Fermi Level Calculator
Fermi energy of the 3D free electron gas at absolute zero — from electron density n, or vice versa. Complete with Fermi temperature, Fermi velocity, and Fermi momentum.
Calculator
Result
Langkah perhitungan
Verified example: copper
- n = 8,49×10²⁸ m⁻³ (1 elektron valensi per atom Cu; n = ρ·N_A/M)
- k_F = (3π²n)^(1/3) = 13.600.000.000 m⁻¹
- E_F = ħ²k_F²/(2m_e) = 1,13×10⁻¹⁸ J
- E_F = 1,13×10⁻¹⁸ J ÷ 1,602×10⁻¹⁹ J/eV = 7,045 eV
- T_F = E_F/k_B = 81.750 K
- v_F = √(2E_F/m_e) = 1.574.000 m/s
The EF = 7.045 eV matches the Kittel table (7.00 eV) within 1 percent — confirming the formula uses the correct ħ.
Electron density & Fermi energy of common metals
| Logam | n (m⁻³) | EF (eV) | TF (K) | vF (m/s) |
|---|---|---|---|---|
| Natrium | 2,54×10²⁸ | 3,154 | 36.600 | 1.053.000 |
| Tembaga | 8,49×10²⁸ | 7,045 | 81.750 | 1.574.000 |
| Perak | 5,86×10²⁸ | 5,499 | 63.810 | 1.391.000 |
| Emas | 5,9×10²⁸ | 5,527 | 64.140 | 1.394.000 |
| Aluminium | 1,81×10²⁹ | 11,66 | 135.300 | 2.025.000 |
n = Z·ρ·NA/M from each metal’s density and molar mass (Z = valence electrons per atom). The EF free-electron-model result matches Kittel table values within 3 percent.
Model limits (read before using)
- T = 0 mutlak. At room temperature, the Fermi-Dirac distribution edge broadens ~kBT (≈ 0.026 eV) — meaningless for metals because EF is thousands of times larger.
- Elektron bebas. Mass = rest electron mass; crystal lattice, electron–electron interactions, and band structure ignored. Real metals need effective mass — this result is a good order of magnitude, not a precision figure.
- Not for semiconductors. Intrinsic semiconductor Fermi level ≈ mid-gap per Fermi-Dirac statistics on the band structure — a different concept. Do not plug doping densities into this formula.
- n must be in m⁻³. cm⁻³ values common in semiconductor literature must first be multiplied by 10⁶.
Frequently asked questions
What is the Fermi energy (Fermi level)?
The highest electron energy still occupied at absolute zero in the free electron gas model. At T=0, electrons fill the lowest-energy states up to E_F following the Pauli exclusion principle; above E_F all states are empty. For typical metals E_F is a few electronvolts.
Why does the formula use ħ, not h?
Because the free electron energy E = ħ²k²/2m uses the reduced Planck constant ħ = h/2π. Using the full h gives results (2π)² ≈ 39.5 times too large — copper would come out 278 eV while the measured value is about 7 eV. This is a common trap in many popular sources, so always check the constant’s identity.
How do I compute density n from E_F (reverse solver)?
Invert the formula: n = (1/3π²)·(2m_e·E_F/ħ²)^(3/2). This calculator provides an E_F → n mode that does it automatically; both directions are mathematically equivalent — round-trip n → E_F → n returns to the original number.
What is the Fermi temperature T_F?
T_F = E_F/k_B — a way to express the Fermi energy in temperature units. This is NOT a real temperature: metals’ T_F reaches tens of thousands of kelvin, far above room temperature, which means electrons in metals stay quantum-degenerate at everyday temperatures.
How fast do electrons at the Fermi level move?
v_F = √(2E_F/m_e), typically around 10⁶ m/s for metals — about 0.5 percent of light speed. These fastest electrons keep moving fast even at absolute zero; that is a consequence of the Pauli principle, not temperature.
Can this calculator be used for semiconductors?
Not directly. This formula only applies to the 3D free electron gas. In intrinsic semiconductors, the Fermi level lies roughly mid-gap per Fermi-Dirac statistics on the band structure — a conceptually different result. Do not plug doping densities into this formula and claim it is the semiconductor Fermi level.
Where do the metal preset density figures come from?
From n = Z·ρ·N_A/M: density ρ, molar mass M, and Z valence electrons per atom for each metal. Values used: Na 2.54×10²⁸, Cu 8.49×10²⁸, Ag 5.86×10²⁸, Au 5.9×10²⁸, Al 1.81×10²⁹ m⁻³. The Fermi energy computed from these matches the Kittel table within 3 percent.
What are this model’s honesty limits?
The model assumes absolute zero temperature, truly free electrons with the rest electron mass, and ignores the crystal lattice and electron–electron interactions. Real metals need effective mass and band structure. Treat results as a good order of magnitude for simple metals — accurate to a few percent for alkali and noble metals — not precision figures.
Free tool by NRTechStudio