Intrinsic Carrier Concentration Calculator
Calculate intrinsic carrier concentration for semiconductors
Room temperature is approximately 300 K
How to Use This Calculator
Select Material
Choose Silicon, Germanium, or Custom material. For Silicon and Germanium, material constants are automatically loaded. For custom materials, you'll need to enter material properties.
Enter Custom Parameters (if applicable)
For custom materials, input the effective density of states in conduction band (Nc), valence band (Nv), and bandgap energy (Eg) in eV. These are material-specific constants.
Enter Temperature
Input the temperature in Kelvin. Room temperature is 300 K. The intrinsic carrier concentration increases exponentially with temperature.
Calculate and Interpret
The calculator provides the intrinsic carrier concentration, which represents the number of free electrons (and holes) per cubic meter in an undoped semiconductor at thermal equilibrium.
Formula
n_i = √(N_c × N_v) × exp(-E_g / (2 × k_B × T))
Intrinsic Carrier Concentration
Where:
- n_i = Intrinsic carrier concentration (m⁻³)
- N_c = Effective density of states in conduction band (m⁻³)
- N_v = Effective density of states in valence band (m⁻³)
- E_g = Bandgap energy (eV)
- k_B = Boltzmann constant = 8.617 × 10⁻⁵ eV/K
- T = Temperature (Kelvin)
Example (Silicon at 300K):
N_c = 2.8 × 10²⁵ m⁻³
N_v = 1.04 × 10²⁵ m⁻³
E_g = 1.12 eV
T = 300 K
n_i = √(2.8×10²⁵ × 1.04×10²⁵) × exp(-1.12 / (2 × 8.617×10⁻⁵ × 300))
n_i ≈ 1.5 × 10¹⁶ m⁻³
About Intrinsic Carrier Concentration Calculator
The Intrinsic Carrier Concentration Calculator is a specialized tool for semiconductor physics that calculates the number of free charge carriers (electrons and holes) in an undoped (intrinsic) semiconductor at thermal equilibrium. This is a fundamental parameter in semiconductor device design and analysis.
When to Use This Calculator
- Semiconductor Device Design: Determine carrier concentrations for device modeling
- Temperature Analysis: Study how carrier concentration varies with temperature
- Material Characterization: Understand intrinsic properties of semiconductor materials
- Research & Development: Analyze new semiconductor materials and their properties
- Educational Purposes: Learn semiconductor physics and carrier statistics
Why Use Our Calculator?
- ✅ Precise Calculations: Accurate exponential calculations for carrier concentration
- ✅ Multiple Materials: Built-in constants for Silicon and Germanium
- ✅ Custom Materials: Support for any semiconductor material with known properties
- ✅ Temperature Dependent: Shows how carrier concentration changes with temperature
- ✅ Educational Tool: Understand the physics behind semiconductor carrier statistics
Key Concepts
Intrinsic Semiconductor: A pure semiconductor with no intentional dopants. At thermal equilibrium, the number of electrons in the conduction band equals the number of holes in the valence band, and this concentration is the intrinsic carrier concentration (n_i).
Temperature Dependence: Intrinsic carrier concentration increases exponentially with temperature because thermal energy helps electrons overcome the bandgap. This is why semiconductor devices have temperature limits.
Typical Values
- Silicon at 300K: n_i ≈ 1.5 × 10¹⁶ m⁻³ (1.5 × 10¹⁰ cm⁻³)
- Germanium at 300K: n_i ≈ 2.4 × 10¹⁹ m⁻³ (2.4 × 10¹³ cm⁻³)
- Gallium Arsenide at 300K: n_i ≈ 1.8 × 10¹² m⁻³ (1.8 × 10⁶ cm⁻³)
Frequently Asked Questions
What is intrinsic carrier concentration?
Intrinsic carrier concentration (n_i) is the number of free electrons (and holes) per unit volume in an undoped semiconductor at thermal equilibrium. It represents the natural carrier density due to thermal generation of electron-hole pairs across the bandgap.
Why does carrier concentration increase with temperature?
As temperature increases, more electrons gain enough thermal energy to jump from the valence band to the conduction band, creating electron-hole pairs. This exponential increase is described by the exp(-E_g/(2k_B×T)) term in the formula.
How does doping affect carrier concentration?
Doping introduces additional carriers: n-type doping adds electrons, p-type doping adds holes. In doped semiconductors, the carrier concentration is much higher than n_i and is determined by the doping concentration, not the intrinsic concentration (except at very high temperatures).
What are typical values for Nc and Nv?
For Silicon at 300K: Nc ≈ 2.8 × 10²⁵ m⁻³ and Nv ≈ 1.04 × 10²⁵ m⁻³. For Germanium: Nc ≈ 1.04 × 10²⁵ m⁻³ and Nv ≈ 6.0 × 10²⁴ m⁻³. These values depend on effective masses and temperature.
Why is this important for device design?
Intrinsic carrier concentration determines the minimum leakage current in devices and sets limits on device operation at high temperatures. It's also fundamental for understanding pn junctions, where n × p = n_i² at thermal equilibrium (mass-action law).