Fermi-Dirac Distribution Plotter
Fermi-Dirac Distribution Plotter visualizes fermion occupancy probability vs. energy for given temperature and Fermi energy, aiding solid-state physics studies.
Formulas Used in Fermi-Dirac Distribution Plotter
The plotter uses the following formula for the Fermi-Dirac distribution:
Fermi-Dirac Distribution:
\\[ f(E) = \frac{1}{1 + e^{(E – E_F)/(k_B T)}} \\]Where:
- \\( f(E) \\): Occupancy probability (0 to 1)
- \\( E \\): Energy (eV)
- \\( E_F \\): Fermi energy (eV)
- \\( k_B \\): Boltzmann constant (\\( 8.617333262 \times 10^{-5} \, \text{eV/K} \\))
- \\( T \\): Temperature (K)
Example Calculations
Example 1: Room Temperature
Input: Temperature = 300 K, Fermi Energy = 5 eV, Energy Range = 0.5 eV
\\[
k_B T = 8.617333262 \times 10^{-5} \cdot 300 \approx 0.02585 \, \text{eV}
\\]
\\[
f(E_F = 5) = \frac{1}{1 + e^{(5 – 5)/(0.02585)}} = \frac{1}{1 + e^0} = 0.5
\\]
Result: Occupancy at E_F: 0.5, Plot Range: [4.5, 5.5] eV
Example 2: Low Temperature
Input: Temperature = 10 K, Fermi Energy = 5 eV, Energy Range = 0.5 eV
\\[
k_B T = 8.617333262 \times 10^{-5} \cdot 10 \approx 0.0008617 \, \text{eV}
\\]
\\[
f(E_F = 5) = \frac{1}{1 + e^{(5 – 5)/(0.0008617)}} = \frac{1}{1 + e^0} = 0.5
\\]
Result: Occupancy at E_F: 0.5, Plot Range: [4.5, 5.5] eV
Example 3: High Fermi Energy
Input: Temperature = 300 K, Fermi Energy = 8 eV, Energy Range = 1 eV
\\[
k_B T = 8.617333262 \times 10^{-5} \cdot 300 \approx 0.02585 \, \text{eV}
\\]
\\[
f(E_F = 8) = \frac{1}{1 + e^{(8 – 8)/(0.02585)}} = \frac{1}{1 + e^0} = 0.5
\\]
Result: Occupancy at E_F: 0.5, Plot Range: [7, 9] eV