Advanced Fermi Level Position Calculator

Calculator Inputs

Material label

Optional label for your report and chart.

Temperature (K)

Typical room-temperature value is 300 K.

Band gap, E_g (eV)

Energy separation between conduction and valence bands.

Electron affinity, χ (eV)

Used to place E_c on a vacuum-referenced scale.

Intrinsic carrier concentration, n_i (cm^-3)

Enter using standard or scientific notation.

Donor concentration, N_D (cm^-3)

Set to 0 if donor doping is absent.

Acceptor concentration, N_A (cm^-3)

Set to 0 if acceptor doping is absent.

Electron effective mass, m*_e/m₀

Relative effective mass for the conduction band.

Hole effective mass, m*_h/m₀

Relative effective mass for the valence band.

Formula Used

Band-edge placement using electron affinity:

E_c = −χ

E_v = −(χ + E_g)

Intrinsic reference including effective-mass correction:

E_i = (E_c + E_v)/2 + (3/4)kT ln(m*_h/m*_e)

Compensated equilibrium solution:

x = (N_D − N_A)/(2n_i)

E_F − E_i = kT sinh⁻¹(x)

n₀ = n_i[x + √(x² + 1)]

p₀ = n_i[−x + √(x² + 1)]

This implementation assumes thermal equilibrium, full dopant ionization, and a non-degenerate semiconductor model. It is suitable for most educational and engineering estimates.

How to Use This Calculator

Enter a material label if you want a named report.
Provide temperature, band gap, and electron affinity.
Enter intrinsic carrier concentration for the chosen material and temperature.
Fill donor and acceptor concentrations, using zero when absent.
Enter electron and hole effective masses relative to the free-electron mass.
Press Calculate Fermi Level to show the results above the form.
Review E_F, E_i, carrier concentrations, and the energy-level chart.
Use the CSV or PDF buttons to export the current result set.

Example Data Table

Material	T (K)	E_g (eV)	χ (eV)	n_i (cm^-3)	N_D (cm^-3)	N_A (cm^-3)	E_i (eV)	E_F (eV)	E_c − E_F (eV)	n₀ (cm^-3)	p₀ (cm^-3)
Silicon-like sample	300	1.12	4.05	1.0 × 10¹⁰	1.0 × 10¹⁶	1.0 × 10¹⁴	-4.623	-4.266	0.216	9.90 × 10¹⁵	1.01 × 10⁴

These values illustrate a strongly n-type case. Your exact result may vary with the chosen intrinsic concentration and effective masses.

Frequently Asked Questions

1) What does this calculator return?

It returns the intrinsic level, Fermi level, band-edge positions, net doping, carrier concentrations, and distances from the Fermi level to both band edges.

2) Which physical model is used here?

It uses an equilibrium, non-degenerate semiconductor model with full dopant ionization. The compensated doping solution is handled through the inverse hyperbolic sine relationship.

3) Why is electron affinity included?

Electron affinity places the conduction band on a vacuum-referenced energy scale. That lets the tool report E_c, E_v, E_i, and E_F consistently in eV.

4) Can this handle both donors and acceptors together?

Yes. The formula uses N_D − N_A and solves the compensated case directly, so mixed doping conditions are supported without switching calculators.

5) Can I use zero for one dopant type?

Yes. Set N_D or N_A to zero when only one dopant family is present. The calculator will still determine the Fermi shift and carrier concentrations correctly.

6) Why do the effective masses matter?

They slightly shift the intrinsic reference away from the exact midgap. This correction improves the position of E_i when electron and hole effective masses are different.

7) What units should I enter?

Use Kelvin for temperature, electron-volts for band gap and affinity, and cm^-3 for carrier and dopant concentrations. Effective masses are relative to the free-electron mass.

8) When is the result less reliable?

Accuracy falls for heavily degenerate doping, incomplete ionization, strong nonequilibrium conditions, or when your material data do not match the selected temperature.