⁻³. This is relatively small compared to typical doping concentrations (10¹⁵-10²⁰ cm⁻³), so intrinsic effects are usually negligible in doped devices.
How does n_i affect device operation?
At high temperatures, n_i can become comparable to doping levels, causing devices to lose their intended characteristics. This limits maximum operating temperatures.
Is n_i the same for all semiconductors?
No! Different materials have different bandgaps and material constants. Silicon has n_i ≈ 10¹⁰ cm⁻³ at 300K, while GaAs has n_i ≈ 10⁶ cm⁻³ (much lower due to larger bandgap).
What happens when n_i exceeds doping concentration?
The material becomes "intrinsic" - the natural carrier concentration dominates over doping. This typically occurs at very high temperatures and causes device failure.